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

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

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

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(Float64(Float64(3.695354938841876 * Float64(x * 0.27061)) / x) + fma(x, 0.27061, -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[(N[(3.695354938841876 * N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * 0.27061 + -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 72.6%

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

   2. expm1-udef 72.6%

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

3. Applied egg-rr 72.6%

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

4. Taylor expanded in x around inf 49.6%

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

   1. +-commutative 49.6%

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

   2. associate--l+ 49.6%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

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: 98.7% 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 98.3%

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

   1. *-commutative 98.3%

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

4. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.16))) (- x) 2.30753))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.16d0))) then
        tmp = -x
    else
        tmp = 2.30753d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.16):
		tmp = -x
	else:
		tmp = 2.30753
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.16))
		tmp = Float64(-x);
	else
		tmp = 2.30753;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.16)))
		tmp = -x;
	else
		tmp = 2.30753;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.16]], $MachinePrecision]], (-x), 2.30753]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   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 98.1%

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

   3. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

   5. Simplified 98.7%

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

   if -1.05000000000000004 < x < 1.15999999999999992

   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 97.9%

      \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]
   3. Step-by-step derivation

      1. associate--l+ 97.9%

         \[\leadsto \color{blue}{2.30753 + \left(-2.0191289437 \cdot x - x\right)} \]

      2. +-commutative 97.9%

         \[\leadsto \color{blue}{\left(-2.0191289437 \cdot x - x\right) + 2.30753} \]

      3. *-un-lft-identity 97.9%

         \[\leadsto \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right) + 2.30753 \]

      4. distribute-rgt-out-- 97.9%

         \[\leadsto \color{blue}{x \cdot \left(-2.0191289437 - 1\right)} + 2.30753 \]

      5. metadata-eval 97.9%

         \[\leadsto x \cdot \color{blue}{-3.0191289437} + 2.30753 \]

   4. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{x \cdot -3.0191289437 + 2.30753} \]

   5. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]

Alternative 5: 98.0% 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 97.4%

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

3. Final simplification 97.4%

   \[\leadsto 2.30753 - x \]

Alternative 6: 51.1% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

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

Wolfram

code[x_] := 2.30753

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 53.4%

   \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]
3. Step-by-step derivation

   1. associate--l+ 53.4%

      \[\leadsto \color{blue}{2.30753 + \left(-2.0191289437 \cdot x - x\right)} \]

   2. +-commutative 53.4%

      \[\leadsto \color{blue}{\left(-2.0191289437 \cdot x - x\right) + 2.30753} \]

   3. *-un-lft-identity 53.4%

      \[\leadsto \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right) + 2.30753 \]

   4. distribute-rgt-out-- 53.4%

      \[\leadsto \color{blue}{x \cdot \left(-2.0191289437 - 1\right)} + 2.30753 \]

   5. metadata-eval 53.4%

      \[\leadsto x \cdot \color{blue}{-3.0191289437} + 2.30753 \]

4. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{x \cdot -3.0191289437 + 2.30753} \]

5. Taylor expanded in x around 0 44.6%

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

6. Final simplification 44.6%

   \[\leadsto 2.30753 \]

Reproduce

herbie shell --seed 2023334 
(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))