Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 10.6s

Alternatives: 14

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{x}{\frac{x + 1}{1 + \frac{x}{y}}} \]
8. Add Preprocessing

Alternative 2: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -58000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.36:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -58000000000.0)
     (/ x y)
     (if (<= x -2.2e-71)
       t_0
       (if (<= x 2.25e-178)
         x
         (if (<= x 1.6e-159)
           t_0
           (if (<= x 1.6e-110)
             x
             (if (<= x 8.5e-89)
               t_0
               (if (<= x 0.36) (* x (- 1.0 x)) (/ x y))))))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -58000000000.0) {
		tmp = x / y;
	} else if (x <= -2.2e-71) {
		tmp = t_0;
	} else if (x <= 2.25e-178) {
		tmp = x;
	} else if (x <= 1.6e-159) {
		tmp = t_0;
	} else if (x <= 1.6e-110) {
		tmp = x;
	} else if (x <= 8.5e-89) {
		tmp = t_0;
	} else if (x <= 0.36) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / 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)
    if (x <= (-58000000000.0d0)) then
        tmp = x / y
    else if (x <= (-2.2d-71)) then
        tmp = t_0
    else if (x <= 2.25d-178) then
        tmp = x
    else if (x <= 1.6d-159) then
        tmp = t_0
    else if (x <= 1.6d-110) then
        tmp = x
    else if (x <= 8.5d-89) then
        tmp = t_0
    else if (x <= 0.36d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -58000000000.0) {
		tmp = x / y;
	} else if (x <= -2.2e-71) {
		tmp = t_0;
	} else if (x <= 2.25e-178) {
		tmp = x;
	} else if (x <= 1.6e-159) {
		tmp = t_0;
	} else if (x <= 1.6e-110) {
		tmp = x;
	} else if (x <= 8.5e-89) {
		tmp = t_0;
	} else if (x <= 0.36) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -58000000000.0:
		tmp = x / y
	elif x <= -2.2e-71:
		tmp = t_0
	elif x <= 2.25e-178:
		tmp = x
	elif x <= 1.6e-159:
		tmp = t_0
	elif x <= 1.6e-110:
		tmp = x
	elif x <= 8.5e-89:
		tmp = t_0
	elif x <= 0.36:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -58000000000.0)
		tmp = Float64(x / y);
	elseif (x <= -2.2e-71)
		tmp = t_0;
	elseif (x <= 2.25e-178)
		tmp = x;
	elseif (x <= 1.6e-159)
		tmp = t_0;
	elseif (x <= 1.6e-110)
		tmp = x;
	elseif (x <= 8.5e-89)
		tmp = t_0;
	elseif (x <= 0.36)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -58000000000.0)
		tmp = x / y;
	elseif (x <= -2.2e-71)
		tmp = t_0;
	elseif (x <= 2.25e-178)
		tmp = x;
	elseif (x <= 1.6e-159)
		tmp = t_0;
	elseif (x <= 1.6e-110)
		tmp = x;
	elseif (x <= 8.5e-89)
		tmp = t_0;
	elseif (x <= 0.36)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -58000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.2e-71], t$95$0, If[LessEqual[x, 2.25e-178], x, If[LessEqual[x, 1.6e-159], t$95$0, If[LessEqual[x, 1.6e-110], x, If[LessEqual[x, 8.5e-89], t$95$0, If[LessEqual[x, 0.36], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.8e10 or 0.35999999999999999 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.9%

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

   if -5.8e10 < x < -2.19999999999999997e-71 or 2.24999999999999989e-178 < x < 1.6e-159 or 1.60000000000000014e-110 < x < 8.49999999999999937e-89

   1. Initial program 99.7%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 68.5%

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

      1. associate-/l* 68.5%

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

   9. Simplified 68.5%

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

   10. Taylor expanded in x around 0 67.8%

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

       1. un-div-inv 67.9%

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

       2. associate-/r/ 67.8%

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

   12. Applied egg-rr 67.8%

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

   if -2.19999999999999997e-71 < x < 2.24999999999999989e-178 or 1.6e-159 < x < 1.60000000000000014e-110

