Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 9.5s

Alternatives: 11

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: 88.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ t\_0 + t\_0 \cdot \frac{x}{y} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (+ t_0 (* t_0 (/ x y)))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (x + 1.0d0)
    code = t_0 + (t_0 * (x / y))
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return t_0 + (t_0 * (x / y));
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	return t_0 + (t_0 * (x / y))

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	return Float64(t_0 + Float64(t_0 * Float64(x / y)))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = t_0 + (t_0 * (x / y));
end

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. associate-/r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-rgt-in 99.9%

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

   4. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+89} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -35000000000.0)
   (/ x y)
   (if (<= x -2.7e-73)
     (* x (/ x y))
     (if (<= x 2.6e-17)
       x
       (if (or (<= x 1.5e+89) (not (<= x 1.55e+95))) (/ x y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = x / y;
	} else if (x <= -2.7e-73) {
		tmp = x * (x / y);
	} else if (x <= 2.6e-17) {
		tmp = x;
	} else if ((x <= 1.5e+89) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} 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 <= (-35000000000.0d0)) then
        tmp = x / y
    else if (x <= (-2.7d-73)) then
        tmp = x * (x / y)
    else if (x <= 2.6d-17) then
        tmp = x
    else if ((x <= 1.5d+89) .or. (.not. (x <= 1.55d+95))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = x / y;
	} else if (x <= -2.7e-73) {
		tmp = x * (x / y);
	} else if (x <= 2.6e-17) {
		tmp = x;
	} else if ((x <= 1.5e+89) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -35000000000.0:
		tmp = x / y
	elif x <= -2.7e-73:
		tmp = x * (x / y)
	elif x <= 2.6e-17:
		tmp = x
	elif (x <= 1.5e+89) or not (x <= 1.55e+95):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -35000000000.0)
		tmp = Float64(x / y);
	elseif (x <= -2.7e-73)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.6e-17)
		tmp = x;
	elseif ((x <= 1.5e+89) || !(x <= 1.55e+95))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -35000000000.0)
		tmp = x / y;
	elseif (x <= -2.7e-73)
		tmp = x * (x / y);
	elseif (x <= 2.6e-17)
		tmp = x;
	elseif ((x <= 1.5e+89) || ~((x <= 1.55e+95)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -35000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -2.7e-73], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-17], x, If[Or[LessEqual[x, 1.5e+89], N[Not[LessEqual[x, 1.55e+95]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.5e10 or 2.60000000000000003e-17 < x < 1.50000000000000006e89 or 1.5500000000000001e95 < x

   1. Initial program 72.9%

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

   5. Taylor expanded in x around inf 77.6%

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

   if -3.5e10 < x < -2.69999999999999994e-73

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.7%

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

      3. remove-double-neg 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-/r* 99.7%

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

      7. +-commutative 99.7%

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

      8. remove-double-neg 99.7%

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

      9. unsub-neg 99.7%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. distribute-lft-in 75.0%

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

      2. *-rgt-identity 75.0%

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

      3. associate-*r/ 75.3%

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

      4. *-rgt-identity 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in x around 0 67.7%

