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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 6

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, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + \left(\frac{3.695354938841876}{\frac{x}{x \cdot 0.27061}} + \mathsf{fma}\left(0.27061, x, -1\right)\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   (+
    2.30753
    (+ (/ 3.695354938841876 (/ x (* x 0.27061))) (fma 0.27061 x -1.0)))
   (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
  x))

C

double code(double x) {
	return ((2.30753 + ((3.695354938841876 / (x / (x * 0.27061))) + fma(0.27061, x, -1.0))) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(Float64(3.695354938841876 / Float64(x / Float64(x * 0.27061))) + fma(0.27061, x, -1.0))) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(N[(3.695354938841876 / N[(x / N[(x * 0.27061), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.27061 * x + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. expm1-log1p-u 76.9%

      \[\leadsto \frac{2.30753 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 0.27061\right)\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. expm1-udef 76.9%

      \[\leadsto \frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

3. Applied egg-rr 76.9%

   \[\leadsto \frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

4. Taylor expanded in x around inf 50.7%

   \[\leadsto \frac{2.30753 + \color{blue}{\left(\left(e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)} + 3.695354938841876 \cdot \frac{e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}}{x}\right) - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
5. Step-by-step derivation

   1. +-commutative 50.7%

      \[\leadsto \frac{2.30753 + \left(\color{blue}{\left(3.695354938841876 \cdot \frac{e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}}{x} + e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right)} - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. associate--l+ 50.7%

      \[\leadsto \frac{2.30753 + \color{blue}{\left(3.695354938841876 \cdot \frac{e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}}{x} + \left(e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)} - 1\right)\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

6. Simplified 100.0%

   \[\leadsto \frac{2.30753 + \color{blue}{\left(\frac{3.695354938841876}{\frac{x}{0.27061 \cdot x}} + \mathsf{fma}\left(0.27061, x, -1\right)\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

7. Final simplification 100.0%

   \[\leadsto \frac{2.30753 + \left(\frac{3.695354938841876}{\frac{x}{x \cdot 0.27061}} + \mathsf{fma}\left(0.27061, x, -1\right)\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

Alternative 2: 100.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((2.30753 + (-1.0 + (1.0 + (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 + ((-1.0d0) + (1.0d0 + (x * 0.27061d0)))) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(-1.0 + N[(1.0 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. expm1-log1p-u 76.9%

      \[\leadsto \frac{2.30753 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot 0.27061\right)\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. expm1-udef 76.9%

      \[\leadsto \frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

3. Applied egg-rr 76.9%

   \[\leadsto \frac{2.30753 + \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot 0.27061\right)} - 1\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
4. Step-by-step derivation

   1. log1p-udef 76.9%

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

   2. rem-exp-log 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 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 4: 98.5% 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 99.0%

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

   1. *-commutative 99.0%

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

4. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 5: 97.8% 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 98.4%

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

3. Final simplification 98.4%

   \[\leadsto 2.30753 - x \]

Alternative 6: 50.7% 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 98.4%

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

3. Taylor expanded in x around inf 50.3%

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

   1. neg-mul-1 50.3%

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

5. Simplified 50.3%

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

6. Final simplification 50.3%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023318 
(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))