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 90.0%

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

   if 8.49999999999999937e-89 < x < 0.35999999999999999

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 84.2%

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

   6. Taylor expanded in x around 0 84.2%

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

      1. neg-mul-1 84.2%

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

      2. sub-neg 84.2%

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

   8. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.36:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.6e+41)
   (/ x y)
   (if (<= x 2.25e-178)
     (/ x (+ x 1.0))
     (if (<= x 1.6e-159)
       (/ x (/ y x))
       (if (<= x 2.8e-115)
         x
         (if (<= x 5.6e-88)
           (* x (/ x y))
           (if (<= x 0.76) (* x (- 1.0 x)) (/ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.6e+41) {
		tmp = x / y;
	} else if (x <= 2.25e-178) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-159) {
		tmp = x / (y / x);
	} else if (x <= 2.8e-115) {
		tmp = x;
	} else if (x <= 5.6e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.76) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / 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 <= (-8.6d+41)) then
        tmp = x / y
    else if (x <= 2.25d-178) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.6d-159) then
        tmp = x / (y / x)
    else if (x <= 2.8d-115) then
        tmp = x
    else if (x <= 5.6d-88) then
        tmp = x * (x / y)
    else if (x <= 0.76d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.6e+41) {
		tmp = x / y;
	} else if (x <= 2.25e-178) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-159) {
		tmp = x / (y / x);
	} else if (x <= 2.8e-115) {
		tmp = x;
	} else if (x <= 5.6e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.76) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.6e+41:
		tmp = x / y
	elif x <= 2.25e-178:
		tmp = x / (x + 1.0)
	elif x <= 1.6e-159:
		tmp = x / (y / x)
	elif x <= 2.8e-115:
		tmp = x
	elif x <= 5.6e-88:
		tmp = x * (x / y)
	elif x <= 0.76:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.6e+41)
		tmp = Float64(x / y);
	elseif (x <= 2.25e-178)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.6e-159)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 2.8e-115)
		tmp = x;
	elseif (x <= 5.6e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.76)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.6e+41)
		tmp = x / y;
	elseif (x <= 2.25e-178)
		tmp = x / (x + 1.0);
	elseif (x <= 1.6e-159)
		tmp = x / (y / x);
	elseif (x <= 2.8e-115)
		tmp = x;
	elseif (x <= 5.6e-88)
		tmp = x * (x / y);
	elseif (x <= 0.76)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.6e+41], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.25e-178], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-159], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-115], x, If[LessEqual[x, 5.6e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.76], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -8.60000000000000048e41 or 0.76000000000000001 < x

   1. Initial program 74.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 80.7%

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

   if -8.60000000000000048e41 < x < 2.24999999999999989e-178

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 78.1%

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

   if 2.24999999999999989e-178 < x < 1.6e-159

   1. Initial program 99.3%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.3%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 86.2%

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

      1. associate-/l* 86.2%

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

   9. Simplified 86.2%

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

   10. Taylor expanded in x around 0 86.2%

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

   if 1.6e-159 < x < 2.79999999999999987e-115

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

   if 2.79999999999999987e-115 < x < 5.59999999999999952e-88

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 74.3%

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

      1. associate-/l* 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in x around 0 74.3%

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

       1. un-div-inv 74.3%

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

       2. associate-/r/ 74.3%

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

   12. Applied egg-rr 74.3%

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

   if 5.59999999999999952e-88 < x < 0.76000000000000001

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 84.2%

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

   6. Taylor expanded in x around 0 84.2%

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

      1. neg-mul-1 84.2%

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

      2. sub-neg 84.2%

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

   8. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (/ x y))))
   (if (<= x -8.6e+41)
     (/ x y)
     (if (<= x 2.25e-178)
       (/ x (+ x 1.0))
       (if (<= x 1.6e-159)
         t_0
         (if (<= x 1.65e-110)
           x
           (if (<= x 9.6e-89)
             t_0
             (if (<= x 0.76) (* x (- 1.0 x)) (/ x y)))))))))

