Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.6% → 99.9%

Time: 8.8s

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: 88.6% 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 88.0%

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

   1. associate-/l* 99.9%

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

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: 69.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.4e+101)
   (/ x y)
   (if (<= x -1e+48)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 1.3e-66)
         x
         (if (<= x 5.6e+15)
           (* x (/ x y))
           (if (<= x 1.65e+121) 1.0 (/ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e+101) {
		tmp = x / y;
	} else if (x <= -1e+48) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = x;
	} else if (x <= 5.6e+15) {
		tmp = x * (x / y);
	} else if (x <= 1.65e+121) {
		tmp = 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 <= (-7.4d+101)) then
        tmp = x / y
    else if (x <= (-1d+48)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 1.3d-66) then
        tmp = x
    else if (x <= 5.6d+15) then
        tmp = x * (x / y)
    else if (x <= 1.65d+121) then
        tmp = 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 <= -7.4e+101) {
		tmp = x / y;
	} else if (x <= -1e+48) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.3e-66) {
		tmp = x;
	} else if (x <= 5.6e+15) {
		tmp = x * (x / y);
	} else if (x <= 1.65e+121) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e+101:
		tmp = x / y
	elif x <= -1e+48:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 1.3e-66:
		tmp = x
	elif x <= 5.6e+15:
		tmp = x * (x / y)
	elif x <= 1.65e+121:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e+101)
		tmp = Float64(x / y);
	elseif (x <= -1e+48)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 1.3e-66)
		tmp = x;
	elseif (x <= 5.6e+15)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e+101)
		tmp = x / y;
	elseif (x <= -1e+48)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 1.3e-66)
		tmp = x;
	elseif (x <= 5.6e+15)
		tmp = x * (x / y);
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -1e+48], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.3e-66], x, If[LessEqual[x, 5.6e+15], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+121], 1.0, N[(x / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.3999999999999995e101 or -1.00000000000000004e48 < x < -1 or 1.6499999999999999e121 < x

   1. Initial program 70.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.4%

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

   if -7.3999999999999995e101 < x < -1.00000000000000004e48 or 5.6e15 < x < 1.6499999999999999e121

   1. Initial program 92.5%

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

      1. clear-num 92.5%

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

      2. associate-/r/ 92.5%

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

      3. fma-def 92.5%

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

   3. Applied egg-rr 92.5%

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

   4. Taylor expanded in y around inf 67.9%

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

      1. +-commutative 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in x around inf 67.9%

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

   if -1 < x < 1.2999999999999999e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 75.4%

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

   if 1.2999999999999999e-66 < x < 5.6e15

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 76.1%

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

   5. Taylor expanded in x around 0 63.8%

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

      1. associate-/r/ 63.6%

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

   7. Applied egg-rr 63.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{if}\;x \leq -15200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 21000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (/ x y)) (/ -1.0 y))))
   (if (<= x -15200.0)
     t_0
     (if (<= x 4e-69)
       (/ x (+ x 1.0))
       (if (<= x 21000.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = (1.0 + (x / y)) + (-1.0 / y);
	double tmp;
	if (x <= -15200.0) {
		tmp = t_0;
	} else if (x <= 4e-69) {
		tmp = x / (x + 1.0);
	} else if (x <= 21000.0) {
		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)) + ((-1.0d0) / y)
    if (x <= (-15200.0d0)) then
        tmp = t_0
    else if (x <= 4d-69) then
        tmp = x / (x + 1.0d0)
    else if (x <= 21000.0d0) 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)) + (-1.0 / y);
	double tmp;
	if (x <= -15200.0) {
		tmp = t_0;
	} else if (x <= 4e-69) {
		tmp = x / (x + 1.0);
	} else if (x <= 21000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 + (x / y)) + (-1.0 / y)
	tmp = 0
	if x <= -15200.0:
		tmp = t_0
	elif x <= 4e-69:
		tmp = x / (x + 1.0)
	elif x <= 21000.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y))
	tmp = 0.0
	if (x <= -15200.0)
		tmp = t_0;
	elseif (x <= 4e-69)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 21000.0)
		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)) + (-1.0 / y);
	tmp = 0.0;
	if (x <= -15200.0)
		tmp = t_0;
	elseif (x <= 4e-69)
		tmp = x / (x + 1.0);
	elseif (x <= 21000.0)
		tmp = x / (y + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -15200.0], t$95$0, If[LessEqual[x, 4e-69], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 21000.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -15200 or 21000 < x

