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

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 7

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + \color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + \color{blue}{x \cdot \frac{4481}{100000}}\right)} - x \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   8. lift--.f64 100.0

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;-x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0.2727126142559131 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_0 -200000.0)
     (- x)
     (if (<= t_0 4.0) 2.30753 (- 0.2727126142559131 x)))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = -x;
	} else if (t_0 <= 4.0) {
		tmp = 2.30753;
	} else {
		tmp = 0.2727126142559131 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if (t_0 <= (-200000.0d0)) then
        tmp = -x
    else if (t_0 <= 4.0d0) then
        tmp = 2.30753d0
    else
        tmp = 0.2727126142559131d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = -x;
	} else if (t_0 <= 4.0) {
		tmp = 2.30753;
	} else {
		tmp = 0.2727126142559131 - x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if t_0 <= -200000.0:
		tmp = -x
	elif t_0 <= 4.0:
		tmp = 2.30753
	else:
		tmp = 0.2727126142559131 - x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = Float64(-x);
	elseif (t_0 <= 4.0)
		tmp = 2.30753;
	else
		tmp = Float64(0.2727126142559131 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if (t_0 <= -200000.0)
		tmp = -x;
	elseif (t_0 <= 4.0)
		tmp = 2.30753;
	else
		tmp = 0.2727126142559131 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -200000.0], (-x), If[LessEqual[t$95$0, 4.0], 2.30753, N[(0.2727126142559131 - x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -2e5

   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

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 99.4

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

   5. Simplified 99.4%

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

   if -2e5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000}} \]
   4. Step-by-step derivation

   5. Simplified 97.0%

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

   if 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) 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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{230753}{100000} + \color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + \color{blue}{x \cdot \frac{4481}{100000}}\right)} - x \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      8. lift--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x \]

      2. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(1 \cdot x\right)} + 1} - x \]

      3. lft-mult-inverse N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\color{blue}{\left(\frac{1}{x} \cdot x\right)} \cdot x\right) + 1} - x \]

      4. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right)\right)} + 1} - x \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right) + 1} - x \]

      6. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1} - x \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right) \cdot x} + 1} - x \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)} + 1} - x \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\mathsf{fma}\left(x, \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, 1\right)}} - x \]

      11. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, 1\right)} - x \]

      12. lft-mult-inverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \frac{99229}{100000} \cdot \color{blue}{1}, 1\right)} - x \]

      13. metadata-eval 97.4

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

   7. Simplified 97.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{27061}{99229} \cdot \frac{1}{x} - 1\right)} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{27061}{99229} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{27061}{99229} \cdot \frac{1}{x} + \color{blue}{-1}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{27061}{99229} \cdot \frac{1}{x}\right) \cdot x + -1 \cdot x} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{27061}{99229} \cdot \left(\frac{1}{x} \cdot x\right)} + -1 \cdot x \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{27061}{99229} \cdot \color{blue}{1} + -1 \cdot x \]

      6. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{27061}{99229}} + -1 \cdot x \]

      7. mul-1-neg N/A

         \[\leadsto \frac{27061}{99229} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      8. sub-neg N/A

         \[\leadsto \color{blue}{\frac{27061}{99229} - x} \]

      9. lower--.f64 97.4

         \[\leadsto \color{blue}{0.2727126142559131 - x} \]

   10. Simplified 97.4%

       \[\leadsto \color{blue}{0.2727126142559131 - x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;-x\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (<= t_0 -200000.0) (- x) (if (<= t_0 4.0) 2.30753 (- x)))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = -x;
	} else if (t_0 <= 4.0) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if (t_0 <= (-200000.0d0)) then
        tmp = -x
    else if (t_0 <= 4.0d0) then
        tmp = 2.30753d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = -x;
	} else if (t_0 <= 4.0) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if t_0 <= -200000.0:
		tmp = -x
	elif t_0 <= 4.0:
		tmp = 2.30753
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = Float64(-x);
	elseif (t_0 <= 4.0)
		tmp = 2.30753;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if (t_0 <= -200000.0)
		tmp = -x;
	elseif (t_0 <= 4.0)
		tmp = 2.30753;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -200000.0], (-x), If[LessEqual[t$95$0, 4.0], 2.30753, (-x)]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -2e5 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 98.5

