Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.9%

Time: 9.2s

Alternatives: 11

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: 88.1% 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 87.5%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 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: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;x \leq -650:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= x -650.0)
     t_0
     (if (<= x 2.15e-97)
       (/ x (+ x 1.0))
       (if (<= x 13200000000000.0) (/ (/ x (/ y x)) (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -650.0) {
		tmp = t_0;
	} else if (x <= 2.15e-97) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = (x / (y / x)) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	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 + y) / y
    if (x <= (-650.0d0)) then
        tmp = t_0
    else if (x <= 2.15d-97) then
        tmp = x / (x + 1.0d0)
    else if (x <= 13200000000000.0d0) then
        tmp = (x / (y / x)) / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -650.0) {
		tmp = t_0;
	} else if (x <= 2.15e-97) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = (x / (y / x)) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if x <= -650.0:
		tmp = t_0
	elif x <= 2.15e-97:
		tmp = x / (x + 1.0)
	elif x <= 13200000000000.0:
		tmp = (x / (y / x)) / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (x <= -650.0)
		tmp = t_0;
	elseif (x <= 2.15e-97)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 13200000000000.0)
		tmp = Float64(Float64(x / Float64(y / x)) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (x <= -650.0)
		tmp = t_0;
	elseif (x <= 2.15e-97)
		tmp = x / (x + 1.0);
	elseif (x <= 13200000000000.0)
		tmp = (x / (y / x)) / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -650.0], t$95$0, If[LessEqual[x, 2.15e-97], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13200000000000.0], N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -650 or 1.32e13 < x

   1. Initial program 72.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8%

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

      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 x around inf 99.4%

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

   8. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -650 < x < 2.15e-97

   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 80.1%

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

   if 2.15e-97 < x < 1.32e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 65.5%

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

      1. clear-num 65.3%

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

      2. un-div-inv 65.5%

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

   5. Applied egg-rr 65.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -650:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;x \leq -920:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= x -920.0)
     t_0
     (if (<= x 3.1e-96)
       (/ x (+ x 1.0))
       (if (<= x 13200000000000.0) (/ (* x (/ x y)) (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -920.0) {
		tmp = t_0;
	} else if (x <= 3.1e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	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 + y) / y
    if (x <= (-920.0d0)) then
        tmp = t_0
    else if (x <= 3.1d-96) then
        tmp = x / (x + 1.0d0)
    else if (x <= 13200000000000.0d0) then
        tmp = (x * (x / y)) / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -920.0) {
		tmp = t_0;
	} else if (x <= 3.1e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = (x * (x / y)) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if x <= -920.0:
		tmp = t_0
	elif x <= 3.1e-96:
		tmp = x / (x + 1.0)
	elif x <= 13200000000000.0:
		tmp = (x * (x / y)) / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (x <= -920.0)
		tmp = t_0;
	elseif (x <= 3.1e-96)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 13200000000000.0)
		tmp = Float64(Float64(x * Float64(x / y)) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (x <= -920.0)
		tmp = t_0;
	elseif (x <= 3.1e-96)
		tmp = x / (x + 1.0);
	elseif (x <= 13200000000000.0)
		tmp = (x * (x / y)) / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -920.0], t$95$0, If[LessEqual[x, 3.1e-96], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13200000000000.0], N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -920 or 1.32e13 < x

   1. Initial program 72.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8%

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

      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 x around inf 99.4%

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

   8. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -920 < x < 3.0999999999999999e-96

   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 80.1%

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

   if 3.0999999999999999e-96 < x < 1.32e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 65.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -920:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ \mathbf{if}\;x \leq -180:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)))
   (if (<= x -180.0)
     t_0
     (if (<= x 2e-96)
       (/ x (+ x 1.0))
       (if (<= x 13200000000000.0) (/ x (* y (+ 1.0 (/ 1.0 x)))) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -180.0) {
		tmp = t_0;
	} else if (x <= 2e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = x / (y * (1.0 + (1.0 / x)));
	} else {
		tmp = t_0;
	}
	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 + y) / y
    if (x <= (-180.0d0)) then
        tmp = t_0
    else if (x <= 2d-96) then
        tmp = x / (x + 1.0d0)
    else if (x <= 13200000000000.0d0) then
        tmp = x / (y * (1.0d0 + (1.0d0 / x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double tmp;
	if (x <= -180.0) {
		tmp = t_0;
	} else if (x <= 2e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 13200000000000.0) {
		tmp = x / (y * (1.0 + (1.0 / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	tmp = 0
	if x <= -180.0:
		tmp = t_0
	elif x <= 2e-96:
		tmp = x / (x + 1.0)
	elif x <= 13200000000000.0:
		tmp = x / (y * (1.0 + (1.0 / x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	tmp = 0.0
	if (x <= -180.0)
		tmp = t_0;
	elseif (x <= 2e-96)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 13200000000000.0)
		tmp = Float64(x / Float64(y * Float64(1.0 + Float64(1.0 / x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	tmp = 0.0;
	if (x <= -180.0)
		tmp = t_0;
	elseif (x <= 2e-96)
		tmp = x / (x + 1.0);
	elseif (x <= 13200000000000.0)
		tmp = x / (y * (1.0 + (1.0 / x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -180.0], t$95$0, If[LessEqual[x, 2e-96], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 13200000000000.0], N[(x / N[(y * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -180 or 1.32e13 < x

   1. Initial program 72.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8%

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

      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 x around inf 99.4%

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

   8. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -180 < x < 1.9999999999999998e-96

   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 80.1%

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

   if 1.9999999999999998e-96 < x < 1.32e13

   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. Taylor expanded in y around 0 99.6%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.7%

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

      3. div-inv 99.6%

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

      4. div-inv 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. +-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in y around 0 65.5%

