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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-def 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-def 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

5. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

7. Applied egg-rr 100.0%

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

   1. fma-udef 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot 0.16558885480950444 + 2.254864010426164\right) - \frac{15.532191530167717}{x}} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (- (/ 6.039053782637804 x) (/ 82.23527511657367 (* x x))) x)
   (if (<= x 1.2)
     (+ 2.30753 (* x (+ (* x 1.900161040244073) -3.0191289437)))
     (-
      (/
       1.0
       (-
        (+ (* x 0.16558885480950444) 2.254864010426164)
        (/ 15.532191530167717 x)))
      x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	} else if (x <= 1.2) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	} else {
		tmp = (1.0 / (((x * 0.16558885480950444) + 2.254864010426164) - (15.532191530167717 / x))) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = ((6.039053782637804d0 / x) - (82.23527511657367d0 / (x * x))) - x
    else if (x <= 1.2d0) then
        tmp = 2.30753d0 + (x * ((x * 1.900161040244073d0) + (-3.0191289437d0)))
    else
        tmp = (1.0d0 / (((x * 0.16558885480950444d0) + 2.254864010426164d0) - (15.532191530167717d0 / x))) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	} else if (x <= 1.2) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	} else {
		tmp = (1.0 / (((x * 0.16558885480950444) + 2.254864010426164) - (15.532191530167717 / x))) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x
	elif x <= 1.2:
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437))
	else:
		tmp = (1.0 / (((x * 0.16558885480950444) + 2.254864010426164) - (15.532191530167717 / x))) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(Float64(6.039053782637804 / x) - Float64(82.23527511657367 / Float64(x * x))) - x);
	elseif (x <= 1.2)
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) + -3.0191289437)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(x * 0.16558885480950444) + 2.254864010426164) - Float64(15.532191530167717 / x))) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	elseif (x <= 1.2)
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	else
		tmp = (1.0 / (((x * 0.16558885480950444) + 2.254864010426164) - (15.532191530167717 / x))) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - N[(82.23527511657367 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.2], N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(x * 0.16558885480950444), $MachinePrecision] + 2.254864010426164), $MachinePrecision] - N[(15.532191530167717 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 inf 97.9%

      \[\leadsto \color{blue}{\left(6.039053782637804 \cdot \frac{1}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right)} - x \]
   3. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto \left(\color{blue}{\frac{6.039053782637804 \cdot 1}{x}} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x \]

      2. metadata-eval 97.9%

         \[\leadsto \left(\frac{\color{blue}{6.039053782637804}}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x \]

      3. unpow2 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - 82.23527511657367 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) - x \]

      4. associate-*r/ 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{x \cdot x}}\right) - x \]

      5. metadata-eval 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - \frac{\color{blue}{82.23527511657367}}{x \cdot x}\right) - x \]

   4. Simplified 97.9%

      \[\leadsto \color{blue}{\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right)} - x \]

   if -1.05000000000000004 < x < 1.19999999999999996

   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. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \left(-2.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)\right)} - x \]
   5. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -2.0191289437 \cdot x\right)}\right) - x \]

      2. unpow2 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -2.0191289437 \cdot x\right)\right) - x \]

      3. *-commutative 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \left(x \cdot x\right) + \color{blue}{x \cdot -2.0191289437}\right)\right) - x \]

      4. fma-def 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)}\right) - x \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)\right)} - x \]

   7. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{2.30753 + \left(-3.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -3.0191289437 \cdot x\right)} \]

      2. unpow2 99.0%

         \[\leadsto 2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -3.0191289437 \cdot x\right) \]

      3. associate-*r* 99.0%

         \[\leadsto 2.30753 + \left(\color{blue}{\left(1.900161040244073 \cdot x\right) \cdot x} + -3.0191289437 \cdot x\right) \]

      4. distribute-rgt-out 99.0%

         \[\leadsto 2.30753 + \color{blue}{x \cdot \left(1.900161040244073 \cdot x + -3.0191289437\right)} \]

      5. *-commutative 99.0%

         \[\leadsto 2.30753 + x \cdot \left(\color{blue}{x \cdot 1.900161040244073} + -3.0191289437\right) \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)} \]

   if 1.19999999999999996 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(2.254864010426164 + 0.16558885480950444 \cdot x\right) - 15.532191530167717 \cdot \frac{1}{x}}} - x \]
   7. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \frac{1}{\color{blue}{\left(0.16558885480950444 \cdot x + 2.254864010426164\right)} - 15.532191530167717 \cdot \frac{1}{x}} - x \]

      2. *-commutative 99.8%

         \[\leadsto \frac{1}{\left(\color{blue}{x \cdot 0.16558885480950444} + 2.254864010426164\right) - 15.532191530167717 \cdot \frac{1}{x}} - x \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{1}{\left(x \cdot 0.16558885480950444 + 2.254864010426164\right) - \color{blue}{\frac{15.532191530167717 \cdot 1}{x}}} - x \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{1}{\left(x \cdot 0.16558885480950444 + 2.254864010426164\right) - \frac{\color{blue}{15.532191530167717}}{x}} - x \]

