Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 7.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+14)
   (- 1.0 (/ (- x) y))
   (if (<= x 2e+14) (/ (fma (/ x y) x x) (- x -1.0)) (/ (- (+ y x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+14) {
		tmp = 1.0 - (-x / y);
	} else if (x <= 2e+14) {
		tmp = fma((x / y), x, x) / (x - -1.0);
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+14)
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	elseif (x <= 2e+14)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x - -1.0));
	else
		tmp = Float64(Float64(Float64(y + x) - 1.0) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+14], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+14], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9e14

   1. Initial program 71.3%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 31.1

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 31.1%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 100.0%

       \[\leadsto 1 - \frac{-x}{y} \]

   if -4.9e14 < x < 2e14

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      5. lower-fma.f64 99.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}}{x + 1} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}}{x + 1} \]

   if 2e14 < x

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 37.5

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 37.5%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 + \frac{x}{y}\right) - \color{blue}{\frac{1}{y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+14}:\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -50000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (<= t_0 -50000000000000.0)
     (/ x y)
     (if (<= t_0 5e+18) (/ x (- x -1.0)) (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -50000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e+18) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (x - 1.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    if (t_0 <= (-50000000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 5d+18) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = (x - 1.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if (t_0 <= -50000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 5e+18) {
		tmp = x / (x - -1.0);
	} else {
		tmp = (x - 1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	tmp = 0
	if t_0 <= -50000000000000.0:
		tmp = x / y
	elif t_0 <= 5e+18:
		tmp = x / (x - -1.0)
	else:
		tmp = (x - 1.0) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if (t_0 <= -50000000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 5e+18)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(Float64(x - 1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -50000000000000.0)
		tmp = x / y;
	elseif (t_0 <= 5e+18)
		tmp = x / (x - -1.0);
	else
		tmp = (x - 1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 5e+18], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e13

   1. Initial program 66.3%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 6.5

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 6.5%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 88.4%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot x}{y}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{-1} \cdot x}{y}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right) \]

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \]

      8. lower-/.f64 88.8

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Applied rewrites 88.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -5e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5e18

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{x - \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)} \]

      7. metadata-eval N/A

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

      8. lower--.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if 5e18 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.6%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 69.1

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 69.1%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.1%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{x - 1}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.1%

       \[\leadsto \frac{x - 1}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -50000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1}\\ \mathbf{if}\;t\_0 \leq -50000000000000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (- x -1.0))))
   (if (or (<= t_0 -50000000000000.0) (not (<= t_0 5e-9)))
     (/ x y)
     (* (- 1.0 x) x))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -50000000000000.0) || !(t_0 <= 5e-9)) {
		tmp = x / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x - (-1.0d0))
    if ((t_0 <= (-50000000000000.0d0)) .or. (.not. (t_0 <= 5d-9))) then
        tmp = x / y
    else
        tmp = (1.0d0 - x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	double tmp;
	if ((t_0 <= -50000000000000.0) || !(t_0 <= 5e-9)) {
		tmp = x / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0)
	tmp = 0
	if (t_0 <= -50000000000000.0) or not (t_0 <= 5e-9):
		tmp = x / y
	else:
		tmp = (1.0 - x) * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x - -1.0))
	tmp = 0.0
	if ((t_0 <= -50000000000000.0) || !(t_0 <= 5e-9))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(1.0 - x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x - -1.0);
	tmp = 0.0;
	if ((t_0 <= -50000000000000.0) || ~((t_0 <= 5e-9)))
		tmp = x / y;
	else
		tmp = (1.0 - x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -50000000000000.0], N[Not[LessEqual[t$95$0, 5e-9]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e13 or 5.0000000000000001e-9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 37.9

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 37.9%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot {\left(\sqrt{-1}\right)}^{2}}{y}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot x}}{y}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot x}{y}\right) \]

      4. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{-1} \cdot x}{y}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{-1 \cdot \frac{x}{y}}\right) \]

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \]

      8. lower-/.f64 68.3

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Applied rewrites 68.3%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -5e13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x} + x}{x + 1} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 79.0%

       \[\leadsto \left(1 - x\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq -50000000000000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x - -1} \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(\left(1 - x\right) \cdot x\right) \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (- 1.0 (/ (- 1.0 x) y))
   (if (<= x 1.0) (* (* (- 1.0 x) x) (+ 1.0 (/ x y))) (/ (- (+ y x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else if (x <= 1.0) {
		tmp = ((1.0 - x) * x) * (1.0 + (x / y));
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    else if (x <= 1.0d0) then
        tmp = ((1.0d0 - x) * x) * (1.0d0 + (x / y))
    else
        tmp = ((y + x) - 1.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else if (x <= 1.0) {
		tmp = ((1.0 - x) * x) * (1.0 + (x / y));
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0 - ((1.0 - x) / y)
	elif x <= 1.0:
		tmp = ((1.0 - x) * x) * (1.0 + (x / y))
	else:
		tmp = ((y + x) - 1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	elseif (x <= 1.0)
		tmp = Float64(Float64(Float64(1.0 - x) * x) * Float64(1.0 + Float64(x / y)));
	else
		tmp = Float64(Float64(Float64(y + x) - 1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0 - ((1.0 - x) / y);
	elseif (x <= 1.0)
		tmp = ((1.0 - x) * x) * (1.0 + (x / y));
	else
		tmp = ((y + x) - 1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 31.7

