Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 9.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 100000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 (- INFINITY))
     (/ x y)
     (if (<= t_0 100000000000.0) t_0 (/ (/ x y) (+ 1.0 (/ 1.0 x)))))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / y;
	} else if (t_0 <= 100000000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (1.0 + (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x / y;
	} else if (t_0 <= 100000000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (1.0 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x / y
	elif t_0 <= 100000000000.0:
		tmp = t_0
	else:
		tmp = (x / y) / (1.0 + (1.0 / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / y);
	elseif (t_0 <= 100000000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x / y;
	elseif (t_0 <= 100000000000.0)
		tmp = t_0;
	else
		tmp = (x / y) / (1.0 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 100000000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1)) < -inf.0

   1. Initial program 43.2%

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

   3. Taylor expanded in x around inf 100.0%

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1)) < 1e11

   1. Initial program 100.0%

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

   if 1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1))

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-/l* 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 99.9%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 100000000000:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + \frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 17.5:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 3850:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -520000000000.0)
     (+ 1.0 (/ (+ x -1.0) y))
     (if (<= x 2.8e-42)
       t_0
       (if (<= x 17.5)
         (/ x (+ y (/ y x)))
         (if (<= x 3850.0) t_0 (+ (/ x y) (+ 1.0 (/ -1.0 y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-42) {
		tmp = t_0;
	} else if (x <= 17.5) {
		tmp = x / (y + (y / x));
	} else if (x <= 3850.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) + (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) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-520000000000.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 2.8d-42) then
        tmp = t_0
    else if (x <= 17.5d0) then
        tmp = x / (y + (y / x))
    else if (x <= 3850.0d0) then
        tmp = t_0
    else
        tmp = (x / y) + (1.0d0 + ((-1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-42) {
		tmp = t_0;
	} else if (x <= 17.5) {
		tmp = x / (y + (y / x));
	} else if (x <= 3850.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) + (1.0 + (-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -520000000000.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 2.8e-42:
		tmp = t_0
	elif x <= 17.5:
		tmp = x / (y + (y / x))
	elif x <= 3850.0:
		tmp = t_0
	else:
		tmp = (x / y) + (1.0 + (-1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -520000000000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 2.8e-42)
		tmp = t_0;
	elseif (x <= 17.5)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 3850.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) + Float64(1.0 + Float64(-1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -520000000000.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 2.8e-42)
		tmp = t_0;
	elseif (x <= 17.5)
		tmp = x / (y + (y / x));
	elseif (x <= 3850.0)
		tmp = t_0;
	else
		tmp = (x / y) + (1.0 + (-1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-42], t$95$0, If[LessEqual[x, 17.5], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3850.0], t$95$0, N[(N[(x / y), $MachinePrecision] + N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.2e11

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -5.2e11 < x < 2.79999999999999998e-42 or 17.5 < x < 3850

