Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.6% → 99.9%

Time: 10.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t\_0 + t\_0 \cdot \frac{x}{y} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (+ t_0 (* t_0 (/ x y)))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (x + 1.0d0)
    code = t_0 + (t_0 * (x / y))
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	return t_0 + (t_0 * (x / y))

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	return Float64(t_0 + Float64(t_0 * Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = t_0 + (t_0 * (x / y));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 + N[(t$95$0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 99.7%

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

   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. Step-by-step derivation

   1. associate-/r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. distribute-lft-in 99.9%

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

   5. *-commutative 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. +-commutative 99.9%

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

   8. +-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.39:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 x))))
   (if (<= x -1.0)
     (/ x y)
     (if (<= x 3.5e-186)
       t_0
       (if (<= x 1.9e-150) (* x (/ x y)) (if (<= x 0.39) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3.5e-186) {
		tmp = t_0;
	} else if (x <= 1.9e-150) {
		tmp = x * (x / y);
	} else if (x <= 0.39) {
		tmp = t_0;
	} 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 * (1.0d0 - x)
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 3.5d-186) then
        tmp = t_0
    else if (x <= 1.9d-150) then
        tmp = x * (x / y)
    else if (x <= 0.39d0) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (1.0 - x);
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3.5e-186) {
		tmp = t_0;
	} else if (x <= 1.9e-150) {
		tmp = x * (x / y);
	} else if (x <= 0.39) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (1.0 - x)
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 3.5e-186:
		tmp = t_0
	elif x <= 1.9e-150:
		tmp = x * (x / y)
	elif x <= 0.39:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(1.0 - x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 3.5e-186)
		tmp = t_0;
	elseif (x <= 1.9e-150)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.39)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (1.0 - x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 3.5e-186)
		tmp = t_0;
	elseif (x <= 1.9e-150)
		tmp = x * (x / y);
	elseif (x <= 0.39)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.5e-186], t$95$0, If[LessEqual[x, 1.9e-150], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.39], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.39000000000000001 < 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* 99.8%

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

   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 78.2%

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

   if -1 < x < 3.49999999999999989e-186 or 1.8999999999999999e-150 < x < 0.39000000000000001

   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. Taylor expanded in y around inf 73.2%

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

   6. Taylor expanded in x around 0 73.0%

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

      1. neg-mul-1 73.0%

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

      2. sub-neg 73.0%

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

   8. Simplified 73.0%

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

   if 3.49999999999999989e-186 < x < 1.8999999999999999e-150

   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 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 83.3%

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

   8. Taylor expanded in x around 0 83.3%

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

      1. associate-/r/ 83.5%

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

   10. Applied egg-rr 83.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.39:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1900000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1900000000.0)
     (/ x y)
     (if (<= x 3.5e-186)
       t_0
       (if (<= x 1.9e-150) (* x (/ x y)) (if (<= x 5.8e+16) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1900000000.0) {
		tmp = x / y;
	} else if (x <= 3.5e-186) {
		tmp = t_0;
	} else if (x <= 1.9e-150) {
		tmp = x * (x / y);
	} else if (x <= 5.8e+16) {
		tmp = t_0;
	} 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 + 1.0d0)
    if (x <= (-1900000000.0d0)) then
        tmp = x / y
    else if (x <= 3.5d-186) then
        tmp = t_0
    else if (x <= 1.9d-150) then
        tmp = x * (x / y)
    else if (x <= 5.8d+16) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1900000000.0) {
		tmp = x / y;
	} else if (x <= 3.5e-186) {
		tmp = t_0;
	} else if (x <= 1.9e-150) {
		tmp = x * (x / y);
	} else if (x <= 5.8e+16) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1900000000.0:
		tmp = x / y
	elif x <= 3.5e-186:
		tmp = t_0
	elif x <= 1.9e-150:
		tmp = x * (x / y)
	elif x <= 5.8e+16:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1900000000.0)
		tmp = Float64(x / y);
	elseif (x <= 3.5e-186)
		tmp = t_0;
	elseif (x <= 1.9e-150)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 5.8e+16)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1900000000.0)
		tmp = x / y;
	elseif (x <= 3.5e-186)
		tmp = t_0;
	elseif (x <= 1.9e-150)
		tmp = x * (x / y);
	elseif (x <= 5.8e+16)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1900000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 3.5e-186], t$95$0, If[LessEqual[x, 1.9e-150], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+16], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9e9 or 5.8e16 < 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* 99.8%