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   if -15200 < x < 3.9999999999999999e-69

   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 77.0%

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

   if 3.9999999999999999e-69 < x < 21000

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.1%

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

   5. Taylor expanded in x around 0 74.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15200:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 21000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{if}\;x \leq -1650:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x \cdot t_0}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))) (t_1 (+ (+ 1.0 (/ x y)) (/ -1.0 y))))
   (if (<= x -1650.0)
     t_1
     (if (<= x 1.02e-66) t_0 (if (<= x 1100.0) (/ (* x t_0) y) t_1)))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double t_1 = (1.0 + (x / y)) + (-1.0 / y);
	double tmp;
	if (x <= -1650.0) {
		tmp = t_1;
	} else if (x <= 1.02e-66) {
		tmp = t_0;
	} else if (x <= 1100.0) {
		tmp = (x * t_0) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    t_1 = (1.0d0 + (x / y)) + ((-1.0d0) / y)
    if (x <= (-1650.0d0)) then
        tmp = t_1
    else if (x <= 1.02d-66) then
        tmp = t_0
    else if (x <= 1100.0d0) then
        tmp = (x * t_0) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double t_1 = (1.0 + (x / y)) + (-1.0 / y);
	double tmp;
	if (x <= -1650.0) {
		tmp = t_1;
	} else if (x <= 1.02e-66) {
		tmp = t_0;
	} else if (x <= 1100.0) {
		tmp = (x * t_0) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	t_1 = (1.0 + (x / y)) + (-1.0 / y)
	tmp = 0
	if x <= -1650.0:
		tmp = t_1
	elif x <= 1.02e-66:
		tmp = t_0
	elif x <= 1100.0:
		tmp = (x * t_0) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	t_1 = Float64(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y))
	tmp = 0.0
	if (x <= -1650.0)
		tmp = t_1;
	elseif (x <= 1.02e-66)
		tmp = t_0;
	elseif (x <= 1100.0)
		tmp = Float64(Float64(x * t_0) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	t_1 = (1.0 + (x / y)) + (-1.0 / y);
	tmp = 0.0;
	if (x <= -1650.0)
		tmp = t_1;
	elseif (x <= 1.02e-66)
		tmp = t_0;
	elseif (x <= 1100.0)
		tmp = (x * t_0) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1650.0], t$95$1, If[LessEqual[x, 1.02e-66], t$95$0, If[LessEqual[x, 1100.0], N[(N[(x * t$95$0), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1650 or 1100 < x

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   if -1650 < x < 1.01999999999999996e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   if 1.01999999999999996e-66 < x < 1100

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.4%

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

      1. unpow2 74.4%

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

      2. +-commutative 74.4%

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

      3. times-frac 74.0%

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

      4. +-commutative 74.0%

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

   6. Simplified 74.0%

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

      1. *-commutative 74.0%

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

      2. associate-*r/ 74.4%

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

      3. +-commutative 74.4%

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

   8. Applied egg-rr 74.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1650:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1100:\\ \;\;\;\;\frac{x \cdot \frac{x}{x + 1}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Alternative 5: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.9%