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.9%

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

      1. +-commutative 79.9%

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

   5. Simplified 79.9%

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

   if 2.79999999999999998e-42 < x < 17.5

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. div-sub 99.7%

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

      6. *-inverses 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 71.9%

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

      1. distribute-lft-in 71.9%

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

      2. *-rgt-identity 71.9%

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

      3. associate-*r/ 71.9%

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

      4. *-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

   if 3850 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

   5. Applied egg-rr 97.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 17.5:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 3850:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 19:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + \frac{1}{x}\right)}\\ \mathbf{elif}\;x \leq 22000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -520000000000.0)
     (+ 1.0 (/ (+ x -1.0) y))
     (if (<= x 8.6e-44)
       t_0
       (if (<= x 19.0)
         (/ x (* y (+ 1.0 (/ 1.0 x))))
         (if (<= x 22000.0) t_0 (+ (/ x y) (+ 1.0 (/ -1.0 y)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 8.6e-44) {
		tmp = t_0;
	} else if (x <= 19.0) {
		tmp = x / (y * (1.0 + (1.0 / x)));
	} else if (x <= 22000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) + (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) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-520000000000.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 8.6d-44) then
        tmp = t_0
    else if (x <= 19.0d0) then
        tmp = x / (y * (1.0d0 + (1.0d0 / x)))
    else if (x <= 22000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) + (1.0d0 + ((-1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 8.6e-44) {
		tmp = t_0;
	} else if (x <= 19.0) {
		tmp = x / (y * (1.0 + (1.0 / x)));
	} else if (x <= 22000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) + (1.0 + (-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -520000000000.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 8.6e-44:
		tmp = t_0
	elif x <= 19.0:
		tmp = x / (y * (1.0 + (1.0 / x)))
	elif x <= 22000.0:
		tmp = t_0
	else:
		tmp = (x / y) + (1.0 + (-1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -520000000000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 8.6e-44)
		tmp = t_0;
	elseif (x <= 19.0)
		tmp = Float64(x / Float64(y * Float64(1.0 + Float64(1.0 / x))));
	elseif (x <= 22000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) + Float64(1.0 + Float64(-1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -520000000000.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 8.6e-44)
		tmp = t_0;
	elseif (x <= 19.0)
		tmp = x / (y * (1.0 + (1.0 / x)));
	elseif (x <= 22000.0)
		tmp = t_0;
	else
		tmp = (x / y) + (1.0 + (-1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-44], t$95$0, If[LessEqual[x, 19.0], N[(x / N[(y * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 22000.0], t$95$0, N[(N[(x / y), $MachinePrecision] + N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.2e11

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. unsub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -5.2e11 < x < 8.60000000000000027e-44 or 19 < x < 22000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.9%

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

      1. +-commutative 79.9%

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

   5. Simplified 79.9%

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

   if 8.60000000000000027e-44 < x < 19

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. div-sub 99.7%

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

      6. *-inverses 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 71.9%

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

   if 22000 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

   5. Applied egg-rr 97.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 19:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + \frac{1}{x}\right)}\\ \mathbf{elif}\;x \leq 22000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e+42)
   (/ x y)
   (if (<= x -1.0)
     1.0
     (if (<= x 1.85e-41)
       x
       (if (<= x 17.0) (* x (/ x y)) (if (<= x 10000.0) 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e+42) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.85e-41) {
		tmp = x;
	} else if (x <= 17.0) {
		tmp = x * (x / y);
	} else if (x <= 10000.0) {
		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 <= (-1.6d+42)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.85d-41) then
        tmp = x
    else if (x <= 17.0d0) then
        tmp = x * (x / y)
    else if (x <= 10000.0d0) 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 <= -1.6e+42) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.85e-41) {
		tmp = x;
	} else if (x <= 17.0) {
		tmp = x * (x / y);
	} else if (x <= 10000.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6e+42:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 1.85e-41:
		tmp = x
	elif x <= 17.0:
		tmp = x * (x / y)
	elif x <= 10000.0:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e+42)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.85e-41)
		tmp = x;
	elseif (x <= 17.0)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 10000.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.6e+42)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.85e-41)
		tmp = x;
	elseif (x <= 17.0)
		tmp = x * (x / y);
	elseif (x <= 10000.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e+42], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.85e-41], x, If[LessEqual[x, 17.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 10000.0], 1.0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.60000000000000001e42 or 1e4 < x

   1. Initial program 71.3%

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

   3. Taylor expanded in x around inf 85.7%

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

   if -1.60000000000000001e42 < x < -1 or 17 < x < 1e4

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf 75.7%

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

      1. +-commutative 75.7%

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

   5. Simplified 75.7%

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

      1. clear-num 75.5%

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

      2. inv-pow 75.5%

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

      3. +-commutative 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. unpow-1 75.5%

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

      2. +-commutative 75.5%

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

   9. Simplified 75.5%

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

   10. Taylor expanded in x around inf 54.3%

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

   if -1 < x < 1.8500000000000001e-41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.4%