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 31.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.9%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -1 \cdot x\right)\right)} \cdot \left(1 + \frac{x}{y}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 1 < x

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 38.3

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 38.3%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.5%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 + \frac{x}{y}\right) - \color{blue}{\frac{1}{y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.5%

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

4. Final simplification 98.6%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ x (+ 1.0 x)) (+ 1.0 (/ x y))))

C

double code(double x, double y) {
	return (x / (1.0 + x)) * (1.0 + (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (1.0d0 + x)) * (1.0d0 + (x / y))
end function

Java

public static double code(double x, double y) {
	return (x / (1.0 + x)) * (1.0 + (x / y));
}

Python

def code(x, y):
	return (x / (1.0 + x)) * (1.0 + (x / y))

Julia

function code(x, y)
	return Float64(Float64(x / Float64(1.0 + x)) * Float64(1.0 + Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / (1.0 + x)) * (1.0 + (x / y));
end

Wolfram

code[x_, y_] := N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{x}{1 + x} \cdot \left(1 + \frac{x}{y}\right)} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (- 1.0 (/ (- 1.0 x) y))
   (if (<= x 1.0) (fma (- (/ x y) x) x x) (/ (- (+ y x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else if (x <= 1.0) {
		tmp = fma(((x / y) - x), x, x);
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	elseif (x <= 1.0)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	else
		tmp = Float64(Float64(Float64(y + x) - 1.0) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 31.7

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 31.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.9%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x} + x}{x + 1} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.7%

       \[\leadsto \mathsf{fma}\left(\frac{x}{y} - x, \color{blue}{x}, x\right) \]

   if 1 < x

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 38.3

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 38.3%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 96.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 96.5%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 + \frac{x}{y}\right) - \color{blue}{\frac{1}{y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -98000 \lor \neg \left(x \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -98000.0) (not (<= x 1.75e+14)))
   (- 1.0 (/ (- 1.0 x) y))
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -98000.0) || !(x <= 1.75e+14)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-98000.0d0)) .or. (.not. (x <= 1.75d+14))) then
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -98000.0) || !(x <= 1.75e+14)) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -98000.0) or not (x <= 1.75e+14):
		tmp = 1.0 - ((1.0 - x) / y)
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -98000.0) || !(x <= 1.75e+14))
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -98000.0) || ~((x <= 1.75e+14)))
		tmp = 1.0 - ((1.0 - x) / y);
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -98000.0], N[Not[LessEqual[x, 1.75e+14]], $MachinePrecision]], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -98000 or 1.75e14 < x

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 34.7

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 34.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -98000 < x < 1.75e14

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{x - \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)} \]

      7. metadata-eval N/A

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

      8. lower--.f64 69.6

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

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\frac{x}{x - -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -98000 \lor \neg \left(x \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -98000:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -98000.0)
   (- 1.0 (/ (- 1.0 x) y))
   (if (<= x 1.75e+14) (/ x (- x -1.0)) (/ (- (+ y x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -98000.0) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else if (x <= 1.75e+14) {
		tmp = x / (x - -1.0);
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-98000.0d0)) then
        tmp = 1.0d0 - ((1.0d0 - x) / y)
    else if (x <= 1.75d+14) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = ((y + x) - 1.0d0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -98000.0) {
		tmp = 1.0 - ((1.0 - x) / y);
	} else if (x <= 1.75e+14) {
		tmp = x / (x - -1.0);
	} else {
		tmp = ((y + x) - 1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -98000.0:
		tmp = 1.0 - ((1.0 - x) / y)
	elif x <= 1.75e+14:
		tmp = x / (x - -1.0)
	else:
		tmp = ((y + x) - 1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -98000.0)
		tmp = Float64(1.0 - Float64(Float64(1.0 - x) / y));
	elseif (x <= 1.75e+14)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(Float64(Float64(y + x) - 1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -98000.0)
		tmp = 1.0 - ((1.0 - x) / y);
	elseif (x <= 1.75e+14)
		tmp = x / (x - -1.0);
	else
		tmp = ((y + x) - 1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -98000.0], N[(1.0 - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e+14], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -98000

   1. Initial program 71.8%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 32.2

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 32.2%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.8%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   if -98000 < x < 1.75e14

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{x - \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)} \]