C

double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -8.6e+41) {
		tmp = x / y;
	} else if (x <= 2.25e-178) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-159) {
		tmp = t_0;
	} else if (x <= 1.65e-110) {
		tmp = x;
	} else if (x <= 9.6e-89) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / 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)
    if (x <= (-8.6d+41)) then
        tmp = x / y
    else if (x <= 2.25d-178) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.6d-159) then
        tmp = t_0
    else if (x <= 1.65d-110) then
        tmp = x
    else if (x <= 9.6d-89) then
        tmp = t_0
    else if (x <= 0.76d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x / y);
	double tmp;
	if (x <= -8.6e+41) {
		tmp = x / y;
	} else if (x <= 2.25e-178) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-159) {
		tmp = t_0;
	} else if (x <= 1.65e-110) {
		tmp = x;
	} else if (x <= 9.6e-89) {
		tmp = t_0;
	} else if (x <= 0.76) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x / y)
	tmp = 0
	if x <= -8.6e+41:
		tmp = x / y
	elif x <= 2.25e-178:
		tmp = x / (x + 1.0)
	elif x <= 1.6e-159:
		tmp = t_0
	elif x <= 1.65e-110:
		tmp = x
	elif x <= 9.6e-89:
		tmp = t_0
	elif x <= 0.76:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x / y))
	tmp = 0.0
	if (x <= -8.6e+41)
		tmp = Float64(x / y);
	elseif (x <= 2.25e-178)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.6e-159)
		tmp = t_0;
	elseif (x <= 1.65e-110)
		tmp = x;
	elseif (x <= 9.6e-89)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x / y);
	tmp = 0.0;
	if (x <= -8.6e+41)
		tmp = x / y;
	elseif (x <= 2.25e-178)
		tmp = x / (x + 1.0);
	elseif (x <= 1.6e-159)
		tmp = t_0;
	elseif (x <= 1.65e-110)
		tmp = x;
	elseif (x <= 9.6e-89)
		tmp = t_0;
	elseif (x <= 0.76)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.6e+41], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.25e-178], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-159], t$95$0, If[LessEqual[x, 1.65e-110], x, If[LessEqual[x, 9.6e-89], t$95$0, If[LessEqual[x, 0.76], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.60000000000000048e41 or 0.76000000000000001 < x

   1. Initial program 74.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 80.7%

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

   if -8.60000000000000048e41 < x < 2.24999999999999989e-178

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 78.1%

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

   if 2.24999999999999989e-178 < x < 1.6e-159 or 1.65e-110 < x < 9.60000000000000065e-89

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 79.8%

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

      1. associate-/l* 79.8%

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

   9. Simplified 79.8%

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

   10. Taylor expanded in x around 0 79.8%

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

       1. un-div-inv 79.8%

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

       2. associate-/r/ 79.6%

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

   12. Applied egg-rr 79.6%

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

   if 1.6e-159 < x < 1.65e-110

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

   if 9.60000000000000065e-89 < x < 0.76000000000000001

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 84.2%

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

   6. Taylor expanded in x around 0 84.2%

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

      1. neg-mul-1 84.2%

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

      2. sub-neg 84.2%

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

   8. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot \left(1 - x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-110}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.76:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-19}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+23)
   (/ x y)
   (if (<= x 1.95e-19) (+ x (* x (- (/ x y) x))) (/ x (+ y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+23) {
		tmp = x / y;
	} else if (x <= 1.95e-19) {
		tmp = x + (x * ((x / y) - x));
	} else {
		tmp = x / (y + (y / 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 <= (-5.6d+23)) then
        tmp = x / y
    else if (x <= 1.95d-19) then
        tmp = x + (x * ((x / y) - x))
    else
        tmp = x / (y + (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+23) {
		tmp = x / y;
	} else if (x <= 1.95e-19) {
		tmp = x + (x * ((x / y) - x));
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e+23:
		tmp = x / y
	elif x <= 1.95e-19:
		tmp = x + (x * ((x / y) - x))
	else:
		tmp = x / (y + (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+23)
		tmp = Float64(x / y);
	elseif (x <= 1.95e-19)
		tmp = Float64(x + Float64(x * Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / Float64(y + Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+23)
		tmp = x / y;
	elseif (x <= 1.95e-19)
		tmp = x + (x * ((x / y) - x));
	else
		tmp = x / (y + (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+23], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.95e-19], N[(x + N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6e23

