cbrt(x + 1) - cbrt(x)

Percentage Accurate: 95.8% → 98.9%
Time: 3.9s
Alternatives: 16
Speedup: 1.0×

Specification

?
\[0.01 \leq x \land x \leq 100\]
\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt[3]{x + 1} - \sqrt[3]{x}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 16 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 95.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sqrt[3]{x + 1} - \sqrt[3]{x}
\end{array}

Alternative 1: 98.9% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)} \end{array} \]
(FPCore (x)
 :precision binary64
 (/
  (- (+ 1.0 x) x)
  (fma
   (cbrt x)
   (+ (cbrt (+ 1.0 x)) (cbrt x))
   (pow (exp 0.6666666666666666) (log1p x)))))
double code(double x) {
	return ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), pow(exp(0.6666666666666666), log1p(x)));
}

function code(x)
	return Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), (exp(0.6666666666666666) ^ log1p(x))))
end

code[x_] := N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Exp[0.6666666666666666], $MachinePrecision], N[Log[1 + x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}
\end{array}
Derivation

  1. Initial program 96.0%

    \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-cbrt.f64N/A

      \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

    2. pow1/3N/A

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

    3. lift-+.f64N/A

      \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

    4. flip3-+N/A

      \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

    5. clear-numN/A

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

    6. inv-powN/A

      \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

    7. metadata-evalN/A

      \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

    8. pow-powN/A

      \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

    9. lower-pow.f64N/A

      \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

  4. Applied rewrites96.3%

    \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]

  6. Applied rewrites99.0%

    \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
  7. Add Preprocessing

Alternative 2: 35.5% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0.232:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.06172839506172839, x, -0.1111111111111111\right), x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)\\ \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 0.232)
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)
   (fma
    (fma (fma 0.06172839506172839 x -0.1111111111111111) x 0.3333333333333333)
    x
    (- 1.0 (cbrt x)))))
double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 0.232) {
		tmp = cbrt(pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = fma(fma(fma(0.06172839506172839, x, -0.1111111111111111), x, 0.3333333333333333), x, (1.0 - cbrt(x)));
	}
	return tmp;
}

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 0.232)
		tmp = Float64(cbrt((Float64(x * x) ^ -1.0)) * 0.3333333333333333);
	else
		tmp = fma(fma(fma(0.06172839506172839, x, -0.1111111111111111), x, 0.3333333333333333), x, Float64(1.0 - cbrt(x)));
	end
	return tmp
end

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.232], N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(0.06172839506172839 * x + -0.1111111111111111), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * x + N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0.232:\\
\;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.06172839506172839, x, -0.1111111111111111\right), x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 0.232000000000000012

    1. Initial program 93.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
    4. Step-by-step derivation

      1. *-commutativeN/A

        \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64N/A

        \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. metadata-evalN/A

        \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

      4. associate-*r/N/A

        \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

      5. lower-cbrt.f64N/A

        \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

      6. unpow2N/A

        \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

      7. associate-/r*N/A

        \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

      8. associate-*r/N/A

        \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

      9. lower-/.f64N/A

        \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

      10. associate-*r/N/A

        \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

      11. metadata-evalN/A

        \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

      12. lower-/.f6425.9

        \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

    5. Applied rewrites25.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
    6. Step-by-step derivation

      1. Applied rewrites25.9%

        \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 \]

      if 0.232000000000000012 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

      1. Initial program 98.2%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} + x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right)\right)\right) - \sqrt[3]{x}} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{3} + x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right)\right) + 1\right)} - \sqrt[3]{x} \]

        2. associate--l+N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right)\right) + \left(1 - \sqrt[3]{x}\right)} \]

        3. *-commutativeN/A

          \[\leadsto \color{blue}{\left(\frac{1}{3} + x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right)\right) \cdot x} + \left(1 - \sqrt[3]{x}\right) \]

        4. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right), x, 1 - \sqrt[3]{x}\right)} \]

        5. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{5}{81} \cdot x - \frac{1}{9}\right) + \frac{1}{3}}, x, 1 - \sqrt[3]{x}\right) \]

        6. *-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{5}{81} \cdot x - \frac{1}{9}\right) \cdot x} + \frac{1}{3}, x, 1 - \sqrt[3]{x}\right) \]

        7. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5}{81} \cdot x - \frac{1}{9}, x, \frac{1}{3}\right)}, x, 1 - \sqrt[3]{x}\right) \]

        8. sub-negN/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5}{81} \cdot x + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)}, x, \frac{1}{3}\right), x, 1 - \sqrt[3]{x}\right) \]

        9. metadata-evalN/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{81} \cdot x + \color{blue}{\frac{-1}{9}}, x, \frac{1}{3}\right), x, 1 - \sqrt[3]{x}\right) \]

        10. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5}{81}, x, \frac{-1}{9}\right)}, x, \frac{1}{3}\right), x, 1 - \sqrt[3]{x}\right) \]

        11. lower--.f64N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{81}, x, \frac{-1}{9}\right), x, \frac{1}{3}\right), x, \color{blue}{1 - \sqrt[3]{x}}\right) \]

        12. lower-cbrt.f6442.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.06172839506172839, x, -0.1111111111111111\right), x, 0.3333333333333333\right), x, 1 - \color{blue}{\sqrt[3]{x}}\right) \]

      5. Applied rewrites42.4%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.06172839506172839, x, -0.1111111111111111\right), x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)} \]
    7. Recombined 2 regimes into one program.

