Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.8% → 99.9%

Time: 8.4s

Alternatives: 10

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: 87.8% 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 89.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{y} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -1.3e+79)
     t_0
     (if (<= y -5e+43)
       (/ x (+ y (/ y x)))
       (if (<= y -6e+22) x (if (<= y 3.5e-74) (* (/ x y) t_0) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -1.3e+79) {
		tmp = t_0;
	} else if (y <= -5e+43) {
		tmp = x / (y + (y / x));
	} else if (y <= -6e+22) {
		tmp = x;
	} else if (y <= 3.5e-74) {
		tmp = (x / y) * t_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 / (x + 1.0d0)
    if (y <= (-1.3d+79)) then
        tmp = t_0
    else if (y <= (-5d+43)) then
        tmp = x / (y + (y / x))
    else if (y <= (-6d+22)) then
        tmp = x
    else if (y <= 3.5d-74) then
        tmp = (x / y) * t_0
    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 <= -1.3e+79) {
		tmp = t_0;
	} else if (y <= -5e+43) {
		tmp = x / (y + (y / x));
	} else if (y <= -6e+22) {
		tmp = x;
	} else if (y <= 3.5e-74) {
		tmp = (x / y) * t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -1.3e+79:
		tmp = t_0
	elif y <= -5e+43:
		tmp = x / (y + (y / x))
	elif y <= -6e+22:
		tmp = x
	elif y <= 3.5e-74:
		tmp = (x / y) * t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (y <= -1.3e+79)
		tmp = t_0;
	elseif (y <= -5e+43)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (y <= -6e+22)
		tmp = x;
	elseif (y <= 3.5e-74)
		tmp = Float64(Float64(x / y) * t_0);
	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 <= -1.3e+79)
		tmp = t_0;
	elseif (y <= -5e+43)
		tmp = x / (y + (y / x));
	elseif (y <= -6e+22)
		tmp = x;
	elseif (y <= 3.5e-74)
		tmp = (x / y) * t_0;
	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, -1.3e+79], t$95$0, If[LessEqual[y, -5e+43], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -6e+22], x, If[LessEqual[y, 3.5e-74], N[(N[(x / y), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.30000000000000007e79 or 3.50000000000000015e-74 < y

   1. Initial program 93.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 80.2%

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

   if -1.30000000000000007e79 < y < -5.0000000000000004e43

   1. Initial program 57.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 59.4%

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

      1. clear-num 59.6%

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

      2. un-div-inv 60.0%

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

      3. associate-/l* 71.0%

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

      4. +-commutative 71.0%

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

   7. Applied egg-rr 71.0%

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

   8. Taylor expanded in x around inf 71.0%

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

   if -5.0000000000000004e43 < y < -6e22

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if -6e22 < y < 3.50000000000000015e-74

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 80.6%

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

      1. clear-num 80.6%

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

      2. un-div-inv 80.7%

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

      3. associate-/l* 80.6%

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

      4. +-commutative 80.6%

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

   7. Applied egg-rr 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. *-commutative 80.6%

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

      3. times-frac 80.7%

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

      4. clear-num 80.7%

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

   9. Applied egg-rr 80.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{\left(x + 1\right) \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -3.5e+78)
     t_0
     (if (<= y -5.6e+50)
       (/ x (+ y (/ y x)))
       (if (<= y -3.2e+22)
         x
         (if (<= y 5.8e-74) (/ x (/ (* (+ x 1.0) y) x)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -3.5e+78) {
		tmp = t_0;
	} else if (y <= -5.6e+50) {
		tmp = x / (y + (y / x));
	} else if (y <= -3.2e+22) {
		tmp = x;
	} else if (y <= 5.8e-74) {
		tmp = x / (((x + 1.0) * y) / 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 / (x + 1.0d0)
    if (y <= (-3.5d+78)) then
        tmp = t_0
    else if (y <= (-5.6d+50)) then
        tmp = x / (y + (y / x))
    else if (y <= (-3.2d+22)) then
        tmp = x
    else if (y <= 5.8d-74) then
        tmp = x / (((x + 1.0d0) * y) / x)
    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 <= -3.5e+78) {
		tmp = t_0;
	} else if (y <= -5.6e+50) {
		tmp = x / (y + (y / x));
	} else if (y <= -3.2e+22) {
		tmp = x;
	} else if (y <= 5.8e-74) {
		tmp = x / (((x + 1.0) * y) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -3.5e+78:
		tmp = t_0
	elif y <= -5.6e+50:
		tmp = x / (y + (y / x))
	elif y <= -3.2e+22:
		tmp = x
	elif y <= 5.8e-74:
		tmp = x / (((x + 1.0) * y) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (y <= -3.5e+78)
		tmp = t_0;
	elseif (y <= -5.6e+50)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (y <= -3.2e+22)
		tmp = x;
	elseif (y <= 5.8e-74)
		tmp = Float64(x / Float64(Float64(Float64(x + 1.0) * y) / x));
	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 <= -3.5e+78)
		tmp = t_0;
	elseif (y <= -5.6e+50)
		tmp = x / (y + (y / x));
	elseif (y <= -3.2e+22)
		tmp = x;
	elseif (y <= 5.8e-74)
		tmp = x / (((x + 1.0) * y) / x);
	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, -3.5e+78], t$95$0, If[LessEqual[y, -5.6e+50], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.2e+22], x, If[LessEqual[y, 5.8e-74], N[(x / N[(N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.5000000000000001e78 or 5.8e-74 < y

