sqrt(x - (1-s)^2 * 0.25)^(0.5)

Percentage Accurate: 99.7% → 99.9%
Time: 23.8s
Alternatives: 3
Speedup: 1.9×

Specification

?
\[\left(0 \leq x \land x \leq 0.001\right) \land \left(0.99 \leq s \land s \leq 1\right)\]
\[\begin{array}{l} \\ {\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5} \end{array} \]
(FPCore (x s)
 :precision binary64
 (pow (sqrt (- x (* (pow (- 1.0 s) 2.0) 0.25))) 0.5))
double code(double x, double s) {
	return pow(sqrt((x - (pow((1.0 - s), 2.0) * 0.25))), 0.5);
}

real(8) function code(x, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: s
    code = sqrt((x - (((1.0d0 - s) ** 2.0d0) * 0.25d0))) ** 0.5d0
end function

public static double code(double x, double s) {
	return Math.pow(Math.sqrt((x - (Math.pow((1.0 - s), 2.0) * 0.25))), 0.5);
}

def code(x, s):
	return math.pow(math.sqrt((x - (math.pow((1.0 - s), 2.0) * 0.25))), 0.5)

function code(x, s)
	return sqrt(Float64(x - Float64((Float64(1.0 - s) ^ 2.0) * 0.25))) ^ 0.5
end

function tmp = code(x, s)
	tmp = sqrt((x - (((1.0 - s) ^ 2.0) * 0.25))) ^ 0.5;
end

code[x_, s_] := N[Power[N[Sqrt[N[(x - N[(N[Power[N[(1.0 - s), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision]

\begin{array}{l}

\\
{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5}
\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 3 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: 99.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ {\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5} \end{array} \]
(FPCore (x s)
 :precision binary64
 (pow (sqrt (- x (* (pow (- 1.0 s) 2.0) 0.25))) 0.5))
double code(double x, double s) {
	return pow(sqrt((x - (pow((1.0 - s), 2.0) * 0.25))), 0.5);
}

real(8) function code(x, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: s
    code = sqrt((x - (((1.0d0 - s) ** 2.0d0) * 0.25d0))) ** 0.5d0
end function

public static double code(double x, double s) {
	return Math.pow(Math.sqrt((x - (Math.pow((1.0 - s), 2.0) * 0.25))), 0.5);
}

def code(x, s):
	return math.pow(math.sqrt((x - (math.pow((1.0 - s), 2.0) * 0.25))), 0.5)

function code(x, s)
	return sqrt(Float64(x - Float64((Float64(1.0 - s) ^ 2.0) * 0.25))) ^ 0.5
end

function tmp = code(x, s)
	tmp = sqrt((x - (((1.0 - s) ^ 2.0) * 0.25))) ^ 0.5;
end

code[x_, s_] := N[Power[N[Sqrt[N[(x - N[(N[Power[N[(1.0 - s), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision]

\begin{array}{l}

\\
{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5}
\end{array}

Alternative 1: 99.9% accurate, 1.9× speedup?

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(1 - s, \left(1 - s\right) \cdot -0.25, x\right)\right)}^{0.25} \end{array} \]
(FPCore (x s)
 :precision binary64
 (pow (fma (- 1.0 s) (* (- 1.0 s) -0.25) x) 0.25))
double code(double x, double s) {
	return pow(fma((1.0 - s), ((1.0 - s) * -0.25), x), 0.25);
}

function code(x, s)
	return fma(Float64(1.0 - s), Float64(Float64(1.0 - s) * -0.25), x) ^ 0.25
end

code[x_, s_] := N[Power[N[(N[(1.0 - s), $MachinePrecision] * N[(N[(1.0 - s), $MachinePrecision] * -0.25), $MachinePrecision] + x), $MachinePrecision], 0.25], $MachinePrecision]

\begin{array}{l}

\\
{\left(\mathsf{fma}\left(1 - s, \left(1 - s\right) \cdot -0.25, x\right)\right)}^{0.25}
\end{array}
Derivation