    8. Final simplification35.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0.232:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.06172839506172839, x, -0.1111111111111111\right), x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)\\ \end{array} \]
    9. Add Preprocessing

    Alternative 3: 96.7% accurate, 0.5× speedup?

    \[\begin{array}{l} \\ {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333} - {x}^{0.3333333333333333} \end{array} \]
    (FPCore (x)
 :precision binary64
 (- (pow (exp (- (log1p x))) -0.3333333333333333) (pow x 0.3333333333333333)))
    double code(double x) {
	return pow(exp(-log1p(x)), -0.3333333333333333) - pow(x, 0.3333333333333333);
}

    public static double code(double x) {
	return Math.pow(Math.exp(-Math.log1p(x)), -0.3333333333333333) - Math.pow(x, 0.3333333333333333);
}

    def code(x):
	return math.pow(math.exp(-math.log1p(x)), -0.3333333333333333) - math.pow(x, 0.3333333333333333)

    function code(x)
	return Float64((exp(Float64(-log1p(x))) ^ -0.3333333333333333) - (x ^ 0.3333333333333333))
end

    code[x_] := N[(N[Power[N[Exp[(-N[Log[1 + x], $MachinePrecision])], $MachinePrecision], -0.3333333333333333], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333} - {x}^{0.3333333333333333}
\end{array}
    Derivation

    1. Initial program 96.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-cbrt.f64N/A

        \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

      2. pow1/3N/A

        \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

      3. lift-+.f64N/A

        \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      4. flip3-+N/A

        \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      5. clear-numN/A

        \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      6. inv-powN/A

        \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      7. metadata-evalN/A

        \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

      8. pow-powN/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

      9. lower-pow.f64N/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

    4. Applied rewrites96.3%

      \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]
    5. Step-by-step derivation

      1. lift-cbrt.f64N/A

        \[\leadsto {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\frac{-1}{3}} - \color{blue}{\sqrt[3]{x}} \]

      2. pow1/3N/A

        \[\leadsto {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\frac{-1}{3}} - \color{blue}{{x}^{\frac{1}{3}}} \]

      3. lower-pow.f6496.8

        \[\leadsto {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333} - \color{blue}{{x}^{0.3333333333333333}} \]

    6. Applied rewrites96.8%

      \[\leadsto {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333} - \color{blue}{{x}^{0.3333333333333333}} \]
    7. Add Preprocessing

    Alternative 4: 96.0% accurate, 0.7× speedup?

    \[\begin{array}{l} \\ {\left({\left(1 + x\right)}^{-2}\right)}^{-0.16666666666666666} - \sqrt[3]{x} \end{array} \]
    (FPCore (x)
 :precision binary64
 (- (pow (pow (+ 1.0 x) -2.0) -0.16666666666666666) (cbrt x)))
    double code(double x) {
	return pow(pow((1.0 + x), -2.0), -0.16666666666666666) - cbrt(x);
}

    public static double code(double x) {
	return Math.pow(Math.pow((1.0 + x), -2.0), -0.16666666666666666) - Math.cbrt(x);
}

    function code(x)
	return Float64(((Float64(1.0 + x) ^ -2.0) ^ -0.16666666666666666) - cbrt(x))
end

    code[x_] := N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], -2.0], $MachinePrecision], -0.16666666666666666], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

    \begin{array}{l}

\\
{\left({\left(1 + x\right)}^{-2}\right)}^{-0.16666666666666666} - \sqrt[3]{x}
\end{array}
    Derivation

    1. Initial program 96.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-cbrt.f64N/A

        \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

      2. pow1/3N/A

        \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

      3. lift-+.f64N/A

        \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      4. flip3-+N/A

        \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      5. clear-numN/A

        \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      6. inv-powN/A

        \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      7. metadata-evalN/A

        \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

      8. pow-powN/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

      9. lower-pow.f64N/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

    4. Applied rewrites96.3%

      \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]
    5. Step-by-step derivation

      1. lift-pow.f64N/A

        \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\frac{-1}{3}}} - \sqrt[3]{x} \]

      2. sqr-powN/A

        \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} \cdot {\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)}} - \sqrt[3]{x} \]

      3. pow-prod-downN/A

        \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)}} - \sqrt[3]{x} \]

