Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.8%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% 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: 99.8% 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 92.3%

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

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

Alternative 2: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.8e+67)
   (/ x y)
   (if (<= x -2.0)
     (/ x (+ x 1.0))
     (if (<= x -8e-5)
       (* x (/ x (* y (+ x 1.0))))
       (if (<= x 1.0) (* x (+ 1.0 (* x (+ (/ 1.0 y) -1.0)))) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.8e+67) {
		tmp = x / y;
	} else if (x <= -2.0) {
		tmp = x / (x + 1.0);
	} else if (x <= -8e-5) {
		tmp = x * (x / (y * (x + 1.0)));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + (x * ((1.0 / y) + -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 <= (-9.8d+67)) then
        tmp = x / y
    else if (x <= (-2.0d0)) then
        tmp = x / (x + 1.0d0)
    else if (x <= (-8d-5)) then
        tmp = x * (x / (y * (x + 1.0d0)))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-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 <= -9.8e+67) {
		tmp = x / y;
	} else if (x <= -2.0) {
		tmp = x / (x + 1.0);
	} else if (x <= -8e-5) {
		tmp = x * (x / (y * (x + 1.0)));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.8e+67:
		tmp = x / y
	elif x <= -2.0:
		tmp = x / (x + 1.0)
	elif x <= -8e-5:
		tmp = x * (x / (y * (x + 1.0)))
	elif x <= 1.0:
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.8e+67)
		tmp = Float64(x / y);
	elseif (x <= -2.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= -8e-5)
		tmp = Float64(x * Float64(x / Float64(y * Float64(x + 1.0))));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.8e+67)
		tmp = x / y;
	elseif (x <= -2.0)
		tmp = x / (x + 1.0);
	elseif (x <= -8e-5)
		tmp = x * (x / (y * (x + 1.0)));
	elseif (x <= 1.0)
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.8e+67], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-5], N[(x * N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(x * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.7999999999999998e67 or 1 < x

   1. Initial program 80.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 x around inf 77.6%

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

   if -9.7999999999999998e67 < x < -2

   1. Initial program 90.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 78.6%

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

   if -2 < x < -8.00000000000000065e-5

   1. Initial program 98.4%

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

      1. associate-/l* 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around 0 100.0%

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

   if -8.00000000000000065e-5 < 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 98.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -3.2e+67) (/ x y) (if (<= x 1.8e-33) t_0 (/ t_0 (/ y x))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -3.2e+67) {
		tmp = x / y;
	} else if (x <= 1.8e-33) {
		tmp = t_0;
	} else {
		tmp = t_0 / (y / x);
	}
	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 <= (-3.2d+67)) then
        tmp = x / y
    else if (x <= 1.8d-33) then
        tmp = t_0
    else
        tmp = t_0 / (y / x)
    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 <= -3.2e+67) {
		tmp = x / y;
	} else if (x <= 1.8e-33) {
		tmp = t_0;
	} else {
		tmp = t_0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -3.2e+67:
		tmp = x / y
	elif x <= 1.8e-33:
		tmp = t_0
	else:
		tmp = t_0 / (y / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -3.2e+67)
		tmp = Float64(x / y);
	elseif (x <= 1.8e-33)
		tmp = t_0;
	else
		tmp = Float64(t_0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -3.2e+67)
		tmp = x / y;
	elseif (x <= 1.8e-33)
		tmp = t_0;
	else
		tmp = t_0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+67], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.8e-33], t$95$0, N[(t$95$0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999983e67

   1. Initial program 78.8%

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

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

   if -3.19999999999999983e67 < x < 1.80000000000000017e-33

   1. Initial program 99.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 78.3%

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

   if 1.80000000000000017e-33 < x

   1. Initial program 83.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 70.5%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   6. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. associate-/l* 79.4%

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

      4. *-lft-identity 79.4%

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

      5. associate-*l/ 79.3%

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

      6. unpow2 79.3%

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

      7. +-commutative 79.3%

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

      8. associate-/l* 99.8%

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

      9. *-lft-identity 99.8%

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

      10. associate-*l/ 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*l/ 99.9%

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

      13. *-lft-identity 99.9%

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

      14. +-commutative 99.9%

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

   7. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.7%

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

      5. clear-num 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. associate-/r* 99.6%

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

       3. associate-/r/ 99.5%

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

       4. *-commutative 99.5%

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

       5. associate-*r/ 99.7%

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

       6. *-rgt-identity 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around 0 74.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1}}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+67} \lor \neg \left(x \leq 1.7 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.35e+67) (not (<= x 1.7e+36))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.35e+67) || !(x <= 1.7e+36)) {
		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 <= (-2.35d+67)) .or. (.not. (x <= 1.7d+36))) 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 <= -2.35e+67) || !(x <= 1.7e+36)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.35e+67) or not (x <= 1.7e+36):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.35e+67) || !(x <= 1.7e+36))
		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 <= -2.35e+67) || ~((x <= 1.7e+36)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.35e+67], N[Not[LessEqual[x, 1.7e+36]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.35000000000000009e67 or 1.6999999999999999e36 < 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* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 79.8%

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

   if -2.35000000000000009e67 < x < 1.6999999999999999e36

   1. Initial program 99.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+67} \lor \neg \left(x \leq 1.7 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 106000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 106000000000.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 106000000000.0)) {
		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 <= 106000000000.0d0))) 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 <= 106000000000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 106000000000.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 106000000000.0))
		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 <= 106000000000.0)))
		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, 106000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.06e11 < x

   1. Initial program 80.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 x around inf 73.2%

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

   if -1 < x < 1.06e11

   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 75.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 106000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.8% 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 92.3%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 46.5%

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

6. Final simplification 46.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024078 
(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)))