  1. Initial program 99.7%

    \[{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-pow.f64N/A

      \[\leadsto \color{blue}{{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot \frac{1}{4}}\right)}^{\frac{1}{2}}} \]

    2. lift-sqrt.f64N/A

      \[\leadsto {\color{blue}{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot \frac{1}{4}}\right)}}^{\frac{1}{2}} \]

    3. sqrt-pow2N/A

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

    4. metadata-evalN/A

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

    5. lower-pow.f6499.8

      \[\leadsto \color{blue}{{\left(x - {\left(1 - s\right)}^{2} \cdot 0.25\right)}^{0.25}} \]

    6. lift--.f64N/A

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

    7. sub-negN/A

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

    8. +-commutativeN/A

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

    9. lift-*.f64N/A

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

    10. *-commutativeN/A

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

    11. distribute-lft-neg-inN/A

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

    12. lower-fma.f64N/A

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

    13. metadata-eval99.8

      \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{-0.25}, {\left(1 - s\right)}^{2}, x\right)\right)}^{0.25} \]

  4. Applied rewrites99.8%

    \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(-0.25, {\left(1 - s\right)}^{2}, x\right)\right)}^{0.25}} \]
  5. Step-by-step derivation

    1. lift-fma.f64N/A

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

    2. *-commutativeN/A

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

    3. lift-pow.f64N/A

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

    4. unpow2N/A

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

    5. associate-*l*N/A

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

    6. lower-fma.f64N/A

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

    7. lower-*.f6499.9

      \[\leadsto {\left(\mathsf{fma}\left(1 - s, \color{blue}{\left(1 - s\right) \cdot -0.25}, x\right)\right)}^{0.25} \]

  6. Applied rewrites99.9%

    \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(1 - s, \left(1 - s\right) \cdot -0.25, x\right)\right)}}^{0.25} \]
  7. Add Preprocessing

Alternative 2: 91.8% accurate, 1.9× speedup?

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(-0.25, \mathsf{fma}\left(s - 2, s, 1\right), x\right)\right)}^{0.25} \end{array} \]
(FPCore (x s)
 :precision binary64
 (pow (fma -0.25 (fma (- s 2.0) s 1.0) x) 0.25))
double code(double x, double s) {
	return pow(fma(-0.25, fma((s - 2.0), s, 1.0), x), 0.25);
}

function code(x, s)
	return fma(-0.25, fma(Float64(s - 2.0), s, 1.0), x) ^ 0.25
end

code[x_, s_] := N[Power[N[(-0.25 * N[(N[(s - 2.0), $MachinePrecision] * s + 1.0), $MachinePrecision] + x), $MachinePrecision], 0.25], $MachinePrecision]

\begin{array}{l}

\\
{\left(\mathsf{fma}\left(-0.25, \mathsf{fma}\left(s - 2, s, 1\right), x\right)\right)}^{0.25}
\end{array}
Derivation

  1. Initial program 99.7%

    \[{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-pow.f64N/A

      \[\leadsto \color{blue}{{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot \frac{1}{4}}\right)}^{\frac{1}{2}}} \]

    2. lift-sqrt.f64N/A

      \[\leadsto {\color{blue}{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot \frac{1}{4}}\right)}}^{\frac{1}{2}} \]

    3. sqrt-pow2N/A

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

    4. metadata-evalN/A

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

    5. lower-pow.f6499.8

      \[\leadsto \color{blue}{{\left(x - {\left(1 - s\right)}^{2} \cdot 0.25\right)}^{0.25}} \]

    6. lift--.f64N/A

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

    7. sub-negN/A

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

    8. +-commutativeN/A

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

    9. lift-*.f64N/A

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

    10. *-commutativeN/A

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

    11. distribute-lft-neg-inN/A

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

    12. lower-fma.f64N/A

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

    13. metadata-eval99.8

      \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{-0.25}, {\left(1 - s\right)}^{2}, x\right)\right)}^{0.25} \]