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

   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 87.7%

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

      1. unpow2 87.7%

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

      2. sub-neg 87.7%

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

      3. metadata-eval 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 94.0%

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

Alternative 6: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+101)
   (/ x y)
   (if (<= x -2.05e+48)
     1.0
     (if (<= x -1.0)
       (/ x y)
       (if (<= x 0.00125) x (if (<= x 1.65e+121) 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -2.05e+48) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.00125) {
		tmp = x;
	} else if (x <= 1.65e+121) {
		tmp = 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 <= (-4.9d+101)) then
        tmp = x / y
    else if (x <= (-2.05d+48)) then
        tmp = 1.0d0
    else if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 0.00125d0) then
        tmp = x
    else if (x <= 1.65d+121) then
        tmp = 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 <= -4.9e+101) {
		tmp = x / y;
	} else if (x <= -2.05e+48) {
		tmp = 1.0;
	} else if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 0.00125) {
		tmp = x;
	} else if (x <= 1.65e+121) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e+101:
		tmp = x / y
	elif x <= -2.05e+48:
		tmp = 1.0
	elif x <= -1.0:
		tmp = x / y
	elif x <= 0.00125:
		tmp = x
	elif x <= 1.65e+121:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+101)
		tmp = Float64(x / y);
	elseif (x <= -2.05e+48)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 0.00125)
		tmp = x;
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e+101)
		tmp = x / y;
	elseif (x <= -2.05e+48)
		tmp = 1.0;
	elseif (x <= -1.0)
		tmp = x / y;
	elseif (x <= 0.00125)
		tmp = x;
	elseif (x <= 1.65e+121)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+101], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.05e+48], 1.0, If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 0.00125], x, If[LessEqual[x, 1.65e+121], 1.0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.89999999999999983e101 or -2.0500000000000001e48 < x < -1 or 1.6499999999999999e121 < x

   1. Initial program 70.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.4%

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

   if -4.89999999999999983e101 < x < -2.0500000000000001e48 or 0.00125000000000000003 < x < 1.6499999999999999e121

   1. Initial program 92.9%

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

      1. clear-num 92.9%

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

      2. associate-/r/ 92.8%

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

      3. fma-def 92.8%

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

   3. Applied egg-rr 92.8%

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

   4. Taylor expanded in y around inf 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in x around inf 64.6%

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

   if -1 < x < 0.00125000000000000003

   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 70.8%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -380000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.06:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -380000.0)
     t_0
     (if (<= x 1.3e-66)
       (/ x (+ x 1.0))
       (if (<= x 0.06) (/ 1.0 (/ y (* x x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -380000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.06) {
		tmp = 1.0 / (y / (x * 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 <= (-380000.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-66) then
        tmp = x / (x + 1.0d0)
    else if (x <= 0.06d0) then
        tmp = 1.0d0 / (y / (x * 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 <= -380000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.06) {
		tmp = 1.0 / (y / (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -380000.0:
		tmp = t_0
	elif x <= 1.3e-66:
		tmp = x / (x + 1.0)
	elif x <= 0.06:
		tmp = 1.0 / (y / (x * 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 <= -380000.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 0.06)
		tmp = Float64(1.0 / Float64(y / Float64(x * 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 <= -380000.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = x / (x + 1.0);
	elseif (x <= 0.06)
		tmp = 1.0 / (y / (x * 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, -380000.0], t$95$0, If[LessEqual[x, 1.3e-66], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.06], N[(1.0 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e5 or 0.059999999999999998 < x

   1. Initial program 77.0%

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

      1. clear-num 76.9%

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

      2. associate-/r/ 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. associate-/r* 76.9%

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

      3. fma-udef 77.0%

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

      4. distribute-rgt-in 77.0%

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

      5. associate-*l/ 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. distribute-rgt-in 77.0%

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

      8. +-commutative 77.0%

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

      9. associate-*l/ 77.0%

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

      10. associate-/r/ 77.0%

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

      11. div-inv 76.8%

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

      12. *-commutative 76.8%

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

      13. associate-/r/ 76.9%

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

      14. div-inv 76.9%

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

      15. metadata-eval 76.9%

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

      16. *-rgt-identity 76.9%

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

      17. distribute-lft-in 76.9%

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

      18. *-rgt-identity 76.9%

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

   5. Applied egg-rr 76.9%

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

   6. Taylor expanded in x around inf 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. rgt-mult-inverse 76.4%

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

      3. metadata-eval 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*l* 99.4%

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

      6. rgt-mult-inverse 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -3.8e5 < x < 1.2999999999999999e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   if 1.2999999999999999e-66 < x < 0.059999999999999998