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

      1. associate-/l* 65.4%

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

      2. *-lft-identity 65.4%

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

      3. associate-*l/ 65.4%

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

      4. +-commutative 65.4%

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

      5. distribute-rgt-in 65.4%

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

      6. rgt-mult-inverse 65.4%

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

      7. *-lft-identity 65.4%

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

   10. Simplified 65.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -60:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;x \cdot \frac{t\_1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) y)) (t_1 (/ x (+ x 1.0))))
   (if (<= x -60.0)
     t_0
     (if (<= x 1.7e-100)
       t_1
       (if (<= x 13200000000000.0) (* x (/ t_1 y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = (x + y) / y;
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -60.0) {
		tmp = t_0;
	} else if (x <= 1.7e-100) {
		tmp = t_1;
	} else if (x <= 13200000000000.0) {
		tmp = x * (t_1 / y);
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / y
    t_1 = x / (x + 1.0d0)
    if (x <= (-60.0d0)) then
        tmp = t_0
    else if (x <= 1.7d-100) then
        tmp = t_1
    else if (x <= 13200000000000.0d0) then
        tmp = x * (t_1 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / y;
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -60.0) {
		tmp = t_0;
	} else if (x <= 1.7e-100) {
		tmp = t_1;
	} else if (x <= 13200000000000.0) {
		tmp = x * (t_1 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / y
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -60.0:
		tmp = t_0
	elif x <= 1.7e-100:
		tmp = t_1
	elif x <= 13200000000000.0:
		tmp = x * (t_1 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / y)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -60.0)
		tmp = t_0;
	elseif (x <= 1.7e-100)
		tmp = t_1;
	elseif (x <= 13200000000000.0)
		tmp = Float64(x * Float64(t_1 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / y;
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -60.0)
		tmp = t_0;
	elseif (x <= 1.7e-100)
		tmp = t_1;
	elseif (x <= 13200000000000.0)
		tmp = x * (t_1 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -60.0], t$95$0, If[LessEqual[x, 1.7e-100], t$95$1, If[LessEqual[x, 13200000000000.0], N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -60 or 1.32e13 < x

   1. Initial program 72.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8%

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

      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 x around inf 99.4%

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

   8. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   if -60 < x < 1.69999999999999988e-100

   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 80.1%

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

   if 1.69999999999999988e-100 < x < 1.32e13

   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. Taylor expanded in y around 0 99.6%

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

   6. Taylor expanded in y around 0 65.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 13200000000000:\\ \;\;\;\;x \cdot \frac{\frac{x}{x + 1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.85)))
   (/ (+ x y) y)
   (* x (+ 1.0 (* x (+ (/ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = (x + y) / y;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -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 <= (-1.0d0)) .or. (.not. (x <= 0.85d0))) then
        tmp = (x + y) / y
    else
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = (x + y) / y;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.85):
		tmp = (x + y) / y
	else:
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.85))
		tmp = Float64(Float64(x + y) / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.85)))
		tmp = (x + y) / y;
	else
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 73.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 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 x around inf 97.8%

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

   8. Taylor expanded in y around 0 97.8%

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

      1. +-commutative 97.8%

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

   10. Simplified 97.8%

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

   if -1 < x < 0.849999999999999978

   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 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1020 \lor \neg \left(x \leq 85\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1020.0) (not (<= x 85.0))) (/ (+ x y) y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1020.0) || !(x <= 85.0)) {
		tmp = (x + y) / 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 <= (-1020.0d0)) .or. (.not. (x <= 85.0d0))) then
        tmp = (x + y) / 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 <= -1020.0) || !(x <= 85.0)) {
		tmp = (x + y) / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1020.0) or not (x <= 85.0):
		tmp = (x + y) / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1020.0) || !(x <= 85.0))
		tmp = Float64(Float64(x + y) / 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 <= -1020.0) || ~((x <= 85.0)))
		tmp = (x + y) / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1020.0], N[Not[LessEqual[x, 85.0]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1020 or 85 < x

   1. Initial program 73.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 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 x around inf 97.8%

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

   8. Taylor expanded in y around 0 97.8%

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

      1. +-commutative 97.8%

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

   10. Simplified 97.8%

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

   if -1020 < x < 85

   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 73.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1020 \lor \neg \left(x \leq 85\right):\\ \;\;\;\;\frac{x + y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1080 \lor \neg \left(x \leq 5.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1080.0) (not (<= x 5.5e+116))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1080.0) || !(x <= 5.5e+116)) {
		tmp = 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 <= (-1080.0d0)) .or. (.not. (x <= 5.5d+116))) then
        tmp = 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 <= -1080.0) || !(x <= 5.5e+116)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1080.0) or not (x <= 5.5e+116):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1080.0) || !(x <= 5.5e+116))
		tmp = 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 <= -1080.0) || ~((x <= 5.5e+116)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1080.0], N[Not[LessEqual[x, 5.5e+116]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1080 or 5.50000000000000035e116 < x

   1. Initial program 68.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 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 x around inf 86.9%

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

   if -1080 < x < 5.50000000000000035e116

   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 71.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1080 \lor \neg \left(x \leq 5.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 2.8))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.8)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 2.8d0))) 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 <= -1.0) || !(x <= 2.8)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 2.8):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 2.8))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 2.8)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.7999999999999998 < x

   1. Initial program 73.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 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 x around inf 78.7%

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

   if -1 < x < 2.7999999999999998

   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 71.9%

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

   6. Taylor expanded in x around 0 71.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.8% 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 87.5%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 11: 38.1% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 87.5%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 39.9%

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

6. Taylor expanded in x around 0 39.9%

   \[\leadsto \color{blue}{x} \]
7. 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 2024139 
(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)))