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

   5. Simplified 98.5%

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

   if -2e5 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000}} \]
   4. Step-by-step derivation

   5. Simplified 97.0%

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

Alternative 4: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + \color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + \color{blue}{x \cdot \frac{4481}{100000}}\right)} - x \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   8. lift--.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x \]

   2. *-lft-identity N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(1 \cdot x\right)} + 1} - x \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\color{blue}{\left(\frac{1}{x} \cdot x\right)} \cdot x\right) + 1} - x \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right)\right)} + 1} - x \]

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right) + 1} - x \]

   6. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1} - x \]

   8. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right) \cdot x} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)} + 1} - x \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\mathsf{fma}\left(x, \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, 1\right)}} - x \]

   11. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, 1\right)} - x \]

   12. lft-mult-inverse N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \frac{99229}{100000} \cdot \color{blue}{1}, 1\right)} - x \]

   13. metadata-eval 98.5

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

7. Simplified 98.5%

   \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229, 1\right)}} - x \]
8. Add Preprocessing

Alternative 5: 97.7% accurate, 9.8× 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

   \[\leadsto \color{blue}{\frac{230753}{100000}} - x \]
4. Step-by-step derivation

5. Simplified 97.5%

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

Alternative 6: 50.4% accurate, 39.0× 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000}} \]
4. Step-by-step derivation

5. Simplified 50.3%

   \[\leadsto \color{blue}{2.30753} \]
6. Add Preprocessing

Alternative 7: 9.7% accurate, 39.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.2727126142559131)

C

double code(double x) {
	return 0.2727126142559131;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.2727126142559131

Julia

function code(x)
	return 0.2727126142559131
end

MATLAB

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

Wolfram

code[x_] := 0.2727126142559131

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. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + \color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + \color{blue}{x \cdot \frac{4481}{100000}}\right)} - x \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + \color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   8. lift--.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x \]

   2. *-lft-identity N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(1 \cdot x\right)} + 1} - x \]

   3. lft-mult-inverse N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\color{blue}{\left(\frac{1}{x} \cdot x\right)} \cdot x\right) + 1} - x \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right)\right)} + 1} - x \]

   5. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right) + 1} - x \]

   6. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1} - x \]

   8. associate-*r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right) \cdot x} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right)} + 1} - x \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\mathsf{fma}\left(x, \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, 1\right)}} - x \]

   11. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, 1\right)} - x \]

   12. lft-mult-inverse N/A

       \[\leadsto \frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \frac{99229}{100000} \cdot \color{blue}{1}, 1\right)} - x \]

   13. metadata-eval 98.5

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

7. Simplified 98.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\frac{27061}{99229} \cdot \frac{1}{x} - 1\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{27061}{99229} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(\frac{27061}{99229} \cdot \frac{1}{x} + \color{blue}{-1}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{27061}{99229} \cdot \frac{1}{x}\right) \cdot x + -1 \cdot x} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{27061}{99229} \cdot \left(\frac{1}{x} \cdot x\right)} + -1 \cdot x \]

   5. lft-mult-inverse N/A

      \[\leadsto \frac{27061}{99229} \cdot \color{blue}{1} + -1 \cdot x \]

   6. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{27061}{99229}} + -1 \cdot x \]

   7. mul-1-neg N/A

      \[\leadsto \frac{27061}{99229} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   8. sub-neg N/A

      \[\leadsto \color{blue}{\frac{27061}{99229} - x} \]

   9. lower--.f64 56.8

      \[\leadsto \color{blue}{0.2727126142559131 - x} \]

10. Simplified 56.8%

    \[\leadsto \color{blue}{0.2727126142559131 - x} \]

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{27061}{99229}} \]
12. Step-by-step derivation

13. Simplified 9.6%

    \[\leadsto \color{blue}{0.2727126142559131} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))