Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.8% → 99.9%

Time: 8.1s

Alternatives: 9

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

   1. *-commutative 90.7%

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

   2. associate-/l* 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := y + \frac{y}{x}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{x}}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.000195:\\ \;\;\;\;\frac{x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (+ y (/ y x))))
   (if (<= x -4.7e+28)
     t_0
     (if (<= x -8.5e-68)
       (/ 1.0 (/ t_1 x))
       (if (<= x 8.8e-64) x (if (<= x 0.000195) (/ x t_1) t_0))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -4.7e+28) {
		tmp = t_0;
	} else if (x <= -8.5e-68) {
		tmp = 1.0 / (t_1 / x);
	} else if (x <= 8.8e-64) {
		tmp = x;
	} else if (x <= 0.000195) {
		tmp = x / 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 / y) + 1.0d0
    t_1 = y + (y / x)
    if (x <= (-4.7d+28)) then
        tmp = t_0
    else if (x <= (-8.5d-68)) then
        tmp = 1.0d0 / (t_1 / x)
    else if (x <= 8.8d-64) then
        tmp = x
    else if (x <= 0.000195d0) then
        tmp = x / 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 / y) + 1.0;
	double t_1 = y + (y / x);
	double tmp;
	if (x <= -4.7e+28) {
		tmp = t_0;
	} else if (x <= -8.5e-68) {
		tmp = 1.0 / (t_1 / x);
	} else if (x <= 8.8e-64) {
		tmp = x;
	} else if (x <= 0.000195) {
		tmp = x / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	t_1 = y + (y / x)
	tmp = 0
	if x <= -4.7e+28:
		tmp = t_0
	elif x <= -8.5e-68:
		tmp = 1.0 / (t_1 / x)
	elif x <= 8.8e-64:
		tmp = x
	elif x <= 0.000195:
		tmp = x / t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(y + Float64(y / x))
	tmp = 0.0
	if (x <= -4.7e+28)
		tmp = t_0;
	elseif (x <= -8.5e-68)
		tmp = Float64(1.0 / Float64(t_1 / x));
	elseif (x <= 8.8e-64)
		tmp = x;
	elseif (x <= 0.000195)
		tmp = Float64(x / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	t_1 = y + (y / x);
	tmp = 0.0;
	if (x <= -4.7e+28)
		tmp = t_0;
	elseif (x <= -8.5e-68)
		tmp = 1.0 / (t_1 / x);
	elseif (x <= 8.8e-64)
		tmp = x;
	elseif (x <= 0.000195)
		tmp = x / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+28], t$95$0, If[LessEqual[x, -8.5e-68], N[(1.0 / N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-64], x, If[LessEqual[x, 0.000195], N[(x / t$95$1), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.69999999999999965e28 or 1.94999999999999996e-4 < x

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 97.7%

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

   if -4.69999999999999965e28 < x < -8.50000000000000026e-68

   1. Initial program 99.7%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. add-cbrt-cube 66.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{\left(\frac{x + 1}{\frac{x}{y} + 1} \cdot \frac{x + 1}{\frac{x}{y} + 1}\right) \cdot \frac{x + 1}{\frac{x}{y} + 1}}}} \]

      2. pow3 66.0%

         \[\leadsto \frac{x}{\sqrt[3]{\color{blue}{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]

   8. Applied egg-rr 66.0%

      \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]
   9. Step-by-step derivation

      1. rem-cbrt-cube 99.5%

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

      2. clear-num 99.8%

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

      3. inv-pow 99.8%

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

      4. associate-/l/ 99.7%

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

      5. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. associate-/r* 99.5%

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

       3. *-lft-identity 99.5%

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

       4. associate-*l/ 99.5%

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

       5. distribute-rgt-in 99.5%

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

       6. rgt-mult-inverse 99.5%

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

       7. *-lft-identity 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in y around 0 70.7%