   8. Simplified 99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot 0.16558885480950444 + 2.254864010426164\right) - \frac{15.532191530167717}{x}}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot 0.16558885480950444 + 2.254864010426164\right) - \frac{15.532191530167717}{x}} - x\\ \end{array} \]

Alternative 3: 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 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%

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

Alternative 4: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.56\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.56)))
   (- (/ 6.039053782637804 x) x)
   (+ 2.30753 (* x (+ (* x 1.900161040244073) -3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.56)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.56d0))) then
        tmp = (6.039053782637804d0 / x) - x
    else
        tmp = 2.30753d0 + (x * ((x * 1.900161040244073d0) + (-3.0191289437d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.56)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.56):
		tmp = (6.039053782637804 / x) - x
	else:
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.56))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) + -3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.56)))
		tmp = (6.039053782637804 / x) - x;
	else
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.56]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.5600000000000001 < 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 inf 98.6%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -1.05000000000000004 < x < 1.5600000000000001

   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. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \left(-2.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)\right)} - x \]
   5. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -2.0191289437 \cdot x\right)}\right) - x \]

      2. unpow2 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -2.0191289437 \cdot x\right)\right) - x \]

      3. *-commutative 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \left(x \cdot x\right) + \color{blue}{x \cdot -2.0191289437}\right)\right) - x \]

      4. fma-def 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)}\right) - x \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)\right)} - x \]

   7. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{2.30753 + \left(-3.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -3.0191289437 \cdot x\right)} \]

      2. unpow2 99.0%

         \[\leadsto 2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -3.0191289437 \cdot x\right) \]

      3. associate-*r* 99.0%

         \[\leadsto 2.30753 + \left(\color{blue}{\left(1.900161040244073 \cdot x\right) \cdot x} + -3.0191289437 \cdot x\right) \]

      4. distribute-rgt-out 99.0%

         \[\leadsto 2.30753 + \color{blue}{x \cdot \left(1.900161040244073 \cdot x + -3.0191289437\right)} \]

      5. *-commutative 99.0%

         \[\leadsto 2.30753 + x \cdot \left(\color{blue}{x \cdot 1.900161040244073} + -3.0191289437\right) \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.56\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{1}{x \cdot 0.16558885480950444 + 2.254864010426164} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.2)))
   (- (/ 1.0 (+ (* x 0.16558885480950444) 2.254864010426164)) x)
   (+ 2.30753 (* x (+ (* x 1.900161040244073) -3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.2)) {
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	} else {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = (1.0d0 / ((x * 0.16558885480950444d0) + 2.254864010426164d0)) - x
    else
        tmp = 2.30753d0 + (x * ((x * 1.900161040244073d0) + (-3.0191289437d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.2)) {
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	} else {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.2):
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x
	else:
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.2))
		tmp = Float64(Float64(1.0 / Float64(Float64(x * 0.16558885480950444) + 2.254864010426164)) - x);
	else
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) + -3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.2)))
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	else
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(N[(1.0 / N[(N[(x * 0.16558885480950444), $MachinePrecision] + 2.254864010426164), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.8%

      \[\leadsto \frac{1}{\color{blue}{2.254864010426164 + 0.16558885480950444 \cdot x}} - x \]
   7. Step-by-step derivation

      1. +-commutative 98.8%

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

      2. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   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. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \left(-2.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)\right)} - x \]
   5. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -2.0191289437 \cdot x\right)}\right) - x \]

      2. unpow2 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -2.0191289437 \cdot x\right)\right) - x \]

      3. *-commutative 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \left(x \cdot x\right) + \color{blue}{x \cdot -2.0191289437}\right)\right) - x \]

      4. fma-def 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)}\right) - x \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)\right)} - x \]

   7. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{2.30753 + \left(-3.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -3.0191289437 \cdot x\right)} \]

      2. unpow2 99.0%

         \[\leadsto 2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -3.0191289437 \cdot x\right) \]

      3. associate-*r* 99.0%

         \[\leadsto 2.30753 + \left(\color{blue}{\left(1.900161040244073 \cdot x\right) \cdot x} + -3.0191289437 \cdot x\right) \]

      4. distribute-rgt-out 99.0%

         \[\leadsto 2.30753 + \color{blue}{x \cdot \left(1.900161040244073 \cdot x + -3.0191289437\right)} \]