  4. Applied rewrites99.8%

    \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(-0.25, {\left(1 - s\right)}^{2}, x\right)\right)}^{0.25}} \]

  5. Taylor expanded in s around 0

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

    1. +-commutativeN/A

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

    2. *-commutativeN/A

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

    3. lower-fma.f64N/A

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

    4. lower--.f6492.0

      \[\leadsto {\left(\mathsf{fma}\left(-0.25, \mathsf{fma}\left(\color{blue}{s - 2}, s, 1\right), x\right)\right)}^{0.25} \]

  7. Applied rewrites92.0%

    \[\leadsto {\left(\mathsf{fma}\left(-0.25, \color{blue}{\mathsf{fma}\left(s - 2, s, 1\right)}, x\right)\right)}^{0.25} \]
  8. Add Preprocessing

Alternative 3: 91.7% accurate, 6.2× speedup?

\[\begin{array}{l} \\ \sqrt{\sqrt{x - \mathsf{fma}\left(\mathsf{fma}\left(0.25, s, -0.5\right), s, 0.25\right)}} \end{array} \]
(FPCore (x s)
 :precision binary64
 (sqrt (sqrt (- x (fma (fma 0.25 s -0.5) s 0.25)))))
double code(double x, double s) {
	return sqrt(sqrt((x - fma(fma(0.25, s, -0.5), s, 0.25))));
}

function code(x, s)
	return sqrt(sqrt(Float64(x - fma(fma(0.25, s, -0.5), s, 0.25))))
end

code[x_, s_] := N[Sqrt[N[Sqrt[N[(x - N[(N[(0.25 * s + -0.5), $MachinePrecision] * s + 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\begin{array}{l}

\\
\sqrt{\sqrt{x - \mathsf{fma}\left(\mathsf{fma}\left(0.25, s, -0.5\right), s, 0.25\right)}}
\end{array}
Derivation

  1. Initial program 99.7%

    \[{\left(\sqrt{x - {\left(1 - s\right)}^{2} \cdot 0.25}\right)}^{0.5} \]
  2. Add Preprocessing

  3. Taylor expanded in s around 0

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

    1. +-commutativeN/A

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

    2. *-commutativeN/A

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

    3. lower-fma.f64N/A

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

    4. sub-negN/A

      \[\leadsto {\left(\sqrt{x - \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot s + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, s, \frac{1}{4}\right)}\right)}^{\frac{1}{2}} \]

    5. metadata-evalN/A

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

    6. lower-fma.f6492.0

      \[\leadsto {\left(\sqrt{x - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, s, -0.5\right)}, s, 0.25\right)}\right)}^{0.5} \]

  5. Applied rewrites92.0%

    \[\leadsto {\left(\sqrt{x - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25, s, -0.5\right), s, 0.25\right)}}\right)}^{0.5} \]
  6. Step-by-step derivation

    1. lift-pow.f64N/A

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

    2. unpow1/2N/A

      \[\leadsto \color{blue}{\sqrt{\sqrt{x - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, s, \frac{-1}{2}\right), s, \frac{1}{4}\right)}}} \]

    3. lower-sqrt.f6492.0

      \[\leadsto \color{blue}{\sqrt{\sqrt{x - \mathsf{fma}\left(\mathsf{fma}\left(0.25, s, -0.5\right), s, 0.25\right)}}} \]

  7. Applied rewrites92.0%

    \[\leadsto \color{blue}{\sqrt{\sqrt{x - \mathsf{fma}\left(\mathsf{fma}\left(0.25, s, -0.5\right), s, 0.25\right)}}} \]
  8. Add Preprocessing

Reproduce

?
herbie shell --seed 1 
(FPCore (x s)
  :name "sqrt(x - (1-s)^2 * 0.25)^(0.5)"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 0.001)) (and (<= 0.99 s) (<= s 1.0)))
  (pow (sqrt (- x (* (pow (- 1.0 s) 2.0) 0.25))) 0.5))