Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\\ \frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{t_0}}}{\sqrt[3]{{t_0}^{2}}} - x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.04481 0.99229) 1.0)))
   (- (/ (/ (fma x 0.27061 2.30753) (cbrt t_0)) (cbrt (pow t_0 2.0))) x)))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.04481, 0.99229), 1.0);
	return ((fma(x, 0.27061, 2.30753) / cbrt(t_0)) / cbrt(pow(t_0, 2.0))) - x;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.04481, 0.99229), 1.0)
	return Float64(Float64(Float64(fma(x, 0.27061, 2.30753) / cbrt(t_0)) / cbrt((t_0 ^ 2.0))) - x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(2.30753 + x \cdot 0.27061\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{1 \cdot \left(2.30753 + x \cdot 0.27061\right)}{\color{blue}{\left(\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \cdot \sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}} - x \]

   3. times-frac 100.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} \cdot \sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}}} - x \]

   4. cbrt-unprod 100.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right) \cdot \left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   5. pow2 100.0%

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)\right)}^{2}}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   6. +-commutative 100.0%

      \[\leadsto \frac{1}{\sqrt[3]{{\color{blue}{\left(x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1\right)}}^{2}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   7. fma-def 100.0%

      \[\leadsto \frac{1}{\sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)\right)}}^{2}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   8. +-commutative 100.0%

      \[\leadsto \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)\right)}^{2}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   9. fma-def 100.0%

      \[\leadsto \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)\right)}^{2}}} \cdot \frac{2.30753 + x \cdot 0.27061}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   10. +-commutative 100.0%

       \[\leadsto \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} \cdot \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   11. fma-def 100.0%

       \[\leadsto \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} \cdot \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\sqrt[3]{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   12. +-commutative 100.0%

       \[\leadsto \frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}} - x \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} - x \]
4. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}}} - x \]

   2. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} - x \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}}} - x \]

6. Final simplification 100.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\sqrt[3]{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}}{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)\right)}^{2}}} - x \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Final simplification 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

Alternative 3: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x 0.99229))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * 0.99229d0))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * 0.99229))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * 0.99229))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 98.9%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x \]
3. Step-by-step derivation

   1. *-commutative 98.9%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

4. Simplified 98.9%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

5. Final simplification 98.9%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x \]

Alternative 4: 97.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 98.3%

   \[\leadsto \color{blue}{2.30753} - x \]

3. Final simplification 98.3%

   \[\leadsto 2.30753 - x \]

Alternative 5: 51.6% accurate, 8.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 98.3%

   \[\leadsto \color{blue}{2.30753} - x \]

3. Taylor expanded in x around inf 57.1%

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

   1. neg-mul-1 57.1%

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

5. Simplified 57.1%

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

6. Final simplification 57.1%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))