Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.5% → 99.9%

Time: 7.3s

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: 87.5% 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 87.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 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: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+41)
   (/ x y)
   (if (<= x -1.02e-18)
     (* x (/ 1.0 (+ x 1.0)))
     (if (<= x -5.6e-88)
       (* x (/ x y))
       (if (<= x 3.9e+51) (/ x (+ x 1.0)) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+41) {
		tmp = x / y;
	} else if (x <= -1.02e-18) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= -5.6e-88) {
		tmp = x * (x / y);
	} else if (x <= 3.9e+51) {
		tmp = x / (x + 1.0);
	} 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) :: tmp
    if (x <= (-3.5d+41)) then
        tmp = x / y
    else if (x <= (-1.02d-18)) then
        tmp = x * (1.0d0 / (x + 1.0d0))
    else if (x <= (-5.6d-88)) then
        tmp = x * (x / y)
    else if (x <= 3.9d+51) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+41) {
		tmp = x / y;
	} else if (x <= -1.02e-18) {
		tmp = x * (1.0 / (x + 1.0));
	} else if (x <= -5.6e-88) {
		tmp = x * (x / y);
	} else if (x <= 3.9e+51) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e+41:
		tmp = x / y
	elif x <= -1.02e-18:
		tmp = x * (1.0 / (x + 1.0))
	elif x <= -5.6e-88:
		tmp = x * (x / y)
	elif x <= 3.9e+51:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+41)
		tmp = Float64(x / y);
	elseif (x <= -1.02e-18)
		tmp = Float64(x * Float64(1.0 / Float64(x + 1.0)));
	elseif (x <= -5.6e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 3.9e+51)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e+41)
		tmp = x / y;
	elseif (x <= -1.02e-18)
		tmp = x * (1.0 / (x + 1.0));
	elseif (x <= -5.6e-88)
		tmp = x * (x / y);
	elseif (x <= 3.9e+51)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e+41], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.02e-18], N[(x * N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.6e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+51], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.4999999999999999e41 or 3.89999999999999984e51 < x

   1. Initial program 68.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 81.7%

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

   if -3.4999999999999999e41 < x < -1.02e-18

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around inf 72.4%

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

   if -1.02e-18 < x < -5.59999999999999952e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 75.6%

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

   6. Taylor expanded in x around 0 75.6%

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

   if -5.59999999999999952e-88 < x < 3.89999999999999984e51

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 79.4%

      \[\leadsto \color{blue}{\frac{x}{1 + x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.56 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -3.6e+41)
     (/ x y)
     (if (<= x -1.56e-21)
       t_0
       (if (<= x -1.85e-88) (* x (/ x y)) (if (<= x 3e+51) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -3.6e+41) {
		tmp = x / y;
	} else if (x <= -1.56e-21) {
		tmp = t_0;
	} else if (x <= -1.85e-88) {
		tmp = x * (x / y);
	} else if (x <= 3e+51) {
		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 <= (-3.6d+41)) then
        tmp = x / y
    else if (x <= (-1.56d-21)) then
        tmp = t_0
    else if (x <= (-1.85d-88)) then
        tmp = x * (x / y)
    else if (x <= 3d+51) 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 <= -3.6e+41) {
		tmp = x / y;
	} else if (x <= -1.56e-21) {
		tmp = t_0;
	} else if (x <= -1.85e-88) {
		tmp = x * (x / y);
	} else if (x <= 3e+51) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -3.6e+41:
		tmp = x / y
	elif x <= -1.56e-21:
		tmp = t_0
	elif x <= -1.85e-88:
		tmp = x * (x / y)
	elif x <= 3e+51:
		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 <= -3.6e+41)
		tmp = Float64(x / y);
	elseif (x <= -1.56e-21)
		tmp = t_0;
	elseif (x <= -1.85e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 3e+51)
		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 <= -3.6e+41)
		tmp = x / y;
	elseif (x <= -1.56e-21)
		tmp = t_0;
	elseif (x <= -1.85e-88)
		tmp = x * (x / y);
	elseif (x <= 3e+51)
		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, -3.6e+41], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.56e-21], t$95$0, If[LessEqual[x, -1.85e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+51], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.60000000000000025e41 or 3e51 < x

   1. Initial program 68.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 81.7%

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

   if -3.60000000000000025e41 < x < -1.55999999999999999e-21 or -1.8499999999999999e-88 < x < 3e51

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 78.7%

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

   if -1.55999999999999999e-21 < x < -1.8499999999999999e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 75.6%

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

   6. Taylor expanded in x around 0 75.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.56 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x -5.2e-88)
     (* x (/ x y))
     (if (<= x 0.88) (* x (- 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -5.2e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.88) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= (-5.2d-88)) then
        tmp = x * (x / y)
    else if (x <= 0.88d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= -5.2e-88) {
		tmp = x * (x / y);
	} else if (x <= 0.88) {
		tmp = x * (1.0 - x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= -5.2e-88:
		tmp = x * (x / y)
	elif x <= 0.88:
		tmp = x * (1.0 - x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= -5.2e-88)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 0.88)
		tmp = Float64(x * Float64(1.0 - x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= -5.2e-88)
		tmp = x * (x / y);
	elseif (x <= 0.88)
		tmp = x * (1.0 - x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.88], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.880000000000000004 < x

