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

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(fma(x, 0.27061, 2.30753) * Float64(1.0 / 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[(1.0 / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $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. div-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-define 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -7.0) (not (<= x 6.2)))
   (-
    (/
     (+ 6.039053782637804 (/ (+ (/ 1686.279566230464 x) -82.23527511657367) x))
     x)
    x)
   (+
    2.30753
    (*
     x
     (- (* x (+ 1.900161040244073 (* x -1.7950336306565942))) 3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -7.0) || !(x <= 6.2)) {
		tmp = ((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.0d0)) .or. (.not. (x <= 6.2d0))) then
        tmp = ((6.039053782637804d0 + (((1686.279566230464d0 / x) + (-82.23527511657367d0)) / x)) / x) - x
    else
        tmp = 2.30753d0 + (x * ((x * (1.900161040244073d0 + (x * (-1.7950336306565942d0)))) - 3.0191289437d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -7.0) || !(x <= 6.2)) {
		tmp = ((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -7.0) or not (x <= 6.2):
		tmp = ((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x
	else:
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -7.0) || !(x <= 6.2))
		tmp = Float64(Float64(Float64(6.039053782637804 + Float64(Float64(Float64(1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	else
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * Float64(1.900161040244073 + Float64(x * -1.7950336306565942))) - 3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -7.0) || ~((x <= 6.2)))
		tmp = ((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x;
	else
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -7.0], N[Not[LessEqual[x, 6.2]], $MachinePrecision]], N[(N[(N[(6.039053782637804 + N[(N[(N[(1686.279566230464 / x), $MachinePrecision] + -82.23527511657367), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(2.30753 + N[(x * N[(N[(x * N[(1.900161040244073 + N[(x * -1.7950336306565942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7 or 6.20000000000000018 < 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 99.5%

      \[\leadsto \color{blue}{\frac{\left(6.039053782637804 + \frac{1686.279566230464}{{x}^{2}}\right) - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   4. Step-by-step derivation

      1. associate--l+ 99.5%

         \[\leadsto \frac{\color{blue}{6.039053782637804 + \left(\frac{1686.279566230464}{{x}^{2}} - 82.23527511657367 \cdot \frac{1}{x}\right)}}{x} - x \]

      2. unpow2 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\frac{1686.279566230464}{\color{blue}{x \cdot x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x \]

      3. associate-/r* 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\color{blue}{\frac{\frac{1686.279566230464}{x}}{x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x \]

      4. metadata-eval 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\frac{\frac{\color{blue}{1686.279566230464 \cdot 1}}{x}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x \]

      5. associate-*r/ 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x}}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x \]

      6. associate-*r/ 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}\right)}{x} - x \]

      7. metadata-eval 99.5%

         \[\leadsto \frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \frac{\color{blue}{82.23527511657367}}{x}\right)}{x} - x \]

      8. div-sub 99.5%

         \[\leadsto \frac{6.039053782637804 + \color{blue}{\frac{1686.279566230464 \cdot \frac{1}{x} - 82.23527511657367}{x}}}{x} - x \]

      9. sub-neg 99.5%

         \[\leadsto \frac{6.039053782637804 + \frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x} + \left(-82.23527511657367\right)}}{x}}{x} - x \]

      10. associate-*r/ 99.5%

          \[\leadsto \frac{6.039053782637804 + \frac{\color{blue}{\frac{1686.279566230464 \cdot 1}{x}} + \left(-82.23527511657367\right)}{x}}{x} - x \]

      11. metadata-eval 99.5%

          \[\leadsto \frac{6.039053782637804 + \frac{\frac{\color{blue}{1686.279566230464}}{x} + \left(-82.23527511657367\right)}{x}}{x} - x \]