      4. lift-exp.f64N/A

        \[\leadsto {\left(\color{blue}{e^{-\mathsf{log1p}\left(x\right)}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      5. lift-exp.f64N/A

        \[\leadsto {\left(e^{-\mathsf{log1p}\left(x\right)} \cdot \color{blue}{e^{-\mathsf{log1p}\left(x\right)}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      6. lift-neg.f64N/A

        \[\leadsto {\left(e^{\color{blue}{\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      7. lift-log1p.f64N/A

        \[\leadsto {\left(e^{\mathsf{neg}\left(\color{blue}{\log \left(1 + x\right)}\right)} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      8. neg-logN/A

        \[\leadsto {\left(e^{\color{blue}{\log \left(\frac{1}{1 + x}\right)}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      9. rem-exp-logN/A

        \[\leadsto {\left(\color{blue}{\frac{1}{1 + x}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      10. +-commutativeN/A

        \[\leadsto {\left(\frac{1}{\color{blue}{x + 1}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      11. flip-+N/A

        \[\leadsto {\left(\frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      12. clear-numN/A

        \[\leadsto {\left(\color{blue}{\frac{x - 1}{x \cdot x - 1 \cdot 1}} \cdot e^{-\mathsf{log1p}\left(x\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      13. lift-neg.f64N/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot e^{\color{blue}{\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      14. lift-log1p.f64N/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot e^{\mathsf{neg}\left(\color{blue}{\log \left(1 + x\right)}\right)}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      15. neg-logN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot e^{\color{blue}{\log \left(\frac{1}{1 + x}\right)}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      16. rem-exp-logN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{\frac{1}{1 + x}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      17. +-commutativeN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot \frac{1}{\color{blue}{x + 1}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      18. flip-+N/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot \frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

      19. clear-numN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{\frac{x - 1}{x \cdot x - 1 \cdot 1}}\right)}^{\left(\frac{\frac{-1}{3}}{2}\right)} - \sqrt[3]{x} \]

    6. Applied rewrites96.3%

      \[\leadsto \color{blue}{{\left({\left(1 + x\right)}^{-2}\right)}^{-0.16666666666666666}} - \sqrt[3]{x} \]
    7. Add Preprocessing

    Alternative 5: 96.0% accurate, 0.9× speedup?

    \[\begin{array}{l} \\ {\left(\frac{x - 1}{\mathsf{fma}\left(x, x, -1\right)}\right)}^{-0.3333333333333333} + \left(-\sqrt[3]{x}\right) \end{array} \]
    (FPCore (x)
 :precision binary64
 (+ (pow (/ (- x 1.0) (fma x x -1.0)) -0.3333333333333333) (- (cbrt x))))
    double code(double x) {
	return pow(((x - 1.0) / fma(x, x, -1.0)), -0.3333333333333333) + -cbrt(x);
}

    function code(x)
	return Float64((Float64(Float64(x - 1.0) / fma(x, x, -1.0)) ^ -0.3333333333333333) + Float64(-cbrt(x)))
end

    code[x_] := N[(N[Power[N[(N[(x - 1.0), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision], -0.3333333333333333], $MachinePrecision] + (-N[Power[x, 1/3], $MachinePrecision])), $MachinePrecision]

    \begin{array}{l}

\\
{\left(\frac{x - 1}{\mathsf{fma}\left(x, x, -1\right)}\right)}^{-0.3333333333333333} + \left(-\sqrt[3]{x}\right)
\end{array}
    Derivation

    1. Initial program 96.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift-cbrt.f64N/A

        \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

      2. pow1/3N/A

        \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

      3. lift-+.f64N/A

        \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      4. flip3-+N/A

        \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      5. clear-numN/A

        \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      6. inv-powN/A

        \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

      7. metadata-evalN/A

        \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

      8. pow-powN/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

      9. lower-pow.f64N/A

        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

    4. Applied rewrites96.3%

      \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]
    5. Step-by-step derivation

      1. lift-exp.f64N/A

        \[\leadsto {\color{blue}{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      2. lift-neg.f64N/A

        \[\leadsto {\left(e^{\color{blue}{\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      3. lift-log1p.f64N/A

        \[\leadsto {\left(e^{\mathsf{neg}\left(\color{blue}{\log \left(1 + x\right)}\right)}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      4. neg-logN/A

        \[\leadsto {\left(e^{\color{blue}{\log \left(\frac{1}{1 + x}\right)}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      5. rem-exp-logN/A

        \[\leadsto {\color{blue}{\left(\frac{1}{1 + x}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      6. +-commutativeN/A

        \[\leadsto {\left(\frac{1}{\color{blue}{x + 1}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      7. flip-+N/A

        \[\leadsto {\left(\frac{1}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      8. clear-numN/A

        \[\leadsto {\color{blue}{\left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      9. lower-/.f64N/A

        \[\leadsto {\color{blue}{\left(\frac{x - 1}{x \cdot x - 1 \cdot 1}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      10. lower--.f64N/A

        \[\leadsto {\left(\frac{\color{blue}{x - 1}}{x \cdot x - 1 \cdot 1}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      11. metadata-evalN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x - \color{blue}{1}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      12. sub-negN/A

        \[\leadsto {\left(\frac{x - 1}{\color{blue}{x \cdot x + \left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      13. metadata-evalN/A

        \[\leadsto {\left(\frac{x - 1}{x \cdot x + \color{blue}{-1}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