   1. Initial program 73.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 73.6%

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

   if -1 < x < -5.20000000000000027e-88

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 67.6%

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

   6. Taylor expanded in x around 0 67.4%

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

   if -5.20000000000000027e-88 < x < 0.880000000000000004

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 81.3%

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

   6. Taylor expanded in x around 0 80.6%

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

      1. neg-mul-1 80.6%

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

      2. sub-neg 80.6%

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

   8. Simplified 80.6%

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

Alternative 5: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-30} \lor \neg \left(y \leq 5.8 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.6e-30) (not (<= y 5.8e-24)))
   (/ x (+ x 1.0))
   (* x (/ (/ x y) (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e-30) || !(y <= 5.8e-24)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x * ((x / y) / (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 ((y <= (-4.6d-30)) .or. (.not. (y <= 5.8d-24))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x * ((x / y) / (x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e-30) || !(y <= 5.8e-24)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x * ((x / y) / (x + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.6e-30) or not (y <= 5.8e-24):
		tmp = x / (x + 1.0)
	else:
		tmp = x * ((x / y) / (x + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.6e-30) || !(y <= 5.8e-24))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(Float64(x / y) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.6e-30) || ~((y <= 5.8e-24)))
		tmp = x / (x + 1.0);
	else
		tmp = x * ((x / y) / (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.6e-30], N[Not[LessEqual[y, 5.8e-24]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x / y), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999968e-30 or 5.7999999999999997e-24 < y

   1. Initial program 86.7%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 75.5%

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

   if -4.59999999999999968e-30 < y < 5.7999999999999997e-24

   1. Initial program 89.1%

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

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 80.5%

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

      1. associate-/l* 80.5%

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

   9. Simplified 80.5%

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

       1. associate-/r* 80.5%

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

       2. associate-/r/ 80.5%

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

   11. Applied egg-rr 80.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-30} \lor \neg \left(y \leq 5.8 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{x}{y}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-30} \lor \neg \left(y \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e-30) (not (<= y 5.2e-24)))
   (/ x (+ x 1.0))
   (/ x (+ y (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e-30) || !(y <= 5.2e-24)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4d-30)) .or. (.not. (y <= 5.2d-24))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / (y + (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e-30) || !(y <= 5.2e-24)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e-30) or not (y <= 5.2e-24):
		tmp = x / (x + 1.0)
	else:
		tmp = x / (y + (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e-30) || !(y <= 5.2e-24))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(y + Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e-30) || ~((y <= 5.2e-24)))
		tmp = x / (x + 1.0);
	else
		tmp = x / (y + (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e-30], N[Not[LessEqual[y, 5.2e-24]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4e-30 or 5.2e-24 < y

   1. Initial program 86.7%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 75.5%

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

   if -4e-30 < y < 5.2e-24

   1. Initial program 89.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 0 80.4%

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

   6. Taylor expanded in y around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. unpow2 68.5%

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

      4. associate-*r/ 80.4%

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

      5. associate-/l/ 80.5%

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

      6. *-commutative 80.5%

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

      7. associate-/l/ 80.4%

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

      8. associate-/r/ 80.5%

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

      9. associate-*r/ 80.5%

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

      10. *-commutative 80.5%

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

      11. distribute-rgt-in 75.2%

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

      12. *-lft-identity 75.2%

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

      13. associate-*l/ 75.1%

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

      14. associate-*r* 80.4%

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

      15. rgt-mult-inverse 80.5%

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

      16. *-lft-identity 80.5%

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

      17. *-lft-identity 80.5%

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

   8. Simplified 80.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-30} \lor \neg \left(y \leq 5.2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.82))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.82)) {
		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 <= 0.82d0))) 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 <= 0.82)) {
		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 <= 0.82):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.82))
		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 <= 0.82)))
		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, 0.82]], $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 0.819999999999999951 < x

   1. Initial program 73.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 73.6%

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

   if -1 < x < 0.819999999999999951

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 73.3%

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

   6. Taylor expanded in x around 0 72.7%

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

      1. neg-mul-1 72.7%

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

      2. sub-neg 72.7%

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

   8. Simplified 72.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.82\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.9e-5))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.9e-5)) {
		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 <= 1.9d-5))) 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 <= 1.9e-5)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.9e-5):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.9e-5))
		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 <= 1.9e-5)))
		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, 1.9e-5]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.9000000000000001e-5 < x

   1. Initial program 74.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 72.0%

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

   if -1 < x < 1.9000000000000001e-5

   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 73.5%

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

4. Final simplification 72.8%

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

Alternative 9: 48.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in y around inf 26.9%

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

      1. un-div-inv 27.0%

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

      2. +-commutative 27.0%

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

   7. Applied egg-rr 27.0%

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

   8. Taylor expanded in x around inf 26.6%

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

   if -1 < x < 1

   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 72.1%

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

Alternative 10: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

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

Alternative 11: 14.1% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 87.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. Taylor expanded in y around inf 51.8%

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

   1. un-div-inv 51.8%

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

   2. +-commutative 51.8%

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

7. Applied egg-rr 51.8%

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

8. Taylor expanded in x around inf 14.5%

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

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(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)))