      12. metadata-eval 99.5%

          \[\leadsto \frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + \color{blue}{-82.23527511657367}}{x}}{x} - x \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x}} - x \]

   if -7 < x < 6.20000000000000018

   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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.3\right):\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.3)))
   (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
   (+
    2.30753
    (*
     x
     (- (* x (+ 1.900161040244073 (* x -1.7950336306565942))) 3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.3)) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 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 <= 2.3d0))) then
        tmp = ((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x
    else
        tmp = 2.30753d0 + (x * ((x * (1.900161040244073d0 + (x * (-1.7950336306565942d0)))) - 3.0191289437d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.3)) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else {
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.3):
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x
	else:
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.3))
		tmp = Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x);
	else
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * Float64(1.900161040244073 + Float64(x * -1.7950336306565942))) - 3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.3)))
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	else
		tmp = 2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.3]], $MachinePrecision]], N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], N[(2.30753 + N[(x * N[(N[(x * N[(1.900161040244073 + N[(x * -1.7950336306565942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.2999999999999998 < 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 99.4%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   4. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x \]

   if -1.05000000000000004 < x < 2.2999999999999998

   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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 3.0191289437\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.3\right):\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
   (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / 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 <= 1.15d0))) then
        tmp = ((6.039053782637804d0 - (82.23527511657367d0 / x)) / 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 <= 1.15)) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / 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 <= 1.15):
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / 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 <= 1.15))
		tmp = Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / 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 <= 1.15)))
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / 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, 1.15]], $MachinePrecision]], N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / 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 1.1499999999999999 < 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 99.4%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   4. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x \]

   if -1.05000000000000004 < x < 1.1499999999999999

   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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.6)))
   (- (/ 6.039053782637804 x) x)
   (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.6)) {
		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 <= 1.6d0))) 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 <= 1.6)) {
		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 <= 1.6):
		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 <= 1.6))
		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 <= 1.6)))
		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, 1.6]], $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 1.6000000000000001 < 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 99.3%

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

   if -1.05000000000000004 < x < 1.6000000000000001

   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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.9\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.9)))
   (- (/ 6.039053782637804 x) x)
   (+ 2.30753 (* x -3.0191289437))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.9)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = 2.30753 + (x * -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 <= 2.9d0))) then
        tmp = (6.039053782637804d0 / x) - x
    else
        tmp = 2.30753d0 + (x * (-3.0191289437d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.9)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = 2.30753 + (x * -3.0191289437);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.9):
		tmp = (6.039053782637804 / x) - x
	else:
		tmp = 2.30753 + (x * -3.0191289437)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.9))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(2.30753 + Float64(x * -3.0191289437));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.9)))
		tmp = (6.039053782637804 / x) - x;
	else
		tmp = 2.30753 + (x * -3.0191289437);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.9]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.89999999999999991 < 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 99.3%

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

   if -1.05000000000000004 < x < 2.89999999999999991

   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 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. associate--l+ 99.8%

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

      2. *-commutative 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. distribute-rgt-out-- 99.8%

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

      5. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.9\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (- x)
   (+ 2.30753 (* x -3.0191289437))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753 + (x * -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 <= 1.15d0))) then
        tmp = -x
    else
        tmp = 2.30753d0 + (x * (-3.0191289437d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753 + (x * -3.0191289437);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = -x
	else:
		tmp = 2.30753 + (x * -3.0191289437)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = Float64(2.30753 + Float64(x * -3.0191289437));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = -x;
	else
		tmp = 2.30753 + (x * -3.0191289437);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   7. Simplified 99.0%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. associate--l+ 99.8%

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

      2. *-commutative 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. distribute-rgt-out-- 99.8%

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

      5. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15))) (- x) 2.30753))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = -x
    else
        tmp = 2.30753d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = -x
	else:
		tmp = 2.30753
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = 2.30753;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = -x;
	else
		tmp = 2.30753;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), 2.30753]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < 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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.0%

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

      1. neg-mul-1 99.0%

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

   7. Simplified 99.0%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   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. div-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.6% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

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

Wolfram

code[x_] := 2.30753

Derivation

1. Initial program 100.0%

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

   1. div-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-define 100.0%

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

   4. +-commutative 100.0%

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

   5. fma-define 100.0%

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

   6. +-commutative 100.0%

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

   7. fma-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 48.6%

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

Reproduce

herbie shell --seed 2024185 
(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))