      14. lower-fma.f6496.3

        \[\leadsto {\left(\frac{x - 1}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{-0.3333333333333333} - \sqrt[3]{x} \]

    6. Applied rewrites96.3%

      \[\leadsto {\color{blue}{\left(\frac{x - 1}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{-0.3333333333333333} - \sqrt[3]{x} \]

    7. Final simplification96.3%

      \[\leadsto {\left(\frac{x - 1}{\mathsf{fma}\left(x, x, -1\right)}\right)}^{-0.3333333333333333} + \left(-\sqrt[3]{x}\right) \]
    8. Add Preprocessing

    Alternative 6: 32.3% accurate, 0.9× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
    (FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (- (fma (fma -0.1111111111111111 x 0.3333333333333333) x 1.0) (cbrt x))
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)))
    double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, 1.0) - cbrt(x);
	} else {
		tmp = cbrt(pow((x * x), -1.0)) * 0.3333333333333333;
	}
	return tmp;
}

    function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, 1.0) - cbrt(x));
	else
		tmp = Float64(cbrt((Float64(x * x) ^ -1.0)) * 0.3333333333333333);
	end
	return tmp
end

    code[x_] := If[LessEqual[x, 1.0], N[(N[(N[(-0.1111111111111111 * x + 0.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right) - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if x < 1

      1. Initial program 98.2%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right)\right)} - \sqrt[3]{x} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) + 1\right)} - \sqrt[3]{x} \]

        2. *-commutativeN/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) \cdot x} + 1\right) - \sqrt[3]{x} \]

        3. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{9} \cdot x, x, 1\right)} - \sqrt[3]{x} \]

        4. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{9} \cdot x + \frac{1}{3}}, x, 1\right) - \sqrt[3]{x} \]

        5. lower-fma.f6437.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right)}, x, 1\right) - \sqrt[3]{x} \]

      5. Applied rewrites37.3%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right)} - \sqrt[3]{x} \]

      if 1 < x

      1. Initial program 93.6%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

        \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
      4. Step-by-step derivation

        1. *-commutativeN/A

          \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

        2. lower-*.f64N/A

          \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

        3. metadata-evalN/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

        4. associate-*r/N/A

          \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

        5. lower-cbrt.f64N/A

          \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

        6. unpow2N/A

          \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

        7. associate-/r*N/A

          \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

        8. associate-*r/N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

        9. lower-/.f64N/A

          \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

        10. associate-*r/N/A

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

        11. metadata-evalN/A

          \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

        12. lower-/.f6425.8

          \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

      5. Applied rewrites25.8%

        \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
      6. Step-by-step derivation

        1. Applied rewrites25.8%

          \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 \]
      7. Recombined 2 regimes into one program.

      8. Final simplification31.9%

        \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \end{array} \]
      9. Add Preprocessing

      Alternative 7: 96.1% accurate, 1.0× speedup?

      \[\begin{array}{l} \\ {\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x} \end{array} \]
      (FPCore (x)
 :precision binary64
 (- (pow (+ 1.0 x) 0.3333333333333333) (cbrt x)))
      double code(double x) {
	return pow((1.0 + x), 0.3333333333333333) - cbrt(x);
}

      public static double code(double x) {
	return Math.pow((1.0 + x), 0.3333333333333333) - Math.cbrt(x);
}

      function code(x)
	return Float64((Float64(1.0 + x) ^ 0.3333333333333333) - cbrt(x))
end

      code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

      \begin{array}{l}

\\
{\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x}
\end{array}
      Derivation

      1. Initial program 96.0%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing
      3. Step-by-step derivation

        1. lift-cbrt.f64N/A

          \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

        2. pow1/3N/A

          \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

        3. lift-+.f64N/A

          \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        4. flip3-+N/A

          \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        5. clear-numN/A

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        6. inv-powN/A

          \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        7. metadata-evalN/A

          \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

        8. pow-powN/A

          \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

        9. lower-pow.f64N/A

          \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

      4. Applied rewrites96.3%

        \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]
      5. Step-by-step derivation

        1. lift-pow.f64N/A

          \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{\frac{-1}{3}}} - \sqrt[3]{x} \]