      7. metadata-eval N/A

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

      8. lower--.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if 1.75e14 < x

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 37.5

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 37.5%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 + \frac{x}{y}\right) - \color{blue}{\frac{1}{y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 100.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -98000:\\ \;\;\;\;1 - \frac{1 - x}{y}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -105000.0) (not (<= x 1.75e+14)))
   (- 1.0 (/ (- x) y))
   (/ x (- x -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -105000.0) || !(x <= 1.75e+14)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-105000.0d0)) .or. (.not. (x <= 1.75d+14))) then
        tmp = 1.0d0 - (-x / y)
    else
        tmp = x / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -105000.0) || !(x <= 1.75e+14)) {
		tmp = 1.0 - (-x / y);
	} else {
		tmp = x / (x - -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -105000.0) or not (x <= 1.75e+14):
		tmp = 1.0 - (-x / y)
	else:
		tmp = x / (x - -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -105000.0) || !(x <= 1.75e+14))
		tmp = Float64(1.0 - Float64(Float64(-x) / y));
	else
		tmp = Float64(x / Float64(x - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -105000.0) || ~((x <= 1.75e+14)))
		tmp = 1.0 - (-x / y);
	else
		tmp = x / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -105000.0], N[Not[LessEqual[x, 1.75e+14]], $MachinePrecision]], N[(1.0 - N[((-x) / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -105000 or 1.75e14 < x

   1. Initial program 73.5%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 34.7

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 34.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 99.6%

       \[\leadsto 1 - \frac{-x}{y} \]

   if -105000 < x < 1.75e14

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

      3. rgt-mult-inverse N/A

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

      4. cancel-sign-sub N/A

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

      5. distribute-lft-neg-out N/A

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

      6. rgt-mult-inverse N/A

         \[\leadsto \frac{x}{x - \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)} \]

      7. metadata-eval N/A

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

      8. lower--.f64 69.6

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

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\frac{x}{x - -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000 \lor \neg \left(x \leq 1.75 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (/ (- x 1.0) y) (* (- 1.0 x) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x - 1.0d0) / y
    else
        tmp = (1.0d0 - x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (x - 1.0) / y;
	} else {
		tmp = (1.0 - x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = (x - 1.0) / y
	else:
		tmp = (1.0 - x) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(x - 1.0) / y);
	else
		tmp = Float64(Float64(1.0 - x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = (x - 1.0) / y;
	else
		tmp = (1.0 - x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.5%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{\left(\frac{x}{y}\right)}^{1}} + 1\right)}{x + 1} \]

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({\left(\frac{x}{y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right)}{x + 1} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{\left(\frac{x}{y}\right)}^{2}}} + 1\right)}{x + 1} \]

      5. pow2 N/A

         \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}} + 1\right)}{x + 1} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{\frac{x}{y}} \cdot \sqrt{\frac{x}{y}}} + 1\right)}{x + 1} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{\sqrt{\frac{x}{y}}}, \sqrt{\frac{x}{y}}, 1\right)}{x + 1} \]

      9. lower-sqrt.f64 34.9

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\sqrt{\frac{x}{y}}, \color{blue}{\sqrt{\frac{x}{y}}}, 1\right)}{x + 1} \]

   4. Applied rewrites 34.9%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x}{y}}, \sqrt{\frac{x}{y}}, 1\right)}}{x + 1} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x} + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

      3. distribute-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(-1 \cdot \frac{1 + \frac{{\left(\sqrt{-1}\right)}^{2}}{y}}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{{\left(\sqrt{-1}\right)}^{2}}{y}\right)\right)\right)} \]

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.7%

       \[\leadsto 1 - \color{blue}{\frac{1 - x}{y}} \]

   11. Taylor expanded in y around 0

       \[\leadsto \frac{x - 1}{y} \]
   12. Step-by-step derivation

   13. Applied rewrites 78.4%

       \[\leadsto \frac{x - 1}{y} \]

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x} + x}{x + 1} \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 68.1%

       \[\leadsto \left(1 - x\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- 1.0 x) x))

C

double code(double x, double y) {
	return (1.0 - x) * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) * x
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) * x;
}

Python

def code(x, y):
	return (1.0 - x) * x

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) * x)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) * x;
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   6. lower-*.f64 87.5

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x} + x}{x + 1} \]

4. Applied rewrites 87.5%

   \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 58.4

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 42.2%

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

11. Final simplification 42.2%

    \[\leadsto \left(1 - x\right) \cdot x \]
12. Add Preprocessing

Alternative 12: 39.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 x))

C

double code(double x, double y) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y) {
	return 1.0 * x;
}

Python

def code(x, y):
	return 1.0 * x

Julia

function code(x, y)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + 1 \cdot x}}{x + 1} \]

   4. *-lft-identity N/A

      \[\leadsto \frac{\frac{x}{y} \cdot x + \color{blue}{x}}{x + 1} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

   6. lower-*.f64 87.5

      \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x} + x}{x + 1} \]

4. Applied rewrites 87.5%

   \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot x + x}}{x + 1} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{y} - 1\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 58.4

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 36.4%

    \[\leadsto 1 \cdot x \]

11. Final simplification 36.4%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0))))

C

double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 1.0d0) * (((x / y) + 1.0d0) / (x + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Python

def code(x, y):
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x / 1.0) * Float64(Float64(Float64(x / y) + 1.0) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x / 1.0), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))