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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. neg-mul-1 99.9%

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

   4. neg-mul-1 99.9%

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

   5. distribute-lft-in 99.9%

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

   6. distribute-rgt-in 99.9%

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

   7. *-commutative 99.9%

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

   8. neg-mul-1 99.9%

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

   9. unsub-neg 99.9%

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

3. Simplified 100.0%

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

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 98.8%

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

   1. *-commutative 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 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 99.9%

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

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 5: 50.5% 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 99.9%

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

2. Taylor expanded in x around 0 98.4%

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

3. Taylor expanded in x around inf 51.4%

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

   1. neg-mul-1 51.4%

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

5. Simplified 51.4%

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

6. Final simplification 51.4%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023320 
(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))