        2. lift-exp.f64N/A

          \[\leadsto {\color{blue}{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        3. lift-neg.f64N/A

          \[\leadsto {\left(e^{\color{blue}{\mathsf{neg}\left(\mathsf{log1p}\left(x\right)\right)}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        4. lift-log1p.f64N/A

          \[\leadsto {\left(e^{\mathsf{neg}\left(\color{blue}{\log \left(1 + x\right)}\right)}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        5. neg-logN/A

          \[\leadsto {\left(e^{\color{blue}{\log \left(\frac{1}{1 + x}\right)}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        6. rem-exp-logN/A

          \[\leadsto {\color{blue}{\left(\frac{1}{1 + x}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        7. +-commutativeN/A

          \[\leadsto {\left(\frac{1}{\color{blue}{x + 1}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        8. lift-+.f64N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{x + 1}}\right)}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        9. inv-powN/A

          \[\leadsto {\color{blue}{\left({\left(x + 1\right)}^{-1}\right)}}^{\frac{-1}{3}} - \sqrt[3]{x} \]

        10. pow-powN/A

          \[\leadsto \color{blue}{{\left(x + 1\right)}^{\left(-1 \cdot \frac{-1}{3}\right)}} - \sqrt[3]{x} \]

        11. metadata-evalN/A

          \[\leadsto {\left(x + 1\right)}^{\color{blue}{\frac{1}{3}}} - \sqrt[3]{x} \]

        12. lower-pow.f6496.3

          \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - \sqrt[3]{x} \]

        13. lift-+.f64N/A

          \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        14. +-commutativeN/A

          \[\leadsto {\color{blue}{\left(1 + x\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

        15. lower-+.f6496.3

          \[\leadsto {\color{blue}{\left(1 + x\right)}}^{0.3333333333333333} - \sqrt[3]{x} \]

      6. Applied rewrites96.3%

        \[\leadsto \color{blue}{{\left(1 + x\right)}^{0.3333333333333333}} - \sqrt[3]{x} \]
      7. Add Preprocessing

      Alternative 8: 95.9% accurate, 1.0× speedup?

      \[\begin{array}{l} \\ \sqrt[3]{x + 1} - {x}^{0.3333333333333333} \end{array} \]
      (FPCore (x)
 :precision binary64
 (- (cbrt (+ x 1.0)) (pow x 0.3333333333333333)))
      double code(double x) {
	return cbrt((x + 1.0)) - pow(x, 0.3333333333333333);
}

      public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.pow(x, 0.3333333333333333);
}

      function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - (x ^ 0.3333333333333333))
end

      code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

      \begin{array}{l}

\\
\sqrt[3]{x + 1} - {x}^{0.3333333333333333}
\end{array}
      Derivation

      1. Initial program 96.0%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing
      3. Step-by-step derivation

        1. lift-cbrt.f64N/A

          \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{x}} \]

        2. pow1/3N/A

          \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\frac{1}{3}}} \]

        3. lower-pow.f6496.1

          \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]

      4. Applied rewrites96.1%

        \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
      5. Add Preprocessing

      Alternative 9: 95.8% accurate, 1.0× speedup?

      \[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]
      (FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
      double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

      public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}

      function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

      code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

      \begin{array}{l}

\\
\sqrt[3]{x + 1} - \sqrt[3]{x}
\end{array}
      Derivation

      1. Initial program 96.0%

        \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
      2. Add Preprocessing
      3. Add Preprocessing

      Alternative 10: 32.3% accurate, 1.7× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
      (FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (- (fma (fma -0.1111111111111111 x 0.3333333333333333) x 1.0) (cbrt x))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))
      double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, 1.0) - cbrt(x);
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

      function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, 1.0) - cbrt(x));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

      code[x_] := If[LessEqual[x, 1.0], N[(N[(N[(-0.1111111111111111 * x + 0.3333333333333333), $MachinePrecision] * x + 1.0), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right) - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\


\end{array}
\end{array}
      Derivation
      1. Split input into 2 regimes

      2. if x < 1

        1. Initial program 98.2%

          \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
        2. Add Preprocessing

        3. Taylor expanded in x around 0

          \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right)\right)} - \sqrt[3]{x} \]
        4. Step-by-step derivation

          1. +-commutativeN/A

            \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) + 1\right)} - \sqrt[3]{x} \]

          2. *-commutativeN/A

            \[\leadsto \left(\color{blue}{\left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) \cdot x} + 1\right) - \sqrt[3]{x} \]

          3. lower-fma.f64N/A

            \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{9} \cdot x, x, 1\right)} - \sqrt[3]{x} \]

          4. +-commutativeN/A

            \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{9} \cdot x + \frac{1}{3}}, x, 1\right) - \sqrt[3]{x} \]

          5. lower-fma.f6437.3

            \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right)}, x, 1\right) - \sqrt[3]{x} \]

        5. Applied rewrites37.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1\right)} - \sqrt[3]{x} \]

        if 1 < x

        1. Initial program 93.6%

          \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
        2. Add Preprocessing

        3. Taylor expanded in x around inf

          \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
        4. Step-by-step derivation

