(1-sqrt(1-x*x))/(1+sqrt(1-x*x))

Percentage Accurate: 53.3% → 100.0%
Time: 7.2s
Alternatives: 11
Speedup: 4.9×

Specification

?
\[0 \leq x \land x \leq 1\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - x \cdot x}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* x x))))) (/ (- 1.0 t_0) (+ 1.0 t_0))))
double code(double x) {
	double t_0 = sqrt((1.0 - (x * x)));
	return (1.0 - t_0) / (1.0 + t_0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 - (x * x)))
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 - (x * x)));
	return (1.0 - t_0) / (1.0 + t_0);
}

def code(x):
	t_0 = math.sqrt((1.0 - (x * x)))
	return (1.0 - t_0) / (1.0 + t_0)

function code(x)
	t_0 = sqrt(Float64(1.0 - Float64(x * x)))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

function tmp = code(x)
	t_0 = sqrt((1.0 - (x * x)));
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{1 - x \cdot x}\\
\frac{1 - t\_0}{1 + t\_0}
\end{array}
\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 11 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: 53.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - x \cdot x}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* x x))))) (/ (- 1.0 t_0) (+ 1.0 t_0))))
double code(double x) {
	double t_0 = sqrt((1.0 - (x * x)));
	return (1.0 - t_0) / (1.0 + t_0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 - (x * x)))
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 - (x * x)));
	return (1.0 - t_0) / (1.0 + t_0);
}

def code(x):
	t_0 = math.sqrt((1.0 - (x * x)))
	return (1.0 - t_0) / (1.0 + t_0)

function code(x)
	t_0 = sqrt(Float64(1.0 - Float64(x * x)))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

function tmp = code(x)
	t_0 = sqrt((1.0 - (x * x)));
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{1 - x \cdot x}\\
\frac{1 - t\_0}{1 + t\_0}
\end{array}
\end{array}

Alternative 1: 100.0% accurate, 0.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := {t\_0}^{1.5} + 1\\ t_2 := \sqrt{t\_0}\\ \mathsf{fma}\left(\frac{x \cdot x}{t\_1}, {\left(t\_2 + 1\right)}^{-1}, \left(1 - \mathsf{fma}\left(x, x, t\_2\right)\right) \cdot \frac{1 - t\_2}{t\_1}\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))) (t_1 (+ (pow t_0 1.5) 1.0)) (t_2 (sqrt t_0)))
   (fma
    (/ (* x x) t_1)
    (pow (+ t_2 1.0) -1.0)
    (* (- 1.0 (fma x x t_2)) (/ (- 1.0 t_2) t_1)))))
double code(double x) {
	double t_0 = 1.0 - (x * x);
	double t_1 = pow(t_0, 1.5) + 1.0;
	double t_2 = sqrt(t_0);
	return fma(((x * x) / t_1), pow((t_2 + 1.0), -1.0), ((1.0 - fma(x, x, t_2)) * ((1.0 - t_2) / t_1)));
}

function code(x)
	t_0 = Float64(1.0 - Float64(x * x))
	t_1 = Float64((t_0 ^ 1.5) + 1.0)
	t_2 = sqrt(t_0)
	return fma(Float64(Float64(x * x) / t_1), (Float64(t_2 + 1.0) ^ -1.0), Float64(Float64(1.0 - fma(x, x, t_2)) * Float64(Float64(1.0 - t_2) / t_1)))
end

code[x_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 1.5], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] * N[Power[N[(t$95$2 + 1.0), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(1.0 - N[(x * x + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 1 - x \cdot x\\
t_1 := {t\_0}^{1.5} + 1\\
t_2 := \sqrt{t\_0}\\
\mathsf{fma}\left(\frac{x \cdot x}{t\_1}, {\left(t\_2 + 1\right)}^{-1}, \left(1 - \mathsf{fma}\left(x, x, t\_2\right)\right) \cdot \frac{1 - t\_2}{t\_1}\right)
\end{array}
\end{array}
Derivation

  1. Initial program 52.7%

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

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0 + x \cdot x}{{\left(1 - x \cdot x\right)}^{1.5} + 1}, {\left(\sqrt{1 - x \cdot x} + 1\right)}^{-1}, \left(1 - \mathsf{fma}\left(x, x, \sqrt{1 - x \cdot x}\right)\right) \cdot \frac{1 - \sqrt{1 - x \cdot x}}{{\left(1 - x \cdot x\right)}^{1.5} + 1}\right)} \]