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

       1. distribute-lft-in 70.7%

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

       2. *-rgt-identity 70.7%

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

       3. associate-*r/ 70.9%

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

       4. *-rgt-identity 70.9%

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

   15. Simplified 70.9%

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

   if -8.50000000000000026e-68 < x < 8.7999999999999998e-64

   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 81.0%

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

   if 8.7999999999999998e-64 < x < 1.94999999999999996e-4

   1. Initial program 99.6%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   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 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. add-cbrt-cube 49.5%

         \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{\left(\frac{x + 1}{\frac{x}{y} + 1} \cdot \frac{x + 1}{\frac{x}{y} + 1}\right) \cdot \frac{x + 1}{\frac{x}{y} + 1}}}} \]

      2. pow3 49.6%

         \[\leadsto \frac{x}{\sqrt[3]{\color{blue}{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]

   8. Applied egg-rr 49.6%

      \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]
   9. Step-by-step derivation

      1. rem-cbrt-cube 99.7%

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

      2. clear-num 99.5%

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

      3. inv-pow 99.5%

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

      4. associate-/l/ 99.4%

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

      5. +-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. unpow-1 99.4%

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

       2. associate-/r* 99.1%

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

       3. *-lft-identity 99.1%

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

       4. associate-*l/ 99.2%

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

       5. distribute-rgt-in 99.2%

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

       6. rgt-mult-inverse 99.2%

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

       7. *-lft-identity 99.2%

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

   12. Simplified 99.2%

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

   13. Taylor expanded in y around 0 80.6%

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

       1. distribute-lft-in 80.6%

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

       2. *-rgt-identity 80.6%

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

       3. associate-*r/ 80.5%

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

       4. *-rgt-identity 80.5%

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

   15. Simplified 80.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{y + \frac{y}{x}}{x}}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.000195:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := \frac{x}{y + \frac{y}{x}}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.000195:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (/ x (+ y (/ y x)))))
   (if (<= x -4.7e+28)
     t_0
     (if (<= x -7.4e-68)
       t_1
       (if (<= x 1.8e-66) x (if (<= x 0.000195) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = x / (y + (y / x));
	double tmp;
	if (x <= -4.7e+28) {
		tmp = t_0;
	} else if (x <= -7.4e-68) {
		tmp = t_1;
	} else if (x <= 1.8e-66) {
		tmp = x;
	} else if (x <= 0.000195) {
		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 / y) + 1.0d0
    t_1 = x / (y + (y / x))
    if (x <= (-4.7d+28)) then
        tmp = t_0
    else if (x <= (-7.4d-68)) then
        tmp = t_1
    else if (x <= 1.8d-66) then
        tmp = x
    else if (x <= 0.000195d0) 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 / y) + 1.0;
	double t_1 = x / (y + (y / x));
	double tmp;
	if (x <= -4.7e+28) {
		tmp = t_0;
	} else if (x <= -7.4e-68) {
		tmp = t_1;
	} else if (x <= 1.8e-66) {
		tmp = x;
	} else if (x <= 0.000195) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	t_1 = x / (y + (y / x))
	tmp = 0
	if x <= -4.7e+28:
		tmp = t_0
	elif x <= -7.4e-68:
		tmp = t_1
	elif x <= 1.8e-66:
		tmp = x
	elif x <= 0.000195:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(x / Float64(y + Float64(y / x)))
	tmp = 0.0
	if (x <= -4.7e+28)
		tmp = t_0;
	elseif (x <= -7.4e-68)
		tmp = t_1;
	elseif (x <= 1.8e-66)
		tmp = x;
	elseif (x <= 0.000195)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	t_1 = x / (y + (y / x));
	tmp = 0.0;
	if (x <= -4.7e+28)
		tmp = t_0;
	elseif (x <= -7.4e-68)
		tmp = t_1;
	elseif (x <= 1.8e-66)
		tmp = x;
	elseif (x <= 0.000195)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+28], t$95$0, If[LessEqual[x, -7.4e-68], t$95$1, If[LessEqual[x, 1.8e-66], x, If[LessEqual[x, 0.000195], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.69999999999999965e28 or 1.94999999999999996e-4 < x