          1. *-commutativeN/A

            \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

          2. lower-*.f64N/A

            \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

          3. metadata-evalN/A

            \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

          4. associate-*r/N/A

            \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

          5. lower-cbrt.f64N/A

            \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

          6. unpow2N/A

            \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

          7. associate-/r*N/A

            \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

          8. associate-*r/N/A

            \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

          9. lower-/.f64N/A

            \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

          10. associate-*r/N/A

            \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

          11. metadata-evalN/A

            \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

          12. lower-/.f6425.8

            \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

        5. Applied rewrites25.8%

          \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
        6. Step-by-step derivation

          1. Applied rewrites25.8%

            \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
        7. Recombined 2 regimes into one program.
        8. Add Preprocessing

        Alternative 11: 32.3% accurate, 1.7× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
        (FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (fma (fma -0.1111111111111111 x 0.3333333333333333) x (- 1.0 (cbrt x)))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))
        double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, (1.0 - cbrt(x)));
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

        function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = fma(fma(-0.1111111111111111, x, 0.3333333333333333), x, Float64(1.0 - cbrt(x)));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

        code[x_] := If[LessEqual[x, 1.0], N[(N[(-0.1111111111111111 * x + 0.3333333333333333), $MachinePrecision] * x + N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

        \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if x < 1

          1. Initial program 98.2%

            \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
          2. Add Preprocessing

          3. Taylor expanded in x around 0

            \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right)\right) - \sqrt[3]{x}} \]
          4. Step-by-step derivation

            1. +-commutativeN/A

              \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) + 1\right)} - \sqrt[3]{x} \]

            2. associate--l+N/A

              \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) + \left(1 - \sqrt[3]{x}\right)} \]

            3. *-commutativeN/A

              \[\leadsto \color{blue}{\left(\frac{1}{3} + \frac{-1}{9} \cdot x\right) \cdot x} + \left(1 - \sqrt[3]{x}\right) \]

            4. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{9} \cdot x, x, 1 - \sqrt[3]{x}\right)} \]

            5. +-commutativeN/A

              \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{9} \cdot x + \frac{1}{3}}, x, 1 - \sqrt[3]{x}\right) \]

            6. lower-fma.f64N/A

              \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{9}, x, \frac{1}{3}\right)}, x, 1 - \sqrt[3]{x}\right) \]

            7. lower--.f64N/A

              \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{9}, x, \frac{1}{3}\right), x, \color{blue}{1 - \sqrt[3]{x}}\right) \]

            8. lower-cbrt.f6437.3

              \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1 - \color{blue}{\sqrt[3]{x}}\right) \]

          5. Applied rewrites37.3%

            \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.1111111111111111, x, 0.3333333333333333\right), x, 1 - \sqrt[3]{x}\right)} \]

          if 1 < x

          1. Initial program 93.6%

            \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
          2. Add Preprocessing

          3. Taylor expanded in x around inf

            \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
          4. Step-by-step derivation

            1. *-commutativeN/A

              \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

            2. lower-*.f64N/A

              \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

            3. metadata-evalN/A

              \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

            4. associate-*r/N/A

              \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

            5. lower-cbrt.f64N/A

              \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

            6. unpow2N/A

              \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

            7. associate-/r*N/A

              \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

            8. associate-*r/N/A

              \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

            9. lower-/.f64N/A

              \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

            10. associate-*r/N/A

              \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

            11. metadata-evalN/A

              \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

            12. lower-/.f6425.8

              \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

          5. Applied rewrites25.8%

            \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
          6. Step-by-step derivation

            1. Applied rewrites25.8%

              \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
          7. Recombined 2 regimes into one program.
          8. Add Preprocessing

          Alternative 12: 29.0% accurate, 1.8× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, x, 1\right) - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
          (FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (- (fma 0.3333333333333333 x 1.0) (cbrt x))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))
          double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(0.3333333333333333, x, 1.0) - cbrt(x);
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

          function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(fma(0.3333333333333333, x, 1.0) - cbrt(x));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

          code[x_] := If[LessEqual[x, 1.0], N[(N[(0.3333333333333333 * x + 1.0), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

          \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\mathsf{fma}\left(0.3333333333333333, x, 1\right) - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if x < 1

            1. Initial program 98.2%

              \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
            2. Add Preprocessing

            3. Taylor expanded in x around 0

              \[\leadsto \color{blue}{\left(1 + \frac{1}{3} \cdot x\right)} - \sqrt[3]{x} \]
            4. Step-by-step derivation

              1. +-commutativeN/A

                \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot x + 1\right)} - \sqrt[3]{x} \]

              2. lower-fma.f6431.2

                \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1\right)} - \sqrt[3]{x} \]

            5. Applied rewrites31.2%

              \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1\right)} - \sqrt[3]{x} \]

            if 1 < x

            1. Initial program 93.6%

              \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
            2. Add Preprocessing

            3. Taylor expanded in x around inf

              \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
            4. Step-by-step derivation

              1. *-commutativeN/A

                \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

              2. lower-*.f64N/A

                \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

              3. metadata-evalN/A

                \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

              4. associate-*r/N/A

                \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

              5. lower-cbrt.f64N/A

                \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

              6. unpow2N/A

                \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

              7. associate-/r*N/A

                \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

              8. associate-*r/N/A

                \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

              9. lower-/.f64N/A

                \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

              10. associate-*r/N/A

                \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

              11. metadata-evalN/A

                \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

              12. lower-/.f6425.8

                \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

            5. Applied rewrites25.8%

              \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
            6. Step-by-step derivation

              1. Applied rewrites25.8%

                \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
            7. Recombined 2 regimes into one program.
            8. Add Preprocessing

            Alternative 13: 29.0% accurate, 1.8× speedup?