   1. Initial program 74.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 75.0%

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

   if -5.6e23 < x < 1.94999999999999998e-19

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 94.8%

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

   6. Taylor expanded in y around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

      1. +-commutative 94.8%

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

      2. distribute-rgt-in 94.8%

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

      3. *-commutative 94.8%

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

      4. *-un-lft-identity 94.8%

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

   10. Applied egg-rr 94.8%

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

   if 1.94999999999999998e-19 < x

   1. Initial program 76.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 76.5%

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

      1. associate-/l* 80.6%

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

   9. Simplified 80.6%

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

   10. Taylor expanded in x around inf 80.6%

       \[\leadsto \frac{x}{\color{blue}{y + \frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-19}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+23)
   (/ x y)
   (if (<= x 1.95e-19) (* x (+ 1.0 (- (/ x y) x))) (/ x (+ y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+23) {
		tmp = x / y;
	} else if (x <= 1.95e-19) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / (y + (y / 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 <= (-5.6d+23)) then
        tmp = x / y
    else if (x <= 1.95d-19) then
        tmp = x * (1.0d0 + ((x / y) - x))
    else
        tmp = x / (y + (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+23) {
		tmp = x / y;
	} else if (x <= 1.95e-19) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e+23:
		tmp = x / y
	elif x <= 1.95e-19:
		tmp = x * (1.0 + ((x / y) - x))
	else:
		tmp = x / (y + (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+23)
		tmp = Float64(x / y);
	elseif (x <= 1.95e-19)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / Float64(y + Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+23)
		tmp = x / y;
	elseif (x <= 1.95e-19)
		tmp = x * (1.0 + ((x / y) - x));
	else
		tmp = x / (y + (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+23], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.95e-19], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6e23

   1. Initial program 74.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 75.0%

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

   if -5.6e23 < x < 1.94999999999999998e-19

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 94.8%

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

   6. Taylor expanded in y around inf 94.8%

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

      1. neg-mul-1 94.8%

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

      2. +-commutative 94.8%

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

      3. unsub-neg 94.8%

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

   8. Simplified 94.8%

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

   if 1.94999999999999998e-19 < x

   1. Initial program 76.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 76.5%

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

      1. associate-/l* 80.6%

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

   9. Simplified 80.6%

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

   10. Taylor expanded in x around inf 80.6%

       \[\leadsto \frac{x}{\color{blue}{y + \frac{y}{x}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 9.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -58000000000.0) (not (<= x 9.8e-11)))
   (/ x (+ y (/ y x)))
   (+ x (* x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 9.8e-11)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x + (x * (x / 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 <= (-58000000000.0d0)) .or. (.not. (x <= 9.8d-11))) then
        tmp = x / (y + (y / x))
    else
        tmp = x + (x * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 9.8e-11)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x + (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -58000000000.0) or not (x <= 9.8e-11):
		tmp = x / (y + (y / x))
	else:
		tmp = x + (x * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -58000000000.0) || !(x <= 9.8e-11))
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = Float64(x + Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -58000000000.0) || ~((x <= 9.8e-11)))
		tmp = x / (y + (y / x));
	else
		tmp = x + (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -58000000000.0], N[Not[LessEqual[x, 9.8e-11]], $MachinePrecision]], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e10 or 9.7999999999999998e-11 < x

   1. Initial program 75.8%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 71.3%

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

      1. associate-/l* 78.4%

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

   9. Simplified 78.4%

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

   10. Taylor expanded in x around inf 78.4%

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

   if -5.8e10 < x < 9.7999999999999998e-11

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 98.0%

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

   6. Taylor expanded in y around inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. +-commutative 98.1%

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

      3. unsub-neg 98.1%

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

   8. Simplified 98.1%

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

      1. +-commutative 98.1%

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

      2. distribute-rgt-in 98.1%

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

      3. *-commutative 98.1%

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

      4. *-un-lft-identity 98.1%

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

   10. Applied egg-rr 98.1%

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

   11. Taylor expanded in y around 0 98.0%

       \[\leadsto x \cdot \color{blue}{\frac{x}{y}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 9.8 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 54\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 54.0))) (/ x y) (+ x (* x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 54.0)) {
		tmp = x / y;
	} else {
		tmp = x + (x * (x / 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)) .or. (.not. (x <= 54.0d0))) then
        tmp = x / y
    else
        tmp = x + (x * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 54.0)) {
		tmp = x / y;
	} else {
		tmp = x + (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 54.0):
		tmp = x / y
	else:
		tmp = x + (x * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 54.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(x + Float64(x * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 54.0)))
		tmp = x / y;
	else
		tmp = x + (x * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 54.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 54 < x