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

   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 82.2%

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

   if -1.9e9 < x < 3.49999999999999989e-186 or 1.8999999999999999e-150 < x < 5.8e16

   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. Taylor expanded in y around inf 73.1%

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

   if 3.49999999999999989e-186 < x < 1.8999999999999999e-150

   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 99.4%

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

      2. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in y around 0 83.3%

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

   8. Taylor expanded in x around 0 83.3%

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

      1. associate-/r/ 83.5%

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

   10. Applied egg-rr 83.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1900000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -1.06 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -1.06e-40)
     t_0
     (if (<= y 1.38e-52)
       (/ x (+ y (/ y x)))
       (if (<= y 7e-19) x (if (<= y 1.22e+36) (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -1.06e-40) {
		tmp = t_0;
	} else if (y <= 1.38e-52) {
		tmp = x / (y + (y / x));
	} else if (y <= 7e-19) {
		tmp = x;
	} else if (y <= 1.22e+36) {
		tmp = x / 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) :: tmp
    t_0 = x / (x + 1.0d0)
    if (y <= (-1.06d-40)) then
        tmp = t_0
    else if (y <= 1.38d-52) then
        tmp = x / (y + (y / x))
    else if (y <= 7d-19) then
        tmp = x
    else if (y <= 1.22d+36) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -1.06e-40) {
		tmp = t_0;
	} else if (y <= 1.38e-52) {
		tmp = x / (y + (y / x));
	} else if (y <= 7e-19) {
		tmp = x;
	} else if (y <= 1.22e+36) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -1.06e-40:
		tmp = t_0
	elif y <= 1.38e-52:
		tmp = x / (y + (y / x))
	elif y <= 7e-19:
		tmp = x
	elif y <= 1.22e+36:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (y <= -1.06e-40)
		tmp = t_0;
	elseif (y <= 1.38e-52)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (y <= 7e-19)
		tmp = x;
	elseif (y <= 1.22e+36)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (y <= -1.06e-40)
		tmp = t_0;
	elseif (y <= 1.38e-52)
		tmp = x / (y + (y / x));
	elseif (y <= 7e-19)
		tmp = x;
	elseif (y <= 1.22e+36)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.06e-40], t$95$0, If[LessEqual[y, 1.38e-52], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e-19], x, If[LessEqual[y, 1.22e+36], N[(x / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.06e-40 or 1.21999999999999995e36 < y

   1. Initial program 93.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.8%

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

   if -1.06e-40 < y < 1.38000000000000008e-52

   1. Initial program 90.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 83.5%

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

   8. Taylor expanded in x around inf 83.5%

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

   if 1.38000000000000008e-52 < y < 7.00000000000000031e-19

   1. Initial program 82.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.8%

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

   if 7.00000000000000031e-19 < y < 1.21999999999999995e36

   1. Initial program 48.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 x around inf 75.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-19}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.3 \cdot 10^{-7} \lor \neg \left(x \leq 2.8 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.3e-7) (not (<= x 2.8e-34)))
   (+ (/ x (+ x 1.0)) (/ x y))
   (* x (+ 1.0 (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.3e-7) || !(x <= 2.8e-34)) {
		tmp = (x / (x + 1.0)) + (x / y);
	} else {
		tmp = x * (1.0 + ((x / 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 <= (-9.3d-7)) .or. (.not. (x <= 2.8d-34))) then
        tmp = (x / (x + 1.0d0)) + (x / y)
    else
        tmp = x * (1.0d0 + ((x / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -9.3e-7) || !(x <= 2.8e-34)) {
		tmp = (x / (x + 1.0)) + (x / y);
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.3e-7) or not (x <= 2.8e-34):
		tmp = (x / (x + 1.0)) + (x / y)
	else:
		tmp = x * (1.0 + ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.3e-7) || !(x <= 2.8e-34))
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(x / y));
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -9.3e-7) || ~((x <= 2.8e-34)))
		tmp = (x / (x + 1.0)) + (x / y);
	else
		tmp = x * (1.0 + ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.3e-7], N[Not[LessEqual[x, 2.8e-34]], $MachinePrecision]], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2999999999999999e-7 or 2.79999999999999997e-34 < x