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.1%

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

   5. Taylor expanded in x around 0 74.1%

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

      1. clear-num 73.8%

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

      2. inv-pow 73.8%

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

   7. Applied egg-rr 73.8%

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

      1. unpow-1 73.8%

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

   9. Simplified 73.8%

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

   10. Taylor expanded in x around 0 68.3%

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

       1. unpow2 68.3%

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

   12. Simplified 68.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.06:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 8: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -7600000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -7600000.0)
     t_0
     (if (<= x 9e-67)
       (/ x (+ x 1.0))
       (if (<= x 5.6e+15) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -7600000.0) {
		tmp = t_0;
	} else if (x <= 9e-67) {
		tmp = x / (x + 1.0);
	} else if (x <= 5.6e+15) {
		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 <= (-7600000.0d0)) then
        tmp = t_0
    else if (x <= 9d-67) then
        tmp = x / (x + 1.0d0)
    else if (x <= 5.6d+15) 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 <= -7600000.0) {
		tmp = t_0;
	} else if (x <= 9e-67) {
		tmp = x / (x + 1.0);
	} else if (x <= 5.6e+15) {
		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 <= -7600000.0:
		tmp = t_0
	elif x <= 9e-67:
		tmp = x / (x + 1.0)
	elif x <= 5.6e+15:
		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 <= -7600000.0)
		tmp = t_0;
	elseif (x <= 9e-67)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 5.6e+15)
		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 <= -7600000.0)
		tmp = t_0;
	elseif (x <= 9e-67)
		tmp = x / (x + 1.0);
	elseif (x <= 5.6e+15)
		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, -7600000.0], t$95$0, If[LessEqual[x, 9e-67], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e+15], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.6e6 or 5.6e15 < x

   1. Initial program 76.8%

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

      1. clear-num 76.7%

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

      2. associate-/r/ 76.8%

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

      3. fma-def 76.8%

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

   3. Applied egg-rr 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. associate-/r* 76.8%

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

      3. fma-udef 76.8%

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

      4. distribute-rgt-in 76.8%

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

      5. associate-*l/ 76.8%

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

      6. *-un-lft-identity 76.8%

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

      7. distribute-rgt-in 76.8%

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

      8. +-commutative 76.8%

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

      9. associate-*l/ 76.8%

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

      10. associate-/r/ 76.8%

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

      11. div-inv 76.7%

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

      12. *-commutative 76.7%

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

      13. associate-/r/ 76.7%

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

      14. div-inv 76.7%

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

      15. metadata-eval 76.7%

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

      16. *-rgt-identity 76.7%

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

      17. distribute-lft-in 76.7%

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

      18. *-rgt-identity 76.7%

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

   5. Applied egg-rr 76.7%

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

   6. Taylor expanded in x around inf 76.2%

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

      1. distribute-rgt-in 76.1%

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

      2. rgt-mult-inverse 76.2%

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

      3. metadata-eval 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*l* 99.4%

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

      6. rgt-mult-inverse 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -7.6e6 < x < 9.00000000000000031e-67

   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 77.0%

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

   if 9.00000000000000031e-67 < x < 5.6e15

   1. Initial program 99.4%

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

      1. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 76.1%

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

   5. Taylor expanded in x around 0 76.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7600000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 9: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x 1.3e-66) x (if (<= x 0.14) (* x (/ x y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x;
	} else if (x <= 0.14) {
		tmp = x * (x / y);
	} 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 <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-66) then
        tmp = x
    else if (x <= 0.14d0) then
        tmp = x * (x / y)
    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 <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x;
	} else if (x <= 0.14) {
		tmp = x * (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 1.3e-66:
		tmp = x
	elif x <= 0.14:
		tmp = x * (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = x;
	elseif (x <= 0.14)
		tmp = Float64(x * Float64(x / y));
	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 <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = x;
	elseif (x <= 0.14)
		tmp = x * (x / y);
	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, -1.0], t$95$0, If[LessEqual[x, 1.3e-66], x, If[LessEqual[x, 0.14], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 0.14000000000000001 < x

