Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.3% → 99.9%

Time: 7.0s

Alternatives: 13

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: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 95.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 78.5%

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

      1. clear-num 78.5%

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

      2. associate-/r/ 78.4%

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

      3. fma-def 78.4%

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

   3. Applied egg-rr 78.4%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-+r- 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+l- 99.4%

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

      5. div-sub 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-neg-frac 99.4%

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

   9. Simplified 99.4%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 93.3%

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

      1. unpow2 93.3%

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

      2. sub-neg 93.3%

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

      3. metadata-eval 93.3%

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

   6. Simplified 93.3%

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

   if 1 < x

   1. Initial program 80.2%

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

      1. clear-num 80.1%

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

      2. associate-/r/ 80.2%

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

      3. fma-def 80.2%

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

   3. Applied egg-rr 80.2%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-+r- 98.9%

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

      3. +-commutative 98.9%

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

      4. associate-+l- 98.9%

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

      5. div-sub 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 96.3%

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

Alternative 3: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+75)
   (/ x y)
   (if (<= x -1.0)
     1.0
     (if (<= x 1.4e-96) x (if (<= x 1.0) (* x (/ x y)) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+75) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.4e-96) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d+75)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.4d-96) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = x * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+75) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.4e-96) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+75:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 1.4e-96:
		tmp = x
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+75)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.4e-96)
		tmp = x;
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e+75)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.4e-96)
		tmp = x;
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+75], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.4e-96], x, If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4499999999999999e75 or 1 < x

   1. Initial program 77.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.5%

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

   if -1.4499999999999999e75 < x < -1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 95.5%

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

      1. +-commutative 95.5%

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

      2. associate-+r- 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l- 95.5%

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

      5. div-sub 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in y around inf 85.0%

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

   if -1 < x < 1.40000000000000008e-96

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 81.9%

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

   if 1.40000000000000008e-96 < x < 1

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-/r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+75)
   (/ x y)
   (if (<= x -1.0)
     1.0
     (if (<= x 1.35e-95)
       (- x (* x x))
       (if (<= x 1.0) (* x (/ x y)) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+75) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.35e-95) {
		tmp = x - (x * x);
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d+75)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.35d-95) then
        tmp = x - (x * x)
    else if (x <= 1.0d0) then
        tmp = x * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+75) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.35e-95) {
		tmp = x - (x * x);
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e+75:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 1.35e-95:
		tmp = x - (x * x)
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+75)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.35e-95)
		tmp = Float64(x - Float64(x * x));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+75)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.35e-95)
		tmp = x - (x * x);
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e+75], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.35e-95], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.04999999999999999e75 or 1 < x

   1. Initial program 77.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.5%

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

   if -1.04999999999999999e75 < x < -1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 95.5%

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

      1. +-commutative 95.5%

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

      2. associate-+r- 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l- 95.5%

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

      5. div-sub 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in y around inf 85.0%

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

   if -1 < x < 1.35e-95

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 82.7%

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

   5. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. unsub-neg 81.9%

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

      3. unpow2 81.9%

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

   7. Simplified 81.9%

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

   if 1.35e-95 < x < 1

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-/r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.85e+26)
   (+ 1.0 (/ x y))
   (if (<= x 2e-95)
     (/ x (+ x 1.0))
     (if (<= x 1.0) (/ x (/ y x)) (+ 1.0 (/ (+ x -1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+26) {
		tmp = 1.0 + (x / y);
	} else if (x <= 2e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = 1.0 + ((x + -1.0) / 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 <= (-2.85d+26)) then
        tmp = 1.0d0 + (x / y)
    else if (x <= 2d-95) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = x / (y / x)
    else
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.85e+26) {
		tmp = 1.0 + (x / y);
	} else if (x <= 2e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = 1.0 + ((x + -1.0) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.85e+26:
		tmp = 1.0 + (x / y)
	elif x <= 2e-95:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x / (y / x)
	else:
		tmp = 1.0 + ((x + -1.0) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.85e+26)
		tmp = Float64(1.0 + Float64(x / y));
	elseif (x <= 2e-95)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.85e+26)
		tmp = 1.0 + (x / y);
	elseif (x <= 2e-95)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x / (y / x);
	else
		tmp = 1.0 + ((x + -1.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.85e+26], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-95], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.8500000000000002e26

   1. Initial program 77.1%

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

      1. clear-num 77.0%

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

      2. associate-/r/ 77.0%

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

      3. fma-def 77.0%

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

   3. Applied egg-rr 77.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l- 100.0%

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

      5. div-sub 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   9. Simplified 100.0%

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

   if -2.8500000000000002e26 < x < 1.99999999999999998e-95

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if 1.99999999999999998e-95 < x < 1