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 97.7%

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

   if -4.69999999999999965e28 < x < -7.40000000000000004e-68 or 1.80000000000000006e-66 < x < 1.94999999999999996e-4

   1. Initial program 99.6%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{x}{y} + 1}{x + 1}} \]
   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 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-cbrt-cube 59.0%

         \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{\left(\frac{x + 1}{\frac{x}{y} + 1} \cdot \frac{x + 1}{\frac{x}{y} + 1}\right) \cdot \frac{x + 1}{\frac{x}{y} + 1}}}} \]

      2. pow3 59.0%

         \[\leadsto \frac{x}{\sqrt[3]{\color{blue}{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]

   8. Applied egg-rr 59.0%

      \[\leadsto \frac{x}{\color{blue}{\sqrt[3]{{\left(\frac{x + 1}{\frac{x}{y} + 1}\right)}^{3}}}} \]
   9. Step-by-step derivation

      1. rem-cbrt-cube 99.6%

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

      2. clear-num 99.6%

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

      3. inv-pow 99.6%

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

      4. associate-/l/ 99.6%

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

      5. +-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. unpow-1 99.6%

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

       2. associate-/r* 99.4%

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

       3. *-lft-identity 99.4%

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

       4. associate-*l/ 99.4%

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

       5. distribute-rgt-in 99.3%

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

       6. rgt-mult-inverse 99.4%

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

       7. *-lft-identity 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in y around 0 74.8%

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

       1. distribute-lft-in 74.9%

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

       2. *-rgt-identity 74.9%

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

       3. associate-*r/ 75.0%

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

       4. *-rgt-identity 75.0%

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

   15. Simplified 75.0%

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

   if -7.40000000000000004e-68 < x < 1.80000000000000006e-66

   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 81.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.000195:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.8d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 97.9%

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

   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 x around 0 98.3%

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

4. Final simplification 98.1%

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

Alternative 5: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 42000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -750.0) (not (<= x 42000000.0)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -750.0) || !(x <= 42000000.0)) {
		tmp = (x / y) + 1.0;
	} 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 <= (-750.0d0)) .or. (.not. (x <= 42000000.0d0))) then
        tmp = (x / y) + 1.0d0
    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 <= -750.0) || !(x <= 42000000.0)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -750.0) or not (x <= 42000000.0):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -750.0) || !(x <= 42000000.0))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -750.0) || ~((x <= 42000000.0)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -750.0], N[Not[LessEqual[x, 42000000.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -750 or 4.2e7 < x

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/l* 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 98.2%

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

   if -750 < x < 4.2e7

   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 69.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 42000000\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.55 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -750.0) (not (<= x 1.55e+103))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -750.0) || !(x <= 1.55e+103)) {
		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 <= (-750.0d0)) .or. (.not. (x <= 1.55d+103))) 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 <= -750.0) || !(x <= 1.55e+103)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -750.0) or not (x <= 1.55e+103):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -750.0) || !(x <= 1.55e+103))
		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 <= -750.0) || ~((x <= 1.55e+103)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -750.0], N[Not[LessEqual[x, 1.55e+103]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -750 or 1.5500000000000001e103 < x

   1. Initial program 78.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 72.1%

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

   if -750 < x < 1.5500000000000001e103

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in y around inf 66.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750 \lor \neg \left(x \leq 1.55 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 14200000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 14200000.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 14200000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 14200000.0d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 14200000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 14200000.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 14200000.0))
		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 <= 14200000.0)))
		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, 14200000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.42e7 < x

   1. Initial program 81.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 65.7%

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

   if -1 < x < 1.42e7

   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 67.6%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 14200000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

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

Alternative 9: 37.9% 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 90.7%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 0 35.2%

   \[\leadsto \color{blue}{x} \]
6. 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 2024091 
(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)))