  5. Final simplification100.0%

    \[\leadsto \mathsf{fma}\left(\frac{x \cdot x}{{\left(1 - x \cdot x\right)}^{1.5} + 1}, {\left(\sqrt{1 - x \cdot x} + 1\right)}^{-1}, \left(1 - \mathsf{fma}\left(x, x, \sqrt{1 - x \cdot x}\right)\right) \cdot \frac{1 - \sqrt{1 - x \cdot x}}{{\left(1 - x \cdot x\right)}^{1.5} + 1}\right) \]
  6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := \sqrt{t\_0}\\ \frac{x \cdot x}{{t\_0}^{1.5} + 1} \cdot \frac{\left(2 - x \cdot x\right) - t\_1}{t\_1 + 1} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x))) (t_1 (sqrt t_0)))
   (*
    (/ (* x x) (+ (pow t_0 1.5) 1.0))
    (/ (- (- 2.0 (* x x)) t_1) (+ t_1 1.0)))))
double code(double x) {
	double t_0 = 1.0 - (x * x);
	double t_1 = sqrt(t_0);
	return ((x * x) / (pow(t_0, 1.5) + 1.0)) * (((2.0 - (x * x)) - t_1) / (t_1 + 1.0));
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = 1.0d0 - (x * x)
    t_1 = sqrt(t_0)
    code = ((x * x) / ((t_0 ** 1.5d0) + 1.0d0)) * (((2.0d0 - (x * x)) - t_1) / (t_1 + 1.0d0))
end function

public static double code(double x) {
	double t_0 = 1.0 - (x * x);
	double t_1 = Math.sqrt(t_0);
	return ((x * x) / (Math.pow(t_0, 1.5) + 1.0)) * (((2.0 - (x * x)) - t_1) / (t_1 + 1.0));
}

def code(x):
	t_0 = 1.0 - (x * x)
	t_1 = math.sqrt(t_0)
	return ((x * x) / (math.pow(t_0, 1.5) + 1.0)) * (((2.0 - (x * x)) - t_1) / (t_1 + 1.0))

function code(x)
	t_0 = Float64(1.0 - Float64(x * x))
	t_1 = sqrt(t_0)
	return Float64(Float64(Float64(x * x) / Float64((t_0 ^ 1.5) + 1.0)) * Float64(Float64(Float64(2.0 - Float64(x * x)) - t_1) / Float64(t_1 + 1.0)))
end

function tmp = code(x)
	t_0 = 1.0 - (x * x);
	t_1 = sqrt(t_0);
	tmp = ((x * x) / ((t_0 ^ 1.5) + 1.0)) * (((2.0 - (x * x)) - t_1) / (t_1 + 1.0));
end

code[x_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] / N[(N[Power[t$95$0, 1.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 1 - x \cdot x\\
t_1 := \sqrt{t\_0}\\
\frac{x \cdot x}{{t\_0}^{1.5} + 1} \cdot \frac{\left(2 - x \cdot x\right) - t\_1}{t\_1 + 1}
\end{array}
\end{array}
Derivation

  1. Initial program 52.7%

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

    1. lift-/.f64N/A

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

    2. lift--.f64N/A

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

    3. flip--N/A

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

    4. flip3-+N/A

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

    5. associate-/r/N/A

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

    6. associate-/l*N/A

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

    7. lower-*.f64N/A

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

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\frac{0 + x \cdot x}{{\left(1 - x \cdot x\right)}^{1.5} + 1} \cdot \frac{\left(2 - x \cdot x\right) - \sqrt{1 - x \cdot x}}{\sqrt{1 - x \cdot x} + 1}} \]

  5. Final simplification100.0%

    \[\leadsto \frac{x \cdot x}{{\left(1 - x \cdot x\right)}^{1.5} + 1} \cdot \frac{\left(2 - x \cdot x\right) - \sqrt{1 - x \cdot x}}{\sqrt{1 - x \cdot x} + 1} \]
  6. Add Preprocessing