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

   if 1.8500000000000001e-41 < x < 17

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. div-sub 99.7%

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

      6. *-inverses 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 71.9%

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

      1. distribute-lft-in 71.9%

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

      2. *-rgt-identity 71.9%

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

      3. associate-*r/ 71.9%

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

      4. *-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in x around 0 62.7%

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

      1. associate-/r/ 62.7%

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

   10. Applied egg-rr 62.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 22000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))) (t_1 (/ x (+ x 1.0))))
   (if (<= x -520000000000.0)
     t_0
     (if (<= x 1.8e-41)
       t_1
       (if (<= x 6.2e-16) (* x (/ x y)) (if (<= x 22000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = t_0;
	} else if (x <= 1.8e-41) {
		tmp = t_1;
	} else if (x <= 6.2e-16) {
		tmp = x * (x / y);
	} else if (x <= 22000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((x + (-1.0d0)) / y)
    t_1 = x / (x + 1.0d0)
    if (x <= (-520000000000.0d0)) then
        tmp = t_0
    else if (x <= 1.8d-41) then
        tmp = t_1
    else if (x <= 6.2d-16) then
        tmp = x * (x / y)
    else if (x <= 22000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = t_0;
	} else if (x <= 1.8e-41) {
		tmp = t_1;
	} else if (x <= 6.2e-16) {
		tmp = x * (x / y);
	} else if (x <= 22000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -520000000000.0:
		tmp = t_0
	elif x <= 1.8e-41:
		tmp = t_1
	elif x <= 6.2e-16:
		tmp = x * (x / y)
	elif x <= 22000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -520000000000.0)
		tmp = t_0;
	elseif (x <= 1.8e-41)
		tmp = t_1;
	elseif (x <= 6.2e-16)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 22000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -520000000000.0)
		tmp = t_0;
	elseif (x <= 1.8e-41)
		tmp = t_1;
	elseif (x <= 6.2e-16)
		tmp = x * (x / y);
	elseif (x <= 22000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000000.0], t$95$0, If[LessEqual[x, 1.8e-41], t$95$1, If[LessEqual[x, 6.2e-16], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 22000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e11 or 22000 < x

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf 98.7%

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

   4. Taylor expanded in y around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. neg-mul-1 98.7%

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

      4. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   if -5.2e11 < x < 1.8e-41 or 6.2000000000000002e-16 < x < 22000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.4%

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

      1. +-commutative 79.4%

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

   5. Simplified 79.4%

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

   if 1.8e-41 < x < 6.2000000000000002e-16

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. div-sub 99.6%

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

      6. *-inverses 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 78.1%

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

      1. distribute-lft-in 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. associate-*r/ 78.1%

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

      4. *-rgt-identity 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in x around 0 78.1%

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

      1. associate-/r/ 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 22000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 15:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 6100:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (+ x -1.0) y))) (t_1 (/ x (+ x 1.0))))
   (if (<= x -520000000000.0)
     t_0
     (if (<= x 9.5e-44)
       t_1
       (if (<= x 15.0) (/ x (+ y (/ y x))) (if (<= x 6100.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = t_0;
	} else if (x <= 9.5e-44) {
		tmp = t_1;
	} else if (x <= 15.0) {
		tmp = x / (y + (y / x));
	} else if (x <= 6100.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((x + (-1.0d0)) / y)
    t_1 = x / (x + 1.0d0)
    if (x <= (-520000000000.0d0)) then
        tmp = t_0
    else if (x <= 9.5d-44) then
        tmp = t_1
    else if (x <= 15.0d0) then
        tmp = x / (y + (y / x))
    else if (x <= 6100.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x + -1.0) / y);
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -520000000000.0) {
		tmp = t_0;
	} else if (x <= 9.5e-44) {
		tmp = t_1;
	} else if (x <= 15.0) {
		tmp = x / (y + (y / x));
	} else if (x <= 6100.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x + -1.0) / y)
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -520000000000.0:
		tmp = t_0
	elif x <= 9.5e-44:
		tmp = t_1
	elif x <= 15.0:
		tmp = x / (y + (y / x))
	elif x <= 6100.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x + -1.0) / y))
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -520000000000.0)
		tmp = t_0;
	elseif (x <= 9.5e-44)
		tmp = t_1;
	elseif (x <= 15.0)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	elseif (x <= 6100.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x + -1.0) / y);
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -520000000000.0)
		tmp = t_0;
	elseif (x <= 9.5e-44)
		tmp = t_1;
	elseif (x <= 15.0)
		tmp = x / (y + (y / x));
	elseif (x <= 6100.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000000.0], t$95$0, If[LessEqual[x, 9.5e-44], t$95$1, If[LessEqual[x, 15.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6100.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e11 or 6100 < x

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf 98.7%

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

   4. Taylor expanded in y around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. neg-mul-1 98.7%

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

      4. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   if -5.2e11 < x < 9.49999999999999924e-44 or 15 < x < 6100

