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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 9

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in x around inf 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 100.0%

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

    1. *-commutative 100.0%

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

12. Simplified 100.0%

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

    1. fma-undefine 100.0%

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

14. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   1.0
   (+
    0.4333638132548656
    (* x (+ 0.37920088514346545 (* x -0.025050834237766436)))))
  x))

C

double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (0.4333638132548656d0 + (x * (0.37920088514346545d0 + (x * (-0.025050834237766436d0)))))) - x
end function

Java

public static double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
}

Python

def code(x):
	return (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * Float64(0.37920088514346545 + Float64(x * -0.025050834237766436))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(0.4333638132548656 + N[(x * N[(0.37920088514346545 + N[(x * -0.025050834237766436), $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. Add Preprocessing

3. Taylor expanded in x around inf 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 98.6%

    \[\leadsto \frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + -0.025050834237766436 \cdot x\right)}} - x \]
11. Step-by-step derivation

    1. *-commutative 98.6%

       \[\leadsto \frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + \color{blue}{x \cdot -0.025050834237766436}\right)} - x \]

12. Simplified 98.6%

    \[\leadsto \frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)}} - x \]
13. Add Preprocessing

Alternative 4: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ 1.0 (+ 0.4333638132548656 (* x 0.37920088514346545))) x))

C

double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (0.4333638132548656d0 + (x * 0.37920088514346545d0))) - x
end function

Java

public static double code(double x) {
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
}

Python

def code(x):
	return (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * 0.37920088514346545))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x;
end

Wolfram

code[x_] := N[(N[(1.0 / N[(0.4333638132548656 + N[(x * 0.37920088514346545), $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. Add Preprocessing

3. Taylor expanded in x around inf 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 98.5%

    \[\leadsto \frac{1}{\color{blue}{0.4333638132548656 + 0.37920088514346545 \cdot x}} - x \]

11. Final simplification 98.5%

    \[\leadsto \frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} - x \]
12. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in x around 0 96.9%

   \[\leadsto \color{blue}{2.30753} - x \]
4. Add Preprocessing

Alternative 6: 58.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 0.4434857248047533 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 0.4434857248047533 x))

C

double code(double x) {
	return 0.4434857248047533 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.4434857248047533d0 - x
end function

Java

public static double code(double x) {
	return 0.4434857248047533 - x;
}

Python

def code(x):
	return 0.4434857248047533 - x

Julia

function code(x)
	return Float64(0.4434857248047533 - x)
end

MATLAB

function tmp = code(x)
	tmp = 0.4434857248047533 - x;
end

Wolfram

code[x_] := N[(0.4434857248047533 - 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. Add Preprocessing

3. Taylor expanded in x around inf 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-define 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-define 100.0%

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

7. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around inf 63.3%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(0.16558885480950444 + 2.254864010426164 \cdot \frac{1}{x}\right)}} - x \]
11. Step-by-step derivation

    1. associate-*r/ 63.3%

       \[\leadsto \frac{1}{x \cdot \left(0.16558885480950444 + \color{blue}{\frac{2.254864010426164 \cdot 1}{x}}\right)} - x \]

    2. metadata-eval 63.3%

       \[\leadsto \frac{1}{x \cdot \left(0.16558885480950444 + \frac{\color{blue}{2.254864010426164}}{x}\right)} - x \]

12. Simplified 63.3%

    \[\leadsto \frac{1}{\color{blue}{x \cdot \left(0.16558885480950444 + \frac{2.254864010426164}{x}\right)}} - x \]

13. Taylor expanded in x around 0 62.2%

    \[\leadsto \color{blue}{0.4434857248047533} - x \]
14. Add Preprocessing

Alternative 7: 58.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 0.2727126142559131 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 0.2727126142559131 x))

C

double code(double x) {
	return 0.2727126142559131 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.2727126142559131d0 - x
end function

Java

public static double code(double x) {
	return 0.2727126142559131 - x;
}

Python

def code(x):
	return 0.2727126142559131 - x

Julia

function code(x)
	return Float64(0.2727126142559131 - x)
end

MATLAB

function tmp = code(x)
	tmp = 0.2727126142559131 - x;
end

Wolfram

code[x_] := N[(0.2727126142559131 - 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. Add Preprocessing

3. Taylor expanded in x around 0 97.7%

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

   1. *-commutative 97.7%

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

5. Simplified 97.7%

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

6. Taylor expanded in x around inf 62.0%

   \[\leadsto \color{blue}{0.2727126142559131} - x \]
7. Add Preprocessing

Alternative 8: 52.6% 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. Add Preprocessing

3. Taylor expanded in x around 0 96.9%

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

4. Taylor expanded in x around inf 57.3%

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

   1. neg-mul-1 57.3%

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

6. Simplified 57.3%

   \[\leadsto \color{blue}{-x} \]
7. Add Preprocessing

Alternative 9: 2.2% accurate, 17.0× 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 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. Add Preprocessing

3. Taylor expanded in x around 0 96.9%

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

4. Taylor expanded in x around inf 57.3%

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

   1. neg-mul-1 57.3%

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

6. Simplified 57.3%

   \[\leadsto \color{blue}{-x} \]
7. Step-by-step derivation

   1. neg-sub0 57.3%

      \[\leadsto \color{blue}{0 - x} \]

   2. sub-neg 57.3%

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

   3. add-sqr-sqrt 26.4%

      \[\leadsto 0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

   4. sqrt-unprod 17.2%

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

   5. sqr-neg 17.2%

      \[\leadsto 0 + \sqrt{\color{blue}{x \cdot x}} \]

   6. sqrt-prod 1.4%

      \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

   7. add-sqr-sqrt 2.1%

      \[\leadsto 0 + \color{blue}{x} \]

8. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{0 + x} \]
9. Step-by-step derivation

   1. +-lft-identity 2.1%

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

10. Simplified 2.1%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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))