            \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
            (FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (fma 0.3333333333333333 x (- 1.0 (cbrt x)))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))
            double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(0.3333333333333333, x, (1.0 - cbrt(x)));
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

            function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = fma(0.3333333333333333, x, Float64(1.0 - cbrt(x)));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

            code[x_] := If[LessEqual[x, 1.0], N[(0.3333333333333333 * x + N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

            \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 1:\\
\;\;\;\;\mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right)\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\


\end{array}
\end{array}
            Derivation
            1. Split input into 2 regimes

            2. if x < 1

              1. Initial program 98.2%

                \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
              2. Add Preprocessing

              3. Taylor expanded in x around 0

                \[\leadsto \color{blue}{\left(1 + \frac{1}{3} \cdot x\right) - \sqrt[3]{x}} \]
              4. Step-by-step derivation

                1. +-commutativeN/A

                  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot x + 1\right)} - \sqrt[3]{x} \]

                2. associate--l+N/A

                  \[\leadsto \color{blue}{\frac{1}{3} \cdot x + \left(1 - \sqrt[3]{x}\right)} \]

                3. lower-fma.f64N/A

                  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3}, x, 1 - \sqrt[3]{x}\right)} \]

                4. lower--.f64N/A

                  \[\leadsto \mathsf{fma}\left(\frac{1}{3}, x, \color{blue}{1 - \sqrt[3]{x}}\right) \]

                5. lower-cbrt.f6431.2

                  \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, 1 - \color{blue}{\sqrt[3]{x}}\right) \]

              5. Applied rewrites31.2%

                \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right)} \]

              if 1 < x

              1. Initial program 93.6%

                \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
              2. Add Preprocessing

              3. Taylor expanded in x around inf

                \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
              4. Step-by-step derivation

                1. *-commutativeN/A

                  \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

                2. lower-*.f64N/A

                  \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

                3. metadata-evalN/A

                  \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

                4. associate-*r/N/A

                  \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

                5. lower-cbrt.f64N/A

                  \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

                6. unpow2N/A

                  \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

                7. associate-/r*N/A

                  \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                8. associate-*r/N/A

                  \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                9. lower-/.f64N/A

                  \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                10. associate-*r/N/A

                  \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

                11. metadata-evalN/A

                  \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

                12. lower-/.f6425.8

                  \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

              5. Applied rewrites25.8%

                \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
              6. Step-by-step derivation

                1. Applied rewrites25.8%

                  \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
              7. Recombined 2 regimes into one program.
              8. Add Preprocessing

              Alternative 14: 25.2% accurate, 1.8× speedup?

              \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]
              (FPCore (x)
 :precision binary64
 (if (<= x 0.48)
   (- 1.0 (cbrt x))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))
              double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = 1.0 - cbrt(x);
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

              public static double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = 1.0 - Math.cbrt(x);
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

              function code(x)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64(1.0 - cbrt(x));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

              code[x_] := If[LessEqual[x, 0.48], N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

              \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 0.48:\\
\;\;\;\;1 - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\


\end{array}
\end{array}
              Derivation
              1. Split input into 2 regimes

              2. if x < 0.47999999999999998

                1. Initial program 98.4%

                  \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
                2. Add Preprocessing

                3. Taylor expanded in x around 0

                  \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
                4. Step-by-step derivation

                  1. Applied rewrites24.7%

                    \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]

                  if 0.47999999999999998 < x

                  1. Initial program 94.2%

                    \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
                  2. Add Preprocessing

                  3. Taylor expanded in x around inf

                    \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
                  4. Step-by-step derivation

                    1. *-commutativeN/A

                      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

                    2. lower-*.f64N/A

                      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

                    3. metadata-evalN/A

                      \[\leadsto \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \cdot \frac{1}{3} \]

                    4. associate-*r/N/A

                      \[\leadsto \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

                    5. lower-cbrt.f64N/A

                      \[\leadsto \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \cdot \frac{1}{3} \]

                    6. unpow2N/A

                      \[\leadsto \sqrt[3]{-1 \cdot \frac{-1}{\color{blue}{x \cdot x}}} \cdot \frac{1}{3} \]

                    7. associate-/r*N/A

                      \[\leadsto \sqrt[3]{-1 \cdot \color{blue}{\frac{\frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                    8. associate-*r/N/A