   1. Initial program 76.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 77.8%

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

   if -1 < x < 54

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in y around inf 99.7%

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

      1. neg-mul-1 99.7%

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

      2. +-commutative 99.7%

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

      3. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-rgt-in 99.7%

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

      3. *-commutative 99.7%

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

      4. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 99.4%

       \[\leadsto x \cdot \color{blue}{\frac{x}{y}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 54\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 13.0))) (/ x y) (* x (+ 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 13.0)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 + (x / 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)) .or. (.not. (x <= 13.0d0))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 13.0)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 13.0):
		tmp = x / y
	else:
		tmp = x * (1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 13.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 13.0)))
		tmp = x / y;
	else
		tmp = x * (1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 13.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 13 < x

   1. Initial program 76.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 77.8%

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

   if -1 < x < 13

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.7%

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

   6. Taylor expanded in y around inf 99.7%

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

      1. neg-mul-1 99.7%

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

      2. +-commutative 99.7%

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

      3. unsub-neg 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 99.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 13\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+27} \lor \neg \left(x \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7.6e+27) (not (<= x 8.5e-17))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7.6e+27) || !(x <= 8.5e-17)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - 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 <= (-7.6d+27)) .or. (.not. (x <= 8.5d-17))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -7.6e+27) || !(x <= 8.5e-17)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7.6e+27) or not (x <= 8.5e-17):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7.6e+27) || !(x <= 8.5e-17))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -7.6e+27) || ~((x <= 8.5e-17)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7.6e+27], N[Not[LessEqual[x, 8.5e-17]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.60000000000000043e27 or 8.5e-17 < x

   1. Initial program 74.7%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.8%

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

   if -7.60000000000000043e27 < x < 8.5e-17

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 74.9%

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

   6. Taylor expanded in x around 0 70.9%

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

      1. neg-mul-1 70.9%

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

      2. sub-neg 70.9%

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

   8. Simplified 70.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+27} \lor \neg \left(x \leq 8.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 19\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -58000000000.0) (not (<= x 19.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 19.0)) {
		tmp = x / y;
	} else {
		tmp = 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 <= (-58000000000.0d0)) .or. (.not. (x <= 19.0d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 19.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -58000000000.0) or not (x <= 19.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -58000000000.0) || !(x <= 19.0))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -58000000000.0) || ~((x <= 19.0)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -58000000000.0], N[Not[LessEqual[x, 19.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e10 or 19 < x

   1. Initial program 75.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 78.9%

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

   if -5.8e10 < x < 19

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.1%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 19\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6600000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6600000.0) 1.0 (if (<= x 8.5e-17) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6600000.0) {
		tmp = 1.0;
	} else if (x <= 8.5e-17) {
		tmp = x;
	} else {
		tmp = 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 <= (-6600000.0d0)) then
        tmp = 1.0d0
    else if (x <= 8.5d-17) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6600000.0) {
		tmp = 1.0;
	} else if (x <= 8.5e-17) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6600000.0:
		tmp = 1.0
	elif x <= 8.5e-17:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6600000.0)
		tmp = 1.0;
	elseif (x <= 8.5e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6600000.0)
		tmp = 1.0;
	elseif (x <= 8.5e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6600000.0], 1.0, If[LessEqual[x, 8.5e-17], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -6.6e6 or 8.5e-17 < x

   1. Initial program 76.1%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 24.3%

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

      1. div-inv 24.3%

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

      2. +-commutative 24.3%

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

   7. Applied egg-rr 24.3%

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

   8. Taylor expanded in x around inf 22.9%

      \[\leadsto \color{blue}{1} \]

   if -6.6e6 < x < 8.5e-17

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.6%

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

Alternative 13: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (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)) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto x \cdot \frac{1 + \frac{x}{y}}{x + 1} \]
6. Add Preprocessing

Alternative 14: 14.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 86.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 47.9%

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

   1. div-inv 48.0%

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

   2. +-commutative 48.0%

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

7. Applied egg-rr 48.0%

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

8. Taylor expanded in x around inf 14.4%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer target: 99.9% 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 2024107 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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