Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 97.7%

Time: 6.2s

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: 88.0% 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: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+162} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (or (<= x -1.45e+162) (not (<= x 10.0))) t_0 (/ (* x t_0) (+ x 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if ((x <= -1.45e+162) || !(x <= 10.0)) {
		tmp = t_0;
	} else {
		tmp = (x * t_0) / (x + 1.0);
	}
	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 / y) + 1.0d0
    if ((x <= (-1.45d+162)) .or. (.not. (x <= 10.0d0))) then
        tmp = t_0
    else
        tmp = (x * t_0) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if ((x <= -1.45e+162) || !(x <= 10.0)) {
		tmp = t_0;
	} else {
		tmp = (x * t_0) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if (x <= -1.45e+162) or not (x <= 10.0):
		tmp = t_0
	else:
		tmp = (x * t_0) / (x + 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if ((x <= -1.45e+162) || !(x <= 10.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(x * t_0) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if ((x <= -1.45e+162) || ~((x <= 10.0)))
		tmp = t_0;
	else
		tmp = (x * t_0) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.45e+162], N[Not[LessEqual[x, 10.0]], $MachinePrecision]], t$95$0, N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000003e162 or 10 < x

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1.45000000000000003e162 < x < 10

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+162} \lor \neg \left(x \leq 10\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \end{array} \]

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6.8:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e+78)
   (/ x y)
   (if (<= x -1.6e-8)
     (/ x (+ x 1.0))
     (if (<= x 6.8) (* x (+ (/ x y) 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e+78) {
		tmp = x / y;
	} else if (x <= -1.6e-8) {
		tmp = x / (x + 1.0);
	} else if (x <= 6.8) {
		tmp = x * ((x / 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 <= (-3.6d+78)) then
        tmp = x / y
    else if (x <= (-1.6d-8)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 6.8d0) then
        tmp = x * ((x / 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 <= -3.6e+78) {
		tmp = x / y;
	} else if (x <= -1.6e-8) {
		tmp = x / (x + 1.0);
	} else if (x <= 6.8) {
		tmp = x * ((x / y) + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e+78:
		tmp = x / y
	elif x <= -1.6e-8:
		tmp = x / (x + 1.0)
	elif x <= 6.8:
		tmp = x * ((x / y) + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e+78)
		tmp = Float64(x / y);
	elseif (x <= -1.6e-8)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 6.8)
		tmp = Float64(x * Float64(Float64(x / y) + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e+78)
		tmp = x / y;
	elseif (x <= -1.6e-8)
		tmp = x / (x + 1.0);
	elseif (x <= 6.8)
		tmp = x * ((x / y) + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e+78], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.6e-8], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8], N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000002e78 or 6.79999999999999982 < x

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 79.4%

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

   if -3.6000000000000002e78 < x < -1.6000000000000001e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.9%

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

      1. +-commutative 69.9%

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

   4. Simplified 69.9%

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

   if -1.6000000000000001e-8 < x < 6.79999999999999982

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. div-sub 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.2%

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

      1. associate-/r/ 97.4%

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

      2. /-rgt-identity 97.4%

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

   6. Applied egg-rr 97.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6.8:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+77)
   (/ x y)
   (if (<= x -6.2e-8)
     (/ x (+ x 1.0))
     (if (<= x 5e-9) (* x (+ (/ x y) 1.0)) (/ x (+ y (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+77) {
		tmp = x / y;
	} else if (x <= -6.2e-8) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-9) {
		tmp = x * ((x / y) + 1.0);
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.5d+77)) then
        tmp = x / y
    else if (x <= (-6.2d-8)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 5d-9) then
        tmp = x * ((x / y) + 1.0d0)
    else
        tmp = x / (y + (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+77) {
		tmp = x / y;
	} else if (x <= -6.2e-8) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-9) {
		tmp = x * ((x / y) + 1.0);
	} else {
		tmp = x / (y + (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e+77:
		tmp = x / y
	elif x <= -6.2e-8:
		tmp = x / (x + 1.0)
	elif x <= 5e-9:
		tmp = x * ((x / y) + 1.0)
	else:
		tmp = x / (y + (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+77)
		tmp = Float64(x / y);
	elseif (x <= -6.2e-8)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 5e-9)
		tmp = Float64(x * Float64(Float64(x / y) + 1.0));
	else
		tmp = Float64(x / Float64(y + Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.5e+77)
		tmp = x / y;
	elseif (x <= -6.2e-8)
		tmp = x / (x + 1.0);
	elseif (x <= 5e-9)
		tmp = x * ((x / y) + 1.0);
	else
		tmp = x / (y + (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+77], N[(x / y), $MachinePrecision], If[LessEqual[x, -6.2e-8], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-9], N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.50000000000000018e77

   1. Initial program 66.5%

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

   2. Taylor expanded in x around inf 78.1%

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

   if -8.50000000000000018e77 < x < -6.2e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.9%

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

      1. +-commutative 69.9%

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

   4. Simplified 69.9%

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

   if -6.2e-8 < x < 5.0000000000000001e-9

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. div-sub 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-/r/ 99.9%