Alternative 3: 100.0% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \sqrt{1 - x \cdot x}\\ \frac{x \cdot x}{t\_0 \cdot t\_0} \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- -1.0 (sqrt (- 1.0 (* x x)))))) (/ (* x x) (* t_0 t_0))))
double code(double x) {
	double t_0 = -1.0 - sqrt((1.0 - (x * x)));
	return (x * x) / (t_0 * t_0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (-1.0d0) - sqrt((1.0d0 - (x * x)))
    code = (x * x) / (t_0 * t_0)
end function

public static double code(double x) {
	double t_0 = -1.0 - Math.sqrt((1.0 - (x * x)));
	return (x * x) / (t_0 * t_0);
}

def code(x):
	t_0 = -1.0 - math.sqrt((1.0 - (x * x)))
	return (x * x) / (t_0 * t_0)

function code(x)
	t_0 = Float64(-1.0 - sqrt(Float64(1.0 - Float64(x * x))))
	return Float64(Float64(x * x) / Float64(t_0 * t_0))
end

function tmp = code(x)
	t_0 = -1.0 - sqrt((1.0 - (x * x)));
	tmp = (x * x) / (t_0 * t_0);
end

code[x_] := Block[{t$95$0 = N[(-1.0 - N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(x * x), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := -1 - \sqrt{1 - x \cdot x}\\
\frac{x \cdot x}{t\_0 \cdot t\_0}
\end{array}
\end{array}
Derivation

  1. Initial program 52.7%

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

    1. lift-/.f64N/A

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

    2. frac-2negN/A

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

    3. lift--.f64N/A

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

    4. flip--N/A

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

    5. lift-+.f64N/A

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

    6. distribute-neg-frac2N/A

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

    7. associate-/l/N/A

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

    8. lower-/.f64N/A

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

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\frac{0 + x \cdot x}{\left(-1 - \sqrt{1 - x \cdot x}\right) \cdot \left(-1 - \sqrt{1 - x \cdot x}\right)}} \]

  5. Final simplification100.0%

    \[\leadsto \frac{x \cdot x}{\left(-1 - \sqrt{1 - x \cdot x}\right) \cdot \left(-1 - \sqrt{1 - x \cdot x}\right)} \]
  6. Add Preprocessing

Alternative 4: 99.8% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x, x, 0.25 \cdot x\right) \cdot x \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  (fma
   (* (* (fma (fma 0.0546875 (* x x) 0.078125) (* x x) 0.125) x) x)
   x
   (* 0.25 x))
  x))
double code(double x) {
	return fma(((fma(fma(0.0546875, (x * x), 0.078125), (x * x), 0.125) * x) * x), x, (0.25 * x)) * x;
}

function code(x)
	return Float64(fma(Float64(Float64(fma(fma(0.0546875, Float64(x * x), 0.078125), Float64(x * x), 0.125) * x) * x), x, Float64(0.25 * x)) * x)
end

code[x_] := N[(N[(N[(N[(N[(N[(0.0546875 * N[(x * x), $MachinePrecision] + 0.078125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.25 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x, x, 0.25 \cdot x\right) \cdot x
\end{array}
Derivation

  1. Initial program 52.7%

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

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)} \]
  4. Step-by-step derivation

    1. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}} \]

    2. unpow2N/A

      \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

    3. associate-*r*N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot x\right) \cdot x} \]

    4. lower-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot x\right) \cdot x} \]

  5. Applied rewrites99.7%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]
  6. Step-by-step derivation

    1. Applied rewrites99.7%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x, x, 0.25 \cdot x\right) \cdot x \]
    2. Add Preprocessing

    Alternative 5: 99.8% accurate, 1.2× speedup?

    \[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x \end{array} \]
    (FPCore (x)
 :precision binary64
 (*
  (* (fma (fma (fma 0.0546875 (* x x) 0.078125) (* x x) 0.125) (* x x) 0.25) x)
  x))
    double code(double x) {
	return (fma(fma(fma(0.0546875, (x * x), 0.078125), (x * x), 0.125), (x * x), 0.25) * x) * x;
}

    function code(x)
	return Float64(Float64(fma(fma(fma(0.0546875, Float64(x * x), 0.078125), Float64(x * x), 0.125), Float64(x * x), 0.25) * x) * x)
end

    code[x_] := N[(N[(N[(N[(N[(0.0546875 * N[(x * x), $MachinePrecision] + 0.078125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

    \begin{array}{l}

\\
\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x
\end{array}
    Derivation

    1. Initial program 52.7%

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

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right)} \]
    4. Step-by-step derivation

      1. *-commutativeN/A

        \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{2}} \]

      2. unpow2N/A

        \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r*N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{5}{64} + \frac{7}{128} \cdot {x}^{2}\right)\right)\right) \cdot x\right) \cdot x} \]

    5. Applied rewrites99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0546875, x \cdot x, 0.078125\right), x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]
    6. Add Preprocessing

    Alternative 6: 99.7% accurate, 1.6× speedup?