   1. Initial program 77.0%

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

      1. clear-num 76.9%

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

      2. associate-/r/ 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. associate-/r* 76.9%

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

      3. fma-udef 77.0%

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

      4. distribute-rgt-in 77.0%

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

      5. associate-*l/ 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. distribute-rgt-in 77.0%

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

      8. +-commutative 77.0%

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

      9. associate-*l/ 77.0%

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

      10. associate-/r/ 77.0%

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

      11. div-inv 76.8%

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

      12. *-commutative 76.8%

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

      13. associate-/r/ 76.9%

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

      14. div-inv 76.9%

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

      15. metadata-eval 76.9%

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

      16. *-rgt-identity 76.9%

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

      17. distribute-lft-in 76.9%

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

      18. *-rgt-identity 76.9%

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

   5. Applied egg-rr 76.9%

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

   6. Taylor expanded in x around inf 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. rgt-mult-inverse 76.4%

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

      3. metadata-eval 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*l* 99.4%

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

      6. rgt-mult-inverse 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -1 < x < 1.2999999999999999e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 75.4%

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

   if 1.2999999999999999e-66 < x < 0.14000000000000001

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.1%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. associate-/r/ 68.1%

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

   7. Applied egg-rr 68.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 10: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -390000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.014:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -390000.0)
     t_0
     (if (<= x 2.95e-67)
       (/ x (+ x 1.0))
       (if (<= x 0.014) (* x (/ x y)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -390000.0) {
		tmp = t_0;
	} else if (x <= 2.95e-67) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.014) {
		tmp = x * (x / y);
	} 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 <= (-390000.0d0)) then
        tmp = t_0
    else if (x <= 2.95d-67) then
        tmp = x / (x + 1.0d0)
    else if (x <= 0.014d0) then
        tmp = x * (x / y)
    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 <= -390000.0) {
		tmp = t_0;
	} else if (x <= 2.95e-67) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.014) {
		tmp = x * (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -390000.0:
		tmp = t_0
	elif x <= 2.95e-67:
		tmp = x / (x + 1.0)
	elif x <= 0.014:
		tmp = x * (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -390000.0)
		tmp = t_0;
	elseif (x <= 2.95e-67)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 0.014)
		tmp = Float64(x * Float64(x / y));
	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 <= -390000.0)
		tmp = t_0;
	elseif (x <= 2.95e-67)
		tmp = x / (x + 1.0);
	elseif (x <= 0.014)
		tmp = x * (x / y);
	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, -390000.0], t$95$0, If[LessEqual[x, 2.95e-67], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.014], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9e5 or 0.0140000000000000003 < x

   1. Initial program 77.0%

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

      1. clear-num 76.9%

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

      2. associate-/r/ 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. associate-/r* 76.9%

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

      3. fma-udef 77.0%

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

      4. distribute-rgt-in 77.0%

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

      5. associate-*l/ 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. distribute-rgt-in 77.0%

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

      8. +-commutative 77.0%

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

      9. associate-*l/ 77.0%

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

      10. associate-/r/ 77.0%

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

      11. div-inv 76.8%

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

      12. *-commutative 76.8%

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

      13. associate-/r/ 76.9%

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

      14. div-inv 76.9%

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

      15. metadata-eval 76.9%

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

      16. *-rgt-identity 76.9%

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

      17. distribute-lft-in 76.9%

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

      18. *-rgt-identity 76.9%

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

   5. Applied egg-rr 76.9%

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

   6. Taylor expanded in x around inf 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. rgt-mult-inverse 76.4%

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

      3. metadata-eval 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*l* 99.4%

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

      6. rgt-mult-inverse 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -3.9e5 < x < 2.95e-67