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

   if 1 < x

   1. Initial program 80.2%

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

      1. clear-num 80.1%

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

      2. associate-/r/ 80.2%

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

      3. fma-def 80.2%

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

   3. Applied egg-rr 80.2%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-+r- 98.9%

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

      3. +-commutative 98.9%

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

      4. associate-+l- 98.9%

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

      5. div-sub 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 6: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6400000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -2.85e+26)
     t_0
     (if (<= x 2.5e-97)
       (/ x (+ x 1.0))
       (if (<= x 6400000.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2.85e+26) {
		tmp = t_0;
	} else if (x <= 2.5e-97) {
		tmp = x / (x + 1.0);
	} else if (x <= 6400000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (x <= (-2.85d+26)) then
        tmp = t_0
    else if (x <= 2.5d-97) then
        tmp = x / (x + 1.0d0)
    else if (x <= 6400000.0d0) then
        tmp = x / (y + (y / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2.85e+26) {
		tmp = t_0;
	} else if (x <= 2.5e-97) {
		tmp = x / (x + 1.0);
	} else if (x <= 6400000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -2.85e+26:
		tmp = t_0
	elif x <= 2.5e-97:
		tmp = x / (x + 1.0)
	elif x <= 6400000.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -2.85e+26)
		tmp = t_0;
	elseif (x <= 2.5e-97)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 6400000.0)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -2.85e+26)
		tmp = t_0;
	elseif (x <= 2.5e-97)
		tmp = x / (x + 1.0);
	elseif (x <= 6400000.0)
		tmp = x / (y + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.85e+26], t$95$0, If[LessEqual[x, 2.5e-97], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6400000.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000002e26 or 6.4e6 < x

   1. Initial program 78.6%

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

      1. clear-num 78.5%

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

      2. associate-/r/ 78.5%

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

      3. fma-def 78.5%

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

   3. Applied egg-rr 78.5%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. div-sub 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. distribute-neg-frac 99.9%

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

   9. Simplified 99.9%

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

   if -2.8500000000000002e26 < x < 2.4999999999999998e-97

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if 2.4999999999999998e-97 < x < 6.4e6

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 71.3%

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

   5. Taylor expanded in x around 0 71.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6400000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 7: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e+77)
   (/ x y)
   (if (<= x 7.2e-96) (/ x (+ x 1.0)) (if (<= x 1.0) (* x (/ x y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+77) {
		tmp = x / y;
	} else if (x <= 7.2e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d+77)) then
        tmp = x / y
    else if (x <= 7.2d-96) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = x * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+77) {
		tmp = x / y;
	} else if (x <= 7.2e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.8e+77:
		tmp = x / y
	elif x <= 7.2e-96:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e+77)
		tmp = Float64(x / y);
	elseif (x <= 7.2e-96)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e+77)
		tmp = x / y;
	elseif (x <= 7.2e-96)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e+77], N[(x / y), $MachinePrecision], If[LessEqual[x, 7.2e-96], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e77 or 1 < x

   1. Initial program 77.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.5%

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

   if -1.7999999999999999e77 < x < 7.20000000000000016e-96

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.3%

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

   if 7.20000000000000016e-96 < x < 1

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

      1. associate-/r/ 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+76)
   (/ x y)
   (if (<= x 2.3e-95) (/ x (+ x 1.0)) (if (<= x 1.0) (/ x (/ y x)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+76) {
		tmp = x / y;
	} else if (x <= 2.3e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d+76)) then
        tmp = x / y
    else if (x <= 2.3d-95) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = x / (y / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+76) {
		tmp = x / y;
	} else if (x <= 2.3e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.0) {
		tmp = x / (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e+76:
		tmp = x / y
	elif x <= 2.3e-95:
		tmp = x / (x + 1.0)
	elif x <= 1.0:
		tmp = x / (y / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+76)
		tmp = Float64(x / y);
	elseif (x <= 2.3e-95)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+76)
		tmp = x / y;
	elseif (x <= 2.3e-95)
		tmp = x / (x + 1.0);
	elseif (x <= 1.0)
		tmp = x / (y / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+76], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.3e-95], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999995e76 or 1 < x

   1. Initial program 77.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.5%

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

   if -1.34999999999999995e76 < x < 2.29999999999999999e-95

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.3%

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

   if 2.29999999999999999e-95 < x < 1

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.5e+77)
   (/ x y)
   (if (<= x 1.9e-96)
     (/ x (+ x 1.0))
     (if (<= x 1.3) (/ x (/ y x)) (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+77) {
		tmp = x / y;
	} else if (x <= 1.9e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.3) {
		tmp = x / (y / x);
	} else {
		tmp = (x + -1.0) / 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 <= (-6.5d+77)) then
        tmp = x / y
    else if (x <= 1.9d-96) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.3d0) then
        tmp = x / (y / x)
    else
        tmp = (x + (-1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+77) {
		tmp = x / y;
	} else if (x <= 1.9e-96) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.3) {
		tmp = x / (y / x);
	} else {
		tmp = (x + -1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.5e+77:
		tmp = x / y
	elif x <= 1.9e-96:
		tmp = x / (x + 1.0)
	elif x <= 1.3:
		tmp = x / (y / x)
	else:
		tmp = (x + -1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.5e+77)
		tmp = Float64(x / y);
	elseif (x <= 1.9e-96)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.3)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = Float64(Float64(x + -1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.5e+77)
		tmp = x / y;
	elseif (x <= 1.9e-96)
		tmp = x / (x + 1.0);
	elseif (x <= 1.3)
		tmp = x / (y / x);
	else
		tmp = (x + -1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.5e+77], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.9e-96], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.5e77