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

      2. /-rgt-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-9 < x

   1. Initial program 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 80.5%

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

      1. distribute-lft-in 80.5%

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

      2. *-rgt-identity 80.5%

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

      3. associate-*r/ 80.5%

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

      4. *-rgt-identity 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (or (<= x -1.0) (not (<= x 1.0))) t_0 (* x t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = t_0;
	} else {
		tmp = x * 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 / y) + 1.0d0
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = t_0
    else
        tmp = x * 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 tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = t_0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = t_0
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = t_0;
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = t_0;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], t$95$0, N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.7%

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

      3. +-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. div-sub 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. associate-/r/ 96.3%

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

      2. /-rgt-identity 96.3%

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

   6. Applied egg-rr 96.3%

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

4. Final simplification 97.5%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((x / y) + 1.0) / ((1.0 / 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) / ((1.0d0 / x) - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. *-commutative 86.5%

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

   2. associate-/l* 99.8%

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

   3. +-commutative 99.8%

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

   4. remove-double-neg 99.8%

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

   5. unsub-neg 99.8%

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

   6. div-sub 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. *-inverses 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \frac{\frac{x}{y} + 1}{\frac{1}{x} - -1} \]

Alternative 6: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e+78)
   (/ x y)
   (if (<= x -2000000000.0) 1.0 (if (<= x 4.8) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e+78) {
		tmp = x / y;
	} else if (x <= -2000000000.0) {
		tmp = 1.0;
	} else if (x <= 4.8) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d+78)) then
        tmp = x / y
    else if (x <= (-2000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 4.8d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e+78) {
		tmp = x / y;
	} else if (x <= -2000000000.0) {
		tmp = 1.0;
	} else if (x <= 4.8) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e+78:
		tmp = x / y
	elif x <= -2000000000.0:
		tmp = 1.0
	elif x <= 4.8:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e+78)
		tmp = Float64(x / y);
	elseif (x <= -2000000000.0)
		tmp = 1.0;
	elseif (x <= 4.8)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e+78)
		tmp = x / y;
	elseif (x <= -2000000000.0)
		tmp = 1.0;
	elseif (x <= 4.8)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e+78], N[(x / y), $MachinePrecision], If[LessEqual[x, -2000000000.0], 1.0, If[LessEqual[x, 4.8], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999982e78 or 4.79999999999999982 < x

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 79.4%

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

   if -2.99999999999999982e78 < x < -2e9

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-prod-up 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 75.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - 1}{\frac{1}{{x}^{2}} - 1}} \]

   7. Taylor expanded in x around inf 75.1%

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

   if -2e9 < x < 4.79999999999999982

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+78} \lor \neg \left(x \leq 7\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.6e+78) (not (<= x 7.0))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.6e+78) || !(x <= 7.0)) {
		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 <= (-1.6d+78)) .or. (.not. (x <= 7.0d0))) 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 <= -1.6e+78) || !(x <= 7.0)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.6e+78) or not (x <= 7.0):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.6e+78) || !(x <= 7.0))
		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 <= -1.6e+78) || ~((x <= 7.0)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.6e+78], N[Not[LessEqual[x, 7.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999997e78 or 7 < x

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 79.4%

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

   if -1.59999999999999997e78 < x < 7

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 74.8%

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

      1. +-commutative 74.8%

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

   4. Simplified 74.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+78} \lor \neg \left(x \leq 7\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 8: 47.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2000000000.0) 1.0 (if (<= x 4.8e+52) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = 1.0;
	} else if (x <= 4.8e+52) {
		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 <= (-2000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 4.8d+52) 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 <= -2000000000.0) {
		tmp = 1.0;
	} else if (x <= 4.8e+52) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2000000000.0:
		tmp = 1.0
	elif x <= 4.8e+52:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2000000000.0)
		tmp = 1.0;
	elseif (x <= 4.8e+52)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2000000000.0)
		tmp = 1.0;
	elseif (x <= 4.8e+52)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2000000000.0], 1.0, If[LessEqual[x, 4.8e+52], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2e9 or 4.8e52 < x

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 100.0%

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

      2. metadata-eval 100.0%

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

      3. sub-neg 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

      6. pow-prod-up 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 26.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - 1}{\frac{1}{{x}^{2}} - 1}} \]

   7. Taylor expanded in x around inf 26.3%

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

   if -2e9 < x < 4.8e52

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 71.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 14.2% 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 86.5%

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

   1. *-commutative 86.5%

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

   2. associate-/l* 99.8%

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

   3. +-commutative 99.8%

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

   4. remove-double-neg 99.8%

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

   5. unsub-neg 99.8%

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

   6. div-sub 99.8%

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

   7. distribute-frac-neg 99.8%

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

   8. *-inverses 99.8%

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

   9. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. flip-- 76.2%

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

   2. metadata-eval 76.2%

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

   3. sub-neg 76.2%

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

   4. inv-pow 76.2%

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

   5. inv-pow 76.2%

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

   6. pow-prod-up 76.1%

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

   7. metadata-eval 76.1%

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

   8. metadata-eval 76.1%

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

5. Applied egg-rr 76.1%

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

6. Taylor expanded in y around inf 31.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{x} - 1}{\frac{1}{{x}^{2}} - 1}} \]

7. Taylor expanded in x around inf 14.4%

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

8. Final simplification 14.4%

   \[\leadsto 1 \]

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 2023320 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))