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

Percentage Accurate: 100.0% → 100.0%

Time: 9.7s

Alternatives: 7

Speedup: N/A×

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. un-div-inv N/A

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

   8. /-lowering-/.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. +-commutative N/A

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

   16. accelerator-lowering-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

   1. +-lowering-+.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. accelerator-lowering-fma.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -1000000000:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

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 -1000000000.0) (- 0.0 x) (if (<= t_0 5.0) 2.30753 (- 0.0 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 <= -1000000000.0) {
		tmp = 0.0 - x;
	} else if (t_0 <= 5.0) {
		tmp = 2.30753;
	} else {
		tmp = 0.0 - 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 <= (-1000000000.0d0)) then
        tmp = 0.0d0 - x
    else if (t_0 <= 5.0d0) then
        tmp = 2.30753d0
    else
        tmp = 0.0d0 - 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 <= -1000000000.0) {
		tmp = 0.0 - x;
	} else if (t_0 <= 5.0) {
		tmp = 2.30753;
	} else {
		tmp = 0.0 - 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 <= -1000000000.0:
		tmp = 0.0 - x
	elif t_0 <= 5.0:
		tmp = 2.30753
	else:
		tmp = 0.0 - 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 <= -1000000000.0)
		tmp = Float64(0.0 - x);
	elseif (t_0 <= 5.0)
		tmp = 2.30753;
	else
		tmp = Float64(0.0 - 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 <= -1000000000.0)
		tmp = 0.0 - x;
	elseif (t_0 <= 5.0)
		tmp = 2.30753;
	else
		tmp = 0.0 - 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, -1000000000.0], N[(0.0 - x), $MachinePrecision], If[LessEqual[t$95$0, 5.0], 2.30753, N[(0.0 - x), $MachinePrecision]]]]

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) < -1e9 or 5 < (-.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. neg-sub0 N/A

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

      3. --lowering--.f64 99.9

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

   5. Simplified 99.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -1e9 < (-.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) < 5

   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 97.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -1000000000:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 5:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. un-div-inv N/A

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

   8. /-lowering-/.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. +-commutative N/A

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

   16. accelerator-lowering-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 4: 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. --lowering--.f64 N/A

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

   2. /-lowering-/.f64 N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f64 100.0

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

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 5: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $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. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x \]

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. associate-*l* N/A

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

   8. lft-mult-inverse N/A

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

   9. metadata-eval 99.0

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

5. Simplified 99.0%

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

Alternative 6: 97.6% 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 98.2%

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

Alternative 7: 50.8% 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 52.5%

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

Reproduce

herbie shell --seed 2024196 
(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))