   1. Initial program 93.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 80.2%

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

   if -3.5000000000000001e78 < y < -5.5999999999999996e50

   1. Initial program 57.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 59.4%

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

      1. clear-num 59.6%

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

      2. un-div-inv 60.0%

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

      3. associate-/l* 71.0%

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

      4. +-commutative 71.0%

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

   7. Applied egg-rr 71.0%

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

   8. Taylor expanded in x around inf 71.0%

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

   if -5.5999999999999996e50 < y < -3.2e22

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if -3.2e22 < y < 5.8e-74

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 80.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{\left(x + 1\right) \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{y + x \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= y -9.5e+77)
     t_0
     (if (<= y -5e+50)
       (/ x (+ y (/ y x)))
       (if (<= y -3e+22)
         x
         (if (<= y 5.7e-74) (/ x (/ (+ y (* x y)) x)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (y <= -9.5e+77) {
		tmp = t_0;
	} else if (y <= -5e+50) {
		tmp = x / (y + (y / x));
	} else if (y <= -3e+22) {
		tmp = x;
	} else if (y <= 5.7e-74) {
		tmp = x / ((y + (x * y)) / 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 / (x + 1.0d0)
    if (y <= (-9.5d+77)) then
        tmp = t_0
    else if (y <= (-5d+50)) then
        tmp = x / (y + (y / x))
    else if (y <= (-3d+22)) then
        tmp = x
    else if (y <= 5.7d-74) then
        tmp = x / ((y + (x * y)) / x)
    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 <= -9.5e+77) {
		tmp = t_0;
	} else if (y <= -5e+50) {
		tmp = x / (y + (y / x));
	} else if (y <= -3e+22) {
		tmp = x;
	} else if (y <= 5.7e-74) {
		tmp = x / ((y + (x * y)) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if y <= -9.5e+77:
		tmp = t_0
	elif y <= -5e+50:
		tmp = x / (y + (y / x))
	elif y <= -3e+22:
		tmp = x
	elif y <= 5.7e-74:
		tmp = x / ((y + (x * y)) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (y <= -9.5e+77)
		tmp = t_0;
	elseif (y <= -5e+50)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (y <= -3e+22)
		tmp = x;
	elseif (y <= 5.7e-74)
		tmp = Float64(x / Float64(Float64(y + Float64(x * y)) / x));
	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 <= -9.5e+77)
		tmp = t_0;
	elseif (y <= -5e+50)
		tmp = x / (y + (y / x));
	elseif (y <= -3e+22)
		tmp = x;
	elseif (y <= 5.7e-74)
		tmp = x / ((y + (x * y)) / x);
	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, -9.5e+77], t$95$0, If[LessEqual[y, -5e+50], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3e+22], x, If[LessEqual[y, 5.7e-74], N[(x / N[(N[(y + N[(x * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.4999999999999998e77 or 5.70000000000000025e-74 < y

   1. Initial program 93.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 80.2%

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

   if -9.4999999999999998e77 < y < -5e50

   1. Initial program 57.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 59.4%

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

      1. clear-num 59.6%

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

      2. un-div-inv 60.0%

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

      3. associate-/l* 71.0%

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

      4. +-commutative 71.0%

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

   7. Applied egg-rr 71.0%

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

   8. Taylor expanded in x around inf 71.0%

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

   if -5e50 < y < -3e22

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if -3e22 < y < 5.70000000000000025e-74