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 79.9%

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

      1. +-commutative 79.9%

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

   5. Simplified 79.9%

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

   if 9.49999999999999924e-44 < x < 15

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. div-sub 99.7%

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

      6. *-inverses 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 71.9%

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

      1. distribute-lft-in 71.9%

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

      2. *-rgt-identity 71.9%

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

      3. associate-*r/ 71.9%

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

      4. *-rgt-identity 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 15:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 6100:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 26500000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4.4e+44)
     (/ x y)
     (if (<= x 2e-41)
       t_0
       (if (<= x 5e-18) (* x (/ x y)) (if (<= x 26500000.0) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.4e+44) {
		tmp = x / y;
	} else if (x <= 2e-41) {
		tmp = t_0;
	} else if (x <= 5e-18) {
		tmp = x * (x / y);
	} else if (x <= 26500000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-4.4d+44)) then
        tmp = x / y
    else if (x <= 2d-41) then
        tmp = t_0
    else if (x <= 5d-18) then
        tmp = x * (x / y)
    else if (x <= 26500000.0d0) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.4e+44) {
		tmp = x / y;
	} else if (x <= 2e-41) {
		tmp = t_0;
	} else if (x <= 5e-18) {
		tmp = x * (x / y);
	} else if (x <= 26500000.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4.4e+44:
		tmp = x / y
	elif x <= 2e-41:
		tmp = t_0
	elif x <= 5e-18:
		tmp = x * (x / y)
	elif x <= 26500000.0:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.4e+44)
		tmp = Float64(x / y);
	elseif (x <= 2e-41)
		tmp = t_0;
	elseif (x <= 5e-18)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 26500000.0)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -4.4e+44)
		tmp = x / y;
	elseif (x <= 2e-41)
		tmp = t_0;
	elseif (x <= 5e-18)
		tmp = x * (x / y);
	elseif (x <= 26500000.0)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e+44], N[(x / y), $MachinePrecision], If[LessEqual[x, 2e-41], t$95$0, If[LessEqual[x, 5e-18], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 26500000.0], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.39999999999999991e44 or 2.65e7 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in x around inf 87.6%

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

   if -4.39999999999999991e44 < x < 2.00000000000000001e-41 or 5.00000000000000036e-18 < x < 2.65e7

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf 77.8%

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

      1. +-commutative 77.8%

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

   5. Simplified 77.8%

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

   if 2.00000000000000001e-41 < x < 5.00000000000000036e-18

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. div-sub 99.6%

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

      6. *-inverses 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 78.1%

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

      1. distribute-lft-in 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. associate-*r/ 78.1%

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

      4. *-rgt-identity 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in x around 0 78.1%

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

      1. associate-/r/ 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 26500000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 12800000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -4.3e+42)
     (/ x y)
     (if (<= x 7e-42)
       t_0
       (if (<= x 5e-13)
         (* x (/ x y))
         (if (<= x 12800000.0) t_0 (/ (+ x -1.0) y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.3e+42) {
		tmp = x / y;
	} else if (x <= 7e-42) {
		tmp = t_0;
	} else if (x <= 5e-13) {
		tmp = x * (x / y);
	} else if (x <= 12800000.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-4.3d+42)) then
        tmp = x / y
    else if (x <= 7d-42) then
        tmp = t_0
    else if (x <= 5d-13) then
        tmp = x * (x / y)
    else if (x <= 12800000.0d0) then
        tmp = t_0
    else
        tmp = (x + (-1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -4.3e+42) {
		tmp = x / y;
	} else if (x <= 7e-42) {
		tmp = t_0;
	} else if (x <= 5e-13) {
		tmp = x * (x / y);
	} else if (x <= 12800000.0) {
		tmp = t_0;
	} else {
		tmp = (x + -1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -4.3e+42:
		tmp = x / y
	elif x <= 7e-42:
		tmp = t_0
	elif x <= 5e-13:
		tmp = x * (x / y)
	elif x <= 12800000.0:
		tmp = t_0
	else:
		tmp = (x + -1.0) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.3e+42)
		tmp = Float64(x / y);
	elseif (x <= 7e-42)
		tmp = t_0;
	elseif (x <= 5e-13)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 12800000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x + -1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -4.3e+42)
		tmp = x / y;
	elseif (x <= 7e-42)
		tmp = t_0;
	elseif (x <= 5e-13)
		tmp = x * (x / y);
	elseif (x <= 12800000.0)
		tmp = t_0;
	else
		tmp = (x + -1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e+42], N[(x / y), $MachinePrecision], If[LessEqual[x, 7e-42], t$95$0, If[LessEqual[x, 5e-13], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12800000.0], t$95$0, N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.2999999999999998e42