      5. *-commutative 99.0%

         \[\leadsto 2.30753 + x \cdot \left(\color{blue}{x \cdot 1.900161040244073} + -3.0191289437\right) \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;\frac{1}{x \cdot 0.16558885480950444 + 2.254864010426164} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 0.16558885480950444 + 2.254864010426164} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (- (/ 6.039053782637804 x) (/ 82.23527511657367 (* x x))) x)
   (if (<= x 1.2)
     (+ 2.30753 (* x (+ (* x 1.900161040244073) -3.0191289437)))
     (- (/ 1.0 (+ (* x 0.16558885480950444) 2.254864010426164)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	} else if (x <= 1.2) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	} else {
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = ((6.039053782637804d0 / x) - (82.23527511657367d0 / (x * x))) - x
    else if (x <= 1.2d0) then
        tmp = 2.30753d0 + (x * ((x * 1.900161040244073d0) + (-3.0191289437d0)))
    else
        tmp = (1.0d0 / ((x * 0.16558885480950444d0) + 2.254864010426164d0)) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	} else if (x <= 1.2) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	} else {
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x
	elif x <= 1.2:
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437))
	else:
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(Float64(6.039053782637804 / x) - Float64(82.23527511657367 / Float64(x * x))) - x);
	elseif (x <= 1.2)
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) + -3.0191289437)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(x * 0.16558885480950444) + 2.254864010426164)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = ((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x;
	elseif (x <= 1.2)
		tmp = 2.30753 + (x * ((x * 1.900161040244073) + -3.0191289437));
	else
		tmp = (1.0 / ((x * 0.16558885480950444) + 2.254864010426164)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - N[(82.23527511657367 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.2], N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] + -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(x * 0.16558885480950444), $MachinePrecision] + 2.254864010426164), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 inf 97.9%

      \[\leadsto \color{blue}{\left(6.039053782637804 \cdot \frac{1}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right)} - x \]
   3. Step-by-step derivation

      1. associate-*r/ 97.9%

         \[\leadsto \left(\color{blue}{\frac{6.039053782637804 \cdot 1}{x}} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x \]

      2. metadata-eval 97.9%

         \[\leadsto \left(\frac{\color{blue}{6.039053782637804}}{x} - 82.23527511657367 \cdot \frac{1}{{x}^{2}}\right) - x \]

      3. unpow2 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - 82.23527511657367 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) - x \]

      4. associate-*r/ 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{x \cdot x}}\right) - x \]

      5. metadata-eval 97.9%

         \[\leadsto \left(\frac{6.039053782637804}{x} - \frac{\color{blue}{82.23527511657367}}{x \cdot x}\right) - x \]

   4. Simplified 97.9%

      \[\leadsto \color{blue}{\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right)} - x \]

   if -1.05000000000000004 < x < 1.19999999999999996

   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. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \left(-2.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)\right)} - x \]
   5. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -2.0191289437 \cdot x\right)}\right) - x \]

      2. unpow2 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -2.0191289437 \cdot x\right)\right) - x \]

      3. *-commutative 99.0%

         \[\leadsto \left(2.30753 + \left(1.900161040244073 \cdot \left(x \cdot x\right) + \color{blue}{x \cdot -2.0191289437}\right)\right) - x \]

      4. fma-def 99.0%

         \[\leadsto \left(2.30753 + \color{blue}{\mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)}\right) - x \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\left(2.30753 + \mathsf{fma}\left(1.900161040244073, x \cdot x, x \cdot -2.0191289437\right)\right)} - x \]

   7. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{2.30753 + \left(-3.0191289437 \cdot x + 1.900161040244073 \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 2.30753 + \color{blue}{\left(1.900161040244073 \cdot {x}^{2} + -3.0191289437 \cdot x\right)} \]

      2. unpow2 99.0%

         \[\leadsto 2.30753 + \left(1.900161040244073 \cdot \color{blue}{\left(x \cdot x\right)} + -3.0191289437 \cdot x\right) \]

      3. associate-*r* 99.0%

         \[\leadsto 2.30753 + \left(\color{blue}{\left(1.900161040244073 \cdot x\right) \cdot x} + -3.0191289437 \cdot x\right) \]

      4. distribute-rgt-out 99.0%

         \[\leadsto 2.30753 + \color{blue}{x \cdot \left(1.900161040244073 \cdot x + -3.0191289437\right)} \]

      5. *-commutative 99.0%

         \[\leadsto 2.30753 + x \cdot \left(\color{blue}{x \cdot 1.900161040244073} + -3.0191289437\right) \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)} \]

   if 1.19999999999999996 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.7%

      \[\leadsto \frac{1}{\color{blue}{2.254864010426164 + 0.16558885480950444 \cdot x}} - x \]
   7. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

   8. Simplified 99.7%

      \[\leadsto \frac{1}{\color{blue}{x \cdot 0.16558885480950444 + 2.254864010426164}} - x \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 + -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot 0.16558885480950444 + 2.254864010426164} - x\\ \end{array} \]

Alternative 7: 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 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 97.6%

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

   1. *-commutative 97.6%

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

4. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 8: 97.9% 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 96.7%

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

3. Final simplification 96.7%

   \[\leadsto 2.30753 - x \]

Alternative 9: 52.5% 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 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 96.7%

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

3. Taylor expanded in x around inf 56.9%

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

   1. neg-mul-1 56.9%

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

5. Simplified 56.9%

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

6. Final simplification 56.9%

   \[\leadsto -x \]

Reproduce

herbie shell --seed 2023271 
(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))