   1. Initial program 77.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-commutative 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 98.8%

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

   if -9.2999999999999999e-7 < x < 2.79999999999999997e-34

   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. Taylor expanded in x around 0 99.8%

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

   6. Taylor expanded in y around inf 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. unsub-neg 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.3 \cdot 10^{-7} \lor \neg \left(x \leq 2.8 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.8e-5)
   (/ x (+ y (/ y x)))
   (if (<= x 1.0) (* x (+ 1.0 (- (/ x y) x))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.8e-5) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - 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 <= (-9.8d-5)) then
        tmp = x / (y + (y / x))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x / y) - x))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.8e-5) {
		tmp = x / (y + (y / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.8e-5:
		tmp = x / (y + (y / x))
	elif x <= 1.0:
		tmp = x * (1.0 + ((x / y) - x))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.8e-5)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.8e-5)
		tmp = x / (y + (y / x));
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x / y) - x));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.8e-5], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8e-5

   1. Initial program 81.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 75.8%

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

   8. Taylor expanded in x around inf 80.3%

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

   if -9.8e-5 < x < 1

   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. Taylor expanded in x around 0 99.6%

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

   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%

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

   if 1 < x

   1. Initial program 71.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 78.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{x + 1} \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e-6)
   (* (/ x (+ x 1.0)) (/ x y))
   (if (<= x 1.0) (* x (+ 1.0 (- (/ x y) x))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-6) {
		tmp = (x / (x + 1.0)) * (x / y);
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - 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 <= (-2.9d-6)) then
        tmp = (x / (x + 1.0d0)) * (x / y)
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x / y) - x))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-6) {
		tmp = (x / (x + 1.0)) * (x / y);
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x / y) - x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e-6:
		tmp = (x / (x + 1.0)) * (x / y)
	elif x <= 1.0:
		tmp = x * (1.0 + ((x / y) - x))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e-6)
		tmp = Float64(Float64(x / Float64(x + 1.0)) * Float64(x / y));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e-6)
		tmp = (x / (x + 1.0)) * (x / y);
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x / y) - x));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e-6], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9000000000000002e-6

   1. Initial program 81.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 75.8%

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

      1. associate-/r/ 75.7%

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

      2. associate-*l/ 67.2%

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

      3. +-commutative 67.2%

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

      4. *-commutative 67.2%

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

      5. frac-times 80.3%

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

   9. Applied egg-rr 80.3%

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

   if -2.9000000000000002e-6 < x < 1

   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. Taylor expanded in x around 0 99.6%

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

   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%

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

   if 1 < x

   1. Initial program 71.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 78.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{x + 1} \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.8))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		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 <= (-1.0d0)) .or. (.not. (x <= 0.8d0))) 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 <= -1.0) || !(x <= 0.8)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.8):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		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 <= -1.0) || ~((x <= 0.8)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.8]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < 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* 99.8%

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

   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 78.2%

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

   if -1 < x < 0.80000000000000004

   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. Taylor expanded in y around inf 70.1%

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

   6. Taylor expanded in x around 0 70.0%

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

      1. neg-mul-1 70.0%

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

      2. sub-neg 70.0%

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

   8. Simplified 70.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 5.8 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 5.8e+16))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 5.8e+16)) {
		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 <= 5.8d+16))) 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 <= 5.8e+16)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 5.8e+16):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 5.8e+16))
		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 <= 5.8e+16)))
		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, 5.8e+16]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 5.8e16 < 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* 99.8%

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

   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 79.5%

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

   if -1 < x < 5.8e16

   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. Taylor expanded in x around 0 68.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 5.8 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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 89.3%

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

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

Alternative 11: 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 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 99.7%

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

   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 12: 39.1% 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 89.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 0 40.1%

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

6. Final simplification 40.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 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 2024115 
(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)))