   1. Initial program 75.1%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.1%

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

   if -6.5e77 < x < 1.9e-96

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.3%

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

   if 1.9e-96 < x < 1.30000000000000004

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 70.1%

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

   5. Taylor expanded in x around 0 66.8%

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

   if 1.30000000000000004 < x

   1. Initial program 80.2%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.9%

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

   5. Taylor expanded in y around 0 72.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \]

Alternative 10: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -2.85e+26)
     t_0
     (if (<= x 2.3e-95)
       (/ x (+ x 1.0))
       (if (<= x 1.6e-12) (/ x (/ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2.85e+26) {
		tmp = t_0;
	} else if (x <= 2.3e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-12) {
		tmp = x / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (x <= (-2.85d+26)) then
        tmp = t_0
    else if (x <= 2.3d-95) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1.6d-12) then
        tmp = x / (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2.85e+26) {
		tmp = t_0;
	} else if (x <= 2.3e-95) {
		tmp = x / (x + 1.0);
	} else if (x <= 1.6e-12) {
		tmp = x / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -2.85e+26:
		tmp = t_0
	elif x <= 2.3e-95:
		tmp = x / (x + 1.0)
	elif x <= 1.6e-12:
		tmp = x / (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -2.85e+26)
		tmp = t_0;
	elseif (x <= 2.3e-95)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1.6e-12)
		tmp = Float64(x / Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -2.85e+26)
		tmp = t_0;
	elseif (x <= 2.3e-95)
		tmp = x / (x + 1.0);
	elseif (x <= 1.6e-12)
		tmp = x / (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.85e+26], t$95$0, If[LessEqual[x, 2.3e-95], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-12], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000002e26 or 1.6e-12 < x

   1. Initial program 79.1%

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

      1. clear-num 79.0%

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

      2. associate-/r/ 79.0%

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

      3. fma-def 79.0%

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

   3. Applied egg-rr 79.0%

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

   4. Taylor expanded in x around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r- 98.0%

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

      3. +-commutative 98.0%

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

      4. associate-+l- 98.0%

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

      5. div-sub 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. distribute-neg-frac 98.0%

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

   9. Simplified 98.0%

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

   if -2.8500000000000002e26 < x < 2.29999999999999999e-95

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 83.4%

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

   if 2.29999999999999999e-95 < x < 1.6e-12

   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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 71.9%

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

   5. Taylor expanded in x around 0 71.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 11: 73.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e+78) (/ x y) (if (<= x -1.0) 1.0 (if (<= x 0.45) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e+78) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 0.45) {
		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 <= (-2.2d+78)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 0.45d0) 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 <= -2.2e+78) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 0.45) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e+78:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 0.45:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e+78)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.45)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e+78)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 0.45)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e+78], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 0.45], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.20000000000000014e78 or 0.450000000000000011 < x

   1. Initial program 78.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.1%

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

   if -2.20000000000000014e78 < x < -1

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around inf 95.5%

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

      1. +-commutative 95.5%

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

      2. associate-+r- 95.5%

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

      3. +-commutative 95.5%

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

      4. associate-+l- 95.5%

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

      5. div-sub 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in y around inf 85.0%

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

   if -1 < x < 0.450000000000000011

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 72.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.45:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 12: 50.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.4%

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

      1. clear-num 79.3%

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

      2. associate-/r/ 79.3%

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

      3. fma-def 79.3%

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

   3. Applied egg-rr 79.3%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-+r- 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+l- 99.1%

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

      5. div-sub 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in y around inf 30.4%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 72.1%

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

4. Final simplification 50.8%

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

Alternative 13: 14.8% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 89.4%

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

   1. clear-num 89.3%

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

   2. associate-/r/ 89.4%

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

   3. fma-def 89.4%

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

3. Applied egg-rr 89.4%

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

4. Taylor expanded in x around inf 51.9%

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

   1. +-commutative 51.9%

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

   2. associate-+r- 51.9%

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

   3. +-commutative 51.9%

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

   4. associate-+l- 51.9%

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

   5. div-sub 51.9%

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

6. Simplified 51.9%

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

7. Taylor expanded in y around inf 17.2%

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

8. Final simplification 17.2%

   \[\leadsto 1 \]

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 2023290 
(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)))