                      \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                    9. lower-/.f64N/A

                      \[\leadsto \sqrt[3]{\color{blue}{\frac{-1 \cdot \frac{-1}{x}}{x}}} \cdot \frac{1}{3} \]

                    10. associate-*r/N/A

                      \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{-1 \cdot -1}{x}}}{x}} \cdot \frac{1}{3} \]

                    11. metadata-evalN/A

                      \[\leadsto \sqrt[3]{\frac{\frac{\color{blue}{1}}{x}}{x}} \cdot \frac{1}{3} \]

                    12. lower-/.f6425.0

                      \[\leadsto \sqrt[3]{\frac{\color{blue}{\frac{1}{x}}}{x}} \cdot 0.3333333333333333 \]

                  5. Applied rewrites25.0%

                    \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
                  6. Step-by-step derivation

                    1. Applied rewrites25.0%

                      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
                  7. Recombined 2 regimes into one program.
                  8. Add Preprocessing

                  Alternative 15: 12.9% accurate, 2.0× speedup?

                  \[\begin{array}{l} \\ 1 - \sqrt[3]{x} \end{array} \]
                  (FPCore (x) :precision binary64 (- 1.0 (cbrt x)))
                  double code(double x) {
	return 1.0 - cbrt(x);
}

                  public static double code(double x) {
	return 1.0 - Math.cbrt(x);
}

                  function code(x)
	return Float64(1.0 - cbrt(x))
end

                  code[x_] := N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

                  \begin{array}{l}

\\
1 - \sqrt[3]{x}
\end{array}
                  Derivation

                  1. Initial program 96.0%

                    \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
                  2. Add Preprocessing

                  3. Taylor expanded in x around 0

                    \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
                  4. Step-by-step derivation

                    1. Applied rewrites13.2%

                      \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
                    2. Add Preprocessing

                    Alternative 16: 1.6% accurate, 2.0× speedup?

                    \[\begin{array}{l} \\ -\sqrt[3]{x} \end{array} \]
                    (FPCore (x) :precision binary64 (- (cbrt x)))
                    double code(double x) {
	return -cbrt(x);
}

                    public static double code(double x) {
	return -Math.cbrt(x);
}

                    function code(x)
	return Float64(-cbrt(x))
end

                    code[x_] := (-N[Power[x, 1/3], $MachinePrecision])

                    \begin{array}{l}

\\
-\sqrt[3]{x}
\end{array}
                    Derivation

                    1. Initial program 96.0%

                      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
                    2. Add Preprocessing
                    3. Step-by-step derivation

                      1. lift-cbrt.f64N/A

                        \[\leadsto \color{blue}{\sqrt[3]{x + 1}} - \sqrt[3]{x} \]

                      2. pow1/3N/A

                        \[\leadsto \color{blue}{{\left(x + 1\right)}^{\frac{1}{3}}} - \sqrt[3]{x} \]

                      3. lift-+.f64N/A

                        \[\leadsto {\color{blue}{\left(x + 1\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

                      4. flip3-+N/A

                        \[\leadsto {\color{blue}{\left(\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

                      5. clear-numN/A

                        \[\leadsto {\color{blue}{\left(\frac{1}{\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

                      6. inv-powN/A

                        \[\leadsto {\color{blue}{\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{-1}\right)}}^{\frac{1}{3}} - \sqrt[3]{x} \]

                      7. metadata-evalN/A

                        \[\leadsto {\left({\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}\right)}^{\frac{1}{3}} - \sqrt[3]{x} \]

                      8. pow-powN/A

                        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

                      9. lower-pow.f64N/A

                        \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}{{x}^{3} + {1}^{3}}\right)}^{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot \frac{1}{3}\right)}} - \sqrt[3]{x} \]

                    4. Applied rewrites96.3%

                      \[\leadsto \color{blue}{{\left(e^{-\mathsf{log1p}\left(x\right)}\right)}^{-0.3333333333333333}} - \sqrt[3]{x} \]

                    5. Taylor expanded in x around inf

                      \[\leadsto \color{blue}{-1 \cdot \sqrt[3]{x}} \]
                    6. Step-by-step derivation

                      1. mul-1-negN/A

                        \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

                      2. lower-neg.f64N/A

                        \[\leadsto \color{blue}{-\sqrt[3]{x}} \]

                      3. lower-cbrt.f641.6

                        \[\leadsto -\color{blue}{\sqrt[3]{x}} \]

                    7. Applied rewrites1.6%

                      \[\leadsto \color{blue}{-\sqrt[3]{x}} \]
                    8. Add Preprocessing

                    Reproduce

                    ?
                    herbie shell --seed 1 
(FPCore (x)
  :name "cbrt(x + 1) - cbrt(x)"
  :precision binary64
  :pre (and (<= 0.01 x) (<= x 100.0))
  (- (cbrt (+ x 1.0)) (cbrt x)))