    \[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.078125, x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x \end{array} \]
    (FPCore (x)
 :precision binary64
 (* (* (fma (fma 0.078125 (* x x) 0.125) (* x x) 0.25) x) x))
    double code(double x) {
	return (fma(fma(0.078125, (x * x), 0.125), (x * x), 0.25) * x) * x;
}

    function code(x)
	return Float64(Float64(fma(fma(0.078125, Float64(x * x), 0.125), Float64(x * x), 0.25) * x) * x)
end

    code[x_] := N[(N[(N[(N[(0.078125 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

    \begin{array}{l}

\\
\left(\mathsf{fma}\left(\mathsf{fma}\left(0.078125, x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x
\end{array}
    Derivation

    1. Initial program 52.7%

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

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right)} \]
    4. Step-by-step derivation

      1. *-commutativeN/A

        \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} \]

      2. unpow2N/A

        \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r*N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutativeN/A

        \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right) + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. *-commutativeN/A

        \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64N/A

        \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{5}{64} \cdot {x}^{2}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      9. +-commutativeN/A

        \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{5}{64} \cdot {x}^{2} + \frac{1}{8}}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      10. lower-fma.f64N/A

        \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5}{64}, {x}^{2}, \frac{1}{8}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      11. unpow2N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{64}, \color{blue}{x \cdot x}, \frac{1}{8}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      12. lower-*.f64N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{64}, \color{blue}{x \cdot x}, \frac{1}{8}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      13. unpow2N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{5}{64}, x \cdot x, \frac{1}{8}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      14. lower-*.f6499.6

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.078125, x \cdot x, 0.125\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

    5. Applied rewrites99.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.078125, x \cdot x, 0.125\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]
    6. Add Preprocessing

    Alternative 7: 99.5% accurate, 2.0× speedup?

    \[\begin{array}{l} \\ \mathsf{fma}\left(0.125 \cdot \left(x \cdot x\right), x, 0.25 \cdot x\right) \cdot x \end{array} \]
    (FPCore (x) :precision binary64 (* (fma (* 0.125 (* x x)) x (* 0.25 x)) x))
    double code(double x) {
	return fma((0.125 * (x * x)), x, (0.25 * x)) * x;
}

    function code(x)
	return Float64(fma(Float64(0.125 * Float64(x * x)), x, Float64(0.25 * x)) * x)
end

    code[x_] := N[(N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(0.25 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

    \begin{array}{l}

\\
\mathsf{fma}\left(0.125 \cdot \left(x \cdot x\right), x, 0.25 \cdot x\right) \cdot x
\end{array}
    Derivation

    1. Initial program 52.7%

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

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)} \]
    4. Step-by-step derivation

      1. *-commutativeN/A

        \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2N/A

        \[\leadsto \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r*N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64N/A

        \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutativeN/A

        \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64N/A

        \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      8. unpow2N/A

        \[\leadsto \left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      9. lower-*.f6499.4

        \[\leadsto \left(\mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

    5. Applied rewrites99.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.125, x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]
    6. Step-by-step derivation

      1. Applied rewrites99.4%

        \[\leadsto \mathsf{fma}\left(0.125 \cdot \left(x \cdot x\right), x, 0.25 \cdot x\right) \cdot x \]
      2. Add Preprocessing

      Alternative 8: 99.5% accurate, 2.5× speedup?

      \[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.125, x \cdot x, 0.25\right) \cdot x\right) \cdot x \end{array} \]
      (FPCore (x) :precision binary64 (* (* (fma 0.125 (* x x) 0.25) x) x))
      double code(double x) {
	return (fma(0.125, (x * x), 0.25) * x) * x;
}

      function code(x)
	return Float64(Float64(fma(0.125, Float64(x * x), 0.25) * x) * x)
end

      code[x_] := N[(N[(N[(0.125 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

      \begin{array}{l}

\\
\left(\mathsf{fma}\left(0.125, x \cdot x, 0.25\right) \cdot x\right) \cdot x
\end{array}
      Derivation

      1. Initial program 52.7%

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

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)} \]
      4. Step-by-step derivation

        1. *-commutativeN/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

        2. unpow2N/A

          \[\leadsto \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

        3. associate-*r*N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

        4. lower-*.f64N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

        5. lower-*.f64N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

        6. +-commutativeN/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

        7. lower-fma.f64N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

        8. unpow2N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

        9. lower-*.f6499.4

          \[\leadsto \left(\mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

      5. Applied rewrites99.4%

        \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.125, x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]
      6. Add Preprocessing

      Alternative 9: 98.9% accurate, 4.9× speedup?