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf 86.5%

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

   if -4.2999999999999998e42 < x < 7.0000000000000004e-42 or 4.9999999999999999e-13 < x < 1.28e7

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf 77.8%

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

      1. +-commutative 77.8%

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

   5. Simplified 77.8%

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

   if 7.0000000000000004e-42 < x < 4.9999999999999999e-13

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. div-sub 99.6%

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

      6. *-inverses 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 78.1%

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

      1. distribute-lft-in 78.1%

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

      2. *-rgt-identity 78.1%

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

      3. associate-*r/ 78.1%

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

      4. *-rgt-identity 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in x around 0 78.1%

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

      1. associate-/r/ 78.1%

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

   10. Applied egg-rr 78.1%

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

   if 1.28e7 < x

   1. Initial program 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-/l* 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. div-sub 100.0%

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

      6. *-inverses 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 89.2%

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

      1. distribute-lft-in 89.3%

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

      2. *-rgt-identity 89.3%

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

      3. associate-*r/ 89.3%

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

      4. *-rgt-identity 89.3%

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

   7. Simplified 89.3%

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

   8. Taylor expanded in x around inf 89.2%

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

   9. Taylor expanded in y around 0 89.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 12800000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1520:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+47)
   (/ x y)
   (if (<= x -1.0) 1.0 (if (<= x 1520.0) x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+47) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1520.0) {
		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 <= (-1.05d+47)) then
        tmp = x / y
    else if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1520.0d0) 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 <= -1.05e+47) {
		tmp = x / y;
	} else if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1520.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e+47:
		tmp = x / y
	elif x <= -1.0:
		tmp = 1.0
	elif x <= 1520.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+47)
		tmp = Float64(x / y);
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1520.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+47)
		tmp = x / y;
	elseif (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1520.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e+47], N[(x / y), $MachinePrecision], If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1520.0], x, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e47 or 1520 < x

   1. Initial program 71.3%

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

   3. Taylor expanded in x around inf 85.7%

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

   if -1.05e47 < x < -1

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf 70.1%

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

      1. +-commutative 70.1%

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

   5. Simplified 70.1%

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

      1. clear-num 70.1%

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

      2. inv-pow 70.1%

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

      3. +-commutative 70.1%

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

   7. Applied egg-rr 70.1%

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

      1. unpow-1 70.1%

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

      2. +-commutative 70.1%

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

   9. Simplified 70.1%

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

   10. Taylor expanded in x around inf 60.0%

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

   if -1 < x < 1520

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1520:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.1% accurate, 1.0× 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 74.6%

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

   3. Taylor expanded in y around inf 22.0%

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

      1. +-commutative 22.0%

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

   5. Simplified 22.0%

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

      1. clear-num 22.0%

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

      2. inv-pow 22.0%

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

      3. +-commutative 22.0%

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

   7. Applied egg-rr 22.0%

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

      1. unpow-1 22.0%

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

      2. +-commutative 22.0%

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

   9. Simplified 22.0%

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

   10. Taylor expanded in x around inf 18.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 74.9%

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

4. Final simplification 48.4%

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

Alternative 11: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - N[(-1.0 / x), $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. *-commutative 88.0%

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

   2. associate-/l* 99.8%

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

   3. metadata-eval 99.8%

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

   4. sub-neg 99.8%

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

   5. div-sub 99.8%

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

   6. *-inverses 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 12: 14.1% 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. Add Preprocessing

3. Taylor expanded in y around inf 50.7%

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

   1. +-commutative 50.7%

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

5. Simplified 50.7%

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

   1. clear-num 50.6%

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

   2. inv-pow 50.6%

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

   3. +-commutative 50.6%

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

7. Applied egg-rr 50.6%

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

   1. unpow-1 50.6%

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

   2. +-commutative 50.6%

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

9. Simplified 50.6%

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

10. Taylor expanded in x around inf 10.8%

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

11. Final simplification 10.8%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer target: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(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)))