   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 77.0%

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

   if 2.95e-67 < x < 0.0140000000000000003

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.1%

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

   5. Taylor expanded in x around 0 68.3%

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

      1. associate-/r/ 68.1%

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

   7. Applied egg-rr 68.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -390000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.014:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 11: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -2100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -2100000.0)
     t_0
     (if (<= x 1.3e-66) (/ x (+ x 1.0)) (if (<= x 0.036) (/ x (/ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2100000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.036) {
		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 <= (-2100000.0d0)) then
        tmp = t_0
    else if (x <= 1.3d-66) then
        tmp = x / (x + 1.0d0)
    else if (x <= 0.036d0) 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 <= -2100000.0) {
		tmp = t_0;
	} else if (x <= 1.3e-66) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.036) {
		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 <= -2100000.0:
		tmp = t_0
	elif x <= 1.3e-66:
		tmp = x / (x + 1.0)
	elif x <= 0.036:
		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 <= -2100000.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 0.036)
		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 <= -2100000.0)
		tmp = t_0;
	elseif (x <= 1.3e-66)
		tmp = x / (x + 1.0);
	elseif (x <= 0.036)
		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, -2100000.0], t$95$0, If[LessEqual[x, 1.3e-66], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.036], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e6 or 0.0359999999999999973 < x

   1. Initial program 77.0%

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

      1. clear-num 76.9%

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

      2. associate-/r/ 77.0%

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

      3. fma-def 76.9%

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

   3. Applied egg-rr 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. associate-/r* 76.9%

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

      3. fma-udef 77.0%

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

      4. distribute-rgt-in 77.0%

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

      5. associate-*l/ 77.0%

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

      6. *-un-lft-identity 77.0%

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

      7. distribute-rgt-in 77.0%

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

      8. +-commutative 77.0%

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

      9. associate-*l/ 77.0%

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

      10. associate-/r/ 77.0%

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

      11. div-inv 76.8%

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

      12. *-commutative 76.8%

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

      13. associate-/r/ 76.9%

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

      14. div-inv 76.9%

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

      15. metadata-eval 76.9%

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

      16. *-rgt-identity 76.9%

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

      17. distribute-lft-in 76.9%

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

      18. *-rgt-identity 76.9%

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

   5. Applied egg-rr 76.9%

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

   6. Taylor expanded in x around inf 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. rgt-mult-inverse 76.4%

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

      3. metadata-eval 76.4%

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

      4. *-commutative 76.4%

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

      5. associate-*l* 99.4%

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

      6. rgt-mult-inverse 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. *-un-lft-identity 99.5%

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

      9. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if -2.1e6 < x < 1.2999999999999999e-66

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.0%

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

   if 1.2999999999999999e-66 < x < 0.0359999999999999973

   1. Initial program 99.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 74.1%

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

   5. Taylor expanded in x around 0 68.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2100000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.036:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 12: 49.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -100000000000.0) 1.0 (if (<= x 0.00125) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -100000000000.0) {
		tmp = 1.0;
	} else if (x <= 0.00125) {
		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 <= (-100000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 0.00125d0) 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 <= -100000000000.0) {
		tmp = 1.0;
	} else if (x <= 0.00125) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -100000000000.0:
		tmp = 1.0
	elif x <= 0.00125:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -100000000000.0)
		tmp = 1.0;
	elseif (x <= 0.00125)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -100000000000.0)
		tmp = 1.0;
	elseif (x <= 0.00125)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -100000000000.0], 1.0, If[LessEqual[x, 0.00125], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1e11 or 0.00125000000000000003 < x

   1. Initial program 76.8%

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

      1. clear-num 76.8%

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

      2. associate-/r/ 76.8%

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

      3. fma-def 76.8%

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

   3. Applied egg-rr 76.8%

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

   4. Taylor expanded in y around inf 30.2%

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

      1. +-commutative 30.2%

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

   6. Simplified 30.2%

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

   7. Taylor expanded in x around inf 30.1%

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

   if -1e11 < x < 0.00125000000000000003

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 69.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00125:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 14.5% 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 88.0%

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

   1. clear-num 87.9%

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

   2. associate-/r/ 87.9%

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

   3. fma-def 87.9%

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

3. Applied egg-rr 87.9%

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

4. Taylor expanded in y around inf 50.2%

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

   1. +-commutative 50.2%

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

6. Simplified 50.2%

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

7. Taylor expanded in x around inf 17.2%

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

8. Final simplification 17.2%

   \[\leadsto 1 \]

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 2023297 
(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)))