      \[\begin{array}{l} \\ \left(0.25 \cdot x\right) \cdot x \end{array} \]
      (FPCore (x) :precision binary64 (* (* 0.25 x) x))
      double code(double x) {
	return (0.25 * x) * x;
}

      real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.25d0 * x) * x
end function

      public static double code(double x) {
	return (0.25 * x) * x;
}

      def code(x):
	return (0.25 * x) * x

      function code(x)
	return Float64(Float64(0.25 * x) * x)
end

      function tmp = code(x)
	tmp = (0.25 * x) * x;
end

      code[x_] := N[(N[(0.25 * x), $MachinePrecision] * x), $MachinePrecision]

      \begin{array}{l}

\\
\left(0.25 \cdot x\right) \cdot x
\end{array}
      Derivation

      1. Initial program 52.7%

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

      3. Taylor expanded in x around 0

        \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right)} \]
      4. Step-by-step derivation

        1. *-commutativeN/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

        2. unpow2N/A

          \[\leadsto \left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

        3. associate-*r*N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

        4. lower-*.f64N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

        5. lower-*.f64N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{8} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

        6. +-commutativeN/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

        7. lower-fma.f64N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

        8. unpow2N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{8}, \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

        9. lower-*.f6499.4

          \[\leadsto \left(\mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

      5. Applied rewrites99.4%

        \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.125, x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

      6. Taylor expanded in x around 0

        \[\leadsto \left(\frac{1}{4} \cdot x\right) \cdot x \]
      7. Step-by-step derivation

        1. Applied rewrites98.9%

          \[\leadsto \left(0.25 \cdot x\right) \cdot x \]
        2. Add Preprocessing

        Alternative 10: 98.9% accurate, 4.9× speedup?

        \[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.25 \end{array} \]
        (FPCore (x) :precision binary64 (* (* x x) 0.25))
        double code(double x) {
	return (x * x) * 0.25;
}

        real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.25d0
end function

        public static double code(double x) {
	return (x * x) * 0.25;
}

        def code(x):
	return (x * x) * 0.25

        function code(x)
	return Float64(Float64(x * x) * 0.25)
end

        function tmp = code(x)
	tmp = (x * x) * 0.25;
end

        code[x_] := N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision]

        \begin{array}{l}

\\
\left(x \cdot x\right) \cdot 0.25
\end{array}
        Derivation

        1. Initial program 52.7%

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

        3. Taylor expanded in x around 0

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

          1. *-commutativeN/A

            \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{4}} \]

          2. lower-*.f64N/A

            \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{4}} \]

          3. unpow2N/A

            \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{4} \]

          4. lower-*.f6498.9

            \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.25 \]

        5. Applied rewrites98.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.25} \]
        6. Add Preprocessing

        Alternative 11: 4.2% accurate, 54.0× speedup?

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

        real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

        public static double code(double x) {
	return 1.0;
}

        def code(x):
	return 1.0

        function code(x)
	return 1.0
end

        function tmp = code(x)
	tmp = 1.0;
end

        code[x_] := 1.0

        \begin{array}{l}

\\
1
\end{array}
        Derivation

        1. Initial program 52.7%

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

          1. lift-/.f64N/A

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

          2. lift--.f64N/A

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

          3. flip--N/A

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

          4. flip3-+N/A

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

          5. associate-/r/N/A

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

          6. associate-/l*N/A

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

          7. lower-*.f64N/A

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

        4. Applied rewrites100.0%

          \[\leadsto \color{blue}{\frac{0 + x \cdot x}{{\left(1 - x \cdot x\right)}^{1.5} + 1} \cdot \frac{\left(2 - x \cdot x\right) - \sqrt{1 - x \cdot x}}{\sqrt{1 - x \cdot x} + 1}} \]

        5. Taylor expanded in x around inf

          \[\leadsto \color{blue}{\frac{-1}{{\left(\sqrt{-1}\right)}^{2}}} \]
        6. Step-by-step derivation

          1. unpow2N/A

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

          2. rem-square-sqrtN/A

            \[\leadsto \frac{-1}{\color{blue}{-1}} \]

          3. metadata-eval4.3

            \[\leadsto \color{blue}{1} \]

        7. Applied rewrites4.3%

          \[\leadsto \color{blue}{1} \]
        8. Add Preprocessing

        Reproduce

        ?
        herbie shell --seed 1 
(FPCore (x)
  :name "(1-sqrt(1-x*x))/(1+sqrt(1-x*x))"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 1.0))
  (/ (- 1.0 (sqrt (- 1.0 (* x x)))) (+ 1.0 (sqrt (- 1.0 (* x x))))))