Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H

Percentage Accurate: 99.8% → 99.9%

Time: 4.9s

Alternatives: 2

Speedup: 1.0×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{6} - 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* x (/ x 6.0)) 0.5))

C

double code(double x) {
	return (x * (x / 6.0)) - 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x / 6.0d0)) - 0.5d0
end function

Java

public static double code(double x) {
	return (x * (x / 6.0)) - 0.5;
}

Python

def code(x):
	return (x * (x / 6.0)) - 0.5

Julia

function code(x)
	return Float64(Float64(x * Float64(x / 6.0)) - 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x * (x / 6.0)) - 0.5;
end

Wolfram

code[x_] := N[(N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{x \cdot x - 3}{6} \]
2. Step-by-step derivation

   1. sqr-neg 99.6%

      \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-x\right)} - 3}{6} \]

   2. *-lft-identity 99.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\left(-x\right) \cdot \left(-x\right) - 3}{6}} \]

   3. metadata-eval 99.6%

      \[\leadsto \color{blue}{\left(-1 \cdot -1\right)} \cdot \frac{\left(-x\right) \cdot \left(-x\right) - 3}{6} \]

   4. associate-*r/ 99.6%

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot -1\right) \cdot \left(\left(-x\right) \cdot \left(-x\right) - 3\right)}{6}} \]

   5. associate-/l* 99.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{6}{\left(-x\right) \cdot \left(-x\right) - 3}}} \]

   6. associate-/r/ 99.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{6} \cdot \left(\left(-x\right) \cdot \left(-x\right) - 3\right)} \]

   7. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\left(-x\right) \cdot \left(-x\right) - 3\right) \cdot \frac{-1 \cdot -1}{6}} \]

   8. sqr-neg 99.6%

      \[\leadsto \left(\color{blue}{x \cdot x} - 3\right) \cdot \frac{-1 \cdot -1}{6} \]

   9. fma-neg 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -3\right)} \cdot \frac{-1 \cdot -1}{6} \]

   10. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-3}\right) \cdot \frac{-1 \cdot -1}{6} \]

   11. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, -3\right) \cdot \frac{\color{blue}{1}}{6} \]

   12. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, -3\right) \cdot \color{blue}{0.16666666666666666} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -3\right) \cdot 0.16666666666666666} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. metadata-eval 99.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-3}\right) \cdot 0.16666666666666666 \]

   2. fma-neg 99.6%

      \[\leadsto \color{blue}{\left(x \cdot x - 3\right)} \cdot 0.16666666666666666 \]

   3. metadata-eval 99.6%

      \[\leadsto \left(x \cdot x - 3\right) \cdot \color{blue}{\frac{1}{6}} \]

   4. div-inv 99.6%

      \[\leadsto \color{blue}{\frac{x \cdot x - 3}{6}} \]

   5. div-sub 99.6%

      \[\leadsto \color{blue}{\frac{x \cdot x}{6} - \frac{3}{6}} \]

   6. div-inv 99.6%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{6}} - \frac{3}{6} \]

   7. pow2 99.6%

      \[\leadsto \color{blue}{{x}^{2}} \cdot \frac{1}{6} - \frac{3}{6} \]

   8. metadata-eval 99.6%

      \[\leadsto {x}^{2} \cdot \color{blue}{0.16666666666666666} - \frac{3}{6} \]

   9. metadata-eval 99.6%

      \[\leadsto {x}^{2} \cdot 0.16666666666666666 - \color{blue}{0.5} \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{x}^{2} \cdot 0.16666666666666666 - 0.5} \]
7. Step-by-step derivation

   1. rem-exp-log 99.4%

      \[\leadsto {x}^{2} \cdot \color{blue}{e^{\log 0.16666666666666666}} - 0.5 \]

   2. add-cube-cbrt 99.2%

      \[\leadsto \color{blue}{\left(\sqrt[3]{{x}^{2} \cdot e^{\log 0.16666666666666666}} \cdot \sqrt[3]{{x}^{2} \cdot e^{\log 0.16666666666666666}}\right) \cdot \sqrt[3]{{x}^{2} \cdot e^{\log 0.16666666666666666}}} - 0.5 \]

   3. pow3 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{{x}^{2} \cdot e^{\log 0.16666666666666666}}\right)}^{3}} - 0.5 \]

   4. rem-exp-log 99.3%

      \[\leadsto {\left(\sqrt[3]{{x}^{2} \cdot \color{blue}{0.16666666666666666}}\right)}^{3} - 0.5 \]

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{{x}^{2} \cdot 0.16666666666666666}\right)}^{3}} - 0.5 \]
9. Step-by-step derivation

   1. rem-cube-cbrt 99.6%

      \[\leadsto \color{blue}{{x}^{2} \cdot 0.16666666666666666} - 0.5 \]

   2. metadata-eval 99.6%

      \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{6}} - 0.5 \]

   3. div-inv 99.6%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{6}} - 0.5 \]

   4. unpow2 99.6%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{6} - 0.5 \]

   5. associate-/l* 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}}} - 0.5 \]

10. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}}} - 0.5 \]
11. Step-by-step derivation

    1. associate-/r/ 100.0%

       \[\leadsto \color{blue}{\frac{x}{6} \cdot x} - 0.5 \]

12. Simplified 100.0%

    \[\leadsto \color{blue}{\frac{x}{6} \cdot x} - 0.5 \]

13. Final simplification 100.0%

    \[\leadsto x \cdot \frac{x}{6} - 0.5 \]
14. Add Preprocessing

Alternative 2: 49.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 99.6%

   \[\frac{x \cdot x - 3}{6} \]
2. Step-by-step derivation

   1. sqr-neg 99.6%

      \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-x\right)} - 3}{6} \]

   2. *-lft-identity 99.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\left(-x\right) \cdot \left(-x\right) - 3}{6}} \]

   3. metadata-eval 99.6%

      \[\leadsto \color{blue}{\left(-1 \cdot -1\right)} \cdot \frac{\left(-x\right) \cdot \left(-x\right) - 3}{6} \]

   4. associate-*r/ 99.6%

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot -1\right) \cdot \left(\left(-x\right) \cdot \left(-x\right) - 3\right)}{6}} \]

   5. associate-/l* 99.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{6}{\left(-x\right) \cdot \left(-x\right) - 3}}} \]

   6. associate-/r/ 99.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot -1}{6} \cdot \left(\left(-x\right) \cdot \left(-x\right) - 3\right)} \]

   7. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\left(-x\right) \cdot \left(-x\right) - 3\right) \cdot \frac{-1 \cdot -1}{6}} \]

   8. sqr-neg 99.6%

      \[\leadsto \left(\color{blue}{x \cdot x} - 3\right) \cdot \frac{-1 \cdot -1}{6} \]

   9. fma-neg 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -3\right)} \cdot \frac{-1 \cdot -1}{6} \]

   10. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-3}\right) \cdot \frac{-1 \cdot -1}{6} \]

   11. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, -3\right) \cdot \frac{\color{blue}{1}}{6} \]

   12. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(x, x, -3\right) \cdot \color{blue}{0.16666666666666666} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -3\right) \cdot 0.16666666666666666} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.3%

   \[\leadsto \color{blue}{-0.5} \]

6. Final simplification 52.3%

   \[\leadsto -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))