   1. Initial program 88.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 80.6%

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

      1. clear-num 80.6%

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

      2. un-div-inv 80.7%

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

      3. associate-/l* 80.6%

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

      4. +-commutative 80.6%

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

   7. Applied egg-rr 80.6%

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

   8. Taylor expanded in x around inf 80.7%

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

   9. Taylor expanded in x around 0 80.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{y + x \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x}{y + \frac{y}{x}}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (/ x (+ y (/ y x)))))
   (if (<= y -9.5e+77)
     t_0
     (if (<= y -4.5e+45)
       t_1
       (if (<= y -2.9e+22) x (if (<= y 5.8e-74) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double t_1 = x / (y + (y / x));
	double tmp;
	if (y <= -9.5e+77) {
		tmp = t_0;
	} else if (y <= -4.5e+45) {
		tmp = t_1;
	} else if (y <= -2.9e+22) {
		tmp = x;
	} else if (y <= 5.8e-74) {
		tmp = t_1;
	} 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 / (x + 1.0d0)
    t_1 = x / (y + (y / x))
    if (y <= (-9.5d+77)) then
        tmp = t_0
    else if (y <= (-4.5d+45)) then
        tmp = t_1
    else if (y <= (-2.9d+22)) then
        tmp = x
    else if (y <= 5.8d-74) then
        tmp = t_1
    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 t_1 = x / (y + (y / x));
	double tmp;
	if (y <= -9.5e+77) {
		tmp = t_0;
	} else if (y <= -4.5e+45) {
		tmp = t_1;
	} else if (y <= -2.9e+22) {
		tmp = x;
	} else if (y <= 5.8e-74) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	t_1 = x / (y + (y / x))
	tmp = 0
	if y <= -9.5e+77:
		tmp = t_0
	elif y <= -4.5e+45:
		tmp = t_1
	elif y <= -2.9e+22:
		tmp = x
	elif y <= 5.8e-74:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(x / Float64(y + Float64(y / x)))
	tmp = 0.0
	if (y <= -9.5e+77)
		tmp = t_0;
	elseif (y <= -4.5e+45)
		tmp = t_1;
	elseif (y <= -2.9e+22)
		tmp = x;
	elseif (y <= 5.8e-74)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	t_1 = x / (y + (y / x));
	tmp = 0.0;
	if (y <= -9.5e+77)
		tmp = t_0;
	elseif (y <= -4.5e+45)
		tmp = t_1;
	elseif (y <= -2.9e+22)
		tmp = x;
	elseif (y <= 5.8e-74)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e+77], t$95$0, If[LessEqual[y, -4.5e+45], t$95$1, If[LessEqual[y, -2.9e+22], x, If[LessEqual[y, 5.8e-74], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.4999999999999998e77 or 5.8e-74 < y

   1. Initial program 93.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 80.2%

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

   if -9.4999999999999998e77 < y < -4.4999999999999998e45 or -2.9e22 < y < 5.8e-74

   1. Initial program 85.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 79.1%

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

      1. clear-num 79.1%

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

      2. un-div-inv 79.2%

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

      3. associate-/l* 80.0%

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

      4. +-commutative 80.0%

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

   7. Applied egg-rr 80.0%

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

   8. Taylor expanded in x around inf 80.0%

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

   if -4.4999999999999998e45 < y < -2.9e22

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+24} \lor \neg \left(x \leq 1.1 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -9.2e+24) (not (<= x 1.1e+16))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -9.2e+24) || !(x <= 1.1e+16)) {
		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 <= (-9.2d+24)) .or. (.not. (x <= 1.1d+16))) 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 <= -9.2e+24) || !(x <= 1.1e+16)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -9.2e+24) or not (x <= 1.1e+16):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -9.2e+24) || !(x <= 1.1e+16))
		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 <= -9.2e+24) || ~((x <= 1.1e+16)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -9.2e+24], N[Not[LessEqual[x, 1.1e+16]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999996e24 or 1.1e16 < x

   1. Initial program 77.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 77.4%

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

   if -9.1999999999999996e24 < x < 1.1e16

   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 76.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+24} \lor \neg \left(x \leq 1.1 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.1e+15))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.1e+15)) {
		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 <= 3.1d+15))) 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 <= 3.1e+15)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 3.1e+15):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 3.1e+15))
		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 <= 3.1e+15)))
		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, 3.1e+15]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.1e15 < x

   1. Initial program 79.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 74.4%

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

   if -1 < x < 3.1e15

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -175000000.0) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -175000000.0) {
		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 <= (-175000000.0d0)) 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 <= -175000000.0) {
		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 <= -175000000.0:
		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 <= -175000000.0)
		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 <= -175000000.0)
		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, -175000000.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e8 or 1 < x

   1. Initial program 79.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 26.3%

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

      1. clear-num 26.3%

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

      2. inv-pow 26.3%

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

      3. +-commutative 26.3%

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

   7. Applied egg-rr 26.3%

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

      1. unpow-1 26.3%

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

      2. +-commutative 26.3%

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

   9. Simplified 26.3%

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

   10. Taylor expanded in x around inf 26.1%

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

   if -1.75e8 < 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 75.0%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -175000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 89.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 10: 14.4% 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 89.8%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 52.0%

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

   1. clear-num 51.8%

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

   2. inv-pow 51.8%

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

   3. +-commutative 51.8%

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

7. Applied egg-rr 51.8%

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

   1. unpow-1 51.8%

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

   2. +-commutative 51.8%

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

9. Simplified 51.8%

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

10. Taylor expanded in x around inf 14.5%

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

11. Final simplification 14.5%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer target: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024075 
(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)))