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

      1. associate-/r/ 68.0%

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

   10. Applied egg-rr 68.0%

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

   if -2.69999999999999994e-73 < x < 2.60000000000000003e-17

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 82.7%

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

   if 1.50000000000000006e89 < x < 1.5500000000000001e95

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+89} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.65 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -35000000000.0)
   (/ x y)
   (if (<= x -4.3e-72)
     (* x (/ x y))
     (if (<= x 2.6e-17)
       (/ x (+ x 1.0))
       (if (or (<= x 8.5e+90) (not (<= x 1.65e+95))) (/ x y) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = x / y;
	} else if (x <= -4.3e-72) {
		tmp = x * (x / y);
	} else if (x <= 2.6e-17) {
		tmp = x / (x + 1.0);
	} else if ((x <= 8.5e+90) || !(x <= 1.65e+95)) {
		tmp = x / y;
	} 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 <= (-35000000000.0d0)) then
        tmp = x / y
    else if (x <= (-4.3d-72)) then
        tmp = x * (x / y)
    else if (x <= 2.6d-17) then
        tmp = x / (x + 1.0d0)
    else if ((x <= 8.5d+90) .or. (.not. (x <= 1.65d+95))) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -35000000000.0) {
		tmp = x / y;
	} else if (x <= -4.3e-72) {
		tmp = x * (x / y);
	} else if (x <= 2.6e-17) {
		tmp = x / (x + 1.0);
	} else if ((x <= 8.5e+90) || !(x <= 1.65e+95)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -35000000000.0:
		tmp = x / y
	elif x <= -4.3e-72:
		tmp = x * (x / y)
	elif x <= 2.6e-17:
		tmp = x / (x + 1.0)
	elif (x <= 8.5e+90) or not (x <= 1.65e+95):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -35000000000.0)
		tmp = Float64(x / y);
	elseif (x <= -4.3e-72)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.6e-17)
		tmp = Float64(x / Float64(x + 1.0));
	elseif ((x <= 8.5e+90) || !(x <= 1.65e+95))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -35000000000.0)
		tmp = x / y;
	elseif (x <= -4.3e-72)
		tmp = x * (x / y);
	elseif (x <= 2.6e-17)
		tmp = x / (x + 1.0);
	elseif ((x <= 8.5e+90) || ~((x <= 1.65e+95)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -35000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, -4.3e-72], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-17], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 8.5e+90], N[Not[LessEqual[x, 1.65e+95]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.5e10 or 2.60000000000000003e-17 < x < 8.5000000000000002e90 or 1.6499999999999999e95 < x

   1. Initial program 72.9%

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

   5. Taylor expanded in x around inf 77.6%

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

   if -3.5e10 < x < -4.2999999999999999e-72

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.7%

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

      3. remove-double-neg 99.7%

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

      4. neg-mul-1 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-/r* 99.7%

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

      7. +-commutative 99.7%

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

      8. remove-double-neg 99.7%

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

      9. unsub-neg 99.7%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. distribute-lft-in 75.0%

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

      2. *-rgt-identity 75.0%

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

      3. associate-*r/ 75.3%

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

      4. *-rgt-identity 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in x around 0 67.7%

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

      1. associate-/r/ 68.0%

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

   10. Applied egg-rr 68.0%

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

   if -4.2999999999999999e-72 < x < 2.60000000000000003e-17

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 82.7%

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

   if 8.5000000000000002e90 < x < 1.6499999999999999e95

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.65 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 2.6e-17)
     x
     (if (or (<= x 5.5e+90) (not (<= x 1.55e+95))) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 2.6e-17) {
		tmp = x;
	} else if ((x <= 5.5e+90) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} 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 = x / y
    else if (x <= 2.6d-17) then
        tmp = x
    else if ((x <= 5.5d+90) .or. (.not. (x <= 1.55d+95))) then
        tmp = x / y
    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 = x / y;
	} else if (x <= 2.6e-17) {
		tmp = x;
	} else if ((x <= 5.5e+90) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 2.6e-17:
		tmp = x
	elif (x <= 5.5e+90) or not (x <= 1.55e+95):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 2.6e-17)
		tmp = x;
	elseif ((x <= 5.5e+90) || !(x <= 1.55e+95))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 2.6e-17)
		tmp = x;
	elseif ((x <= 5.5e+90) || ~((x <= 1.55e+95)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.6e-17], x, If[Or[LessEqual[x, 5.5e+90], N[Not[LessEqual[x, 1.55e+95]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 2.60000000000000003e-17 < x < 5.49999999999999999e90 or 1.5500000000000001e95 < x

   1. Initial program 73.3%

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

   5. Taylor expanded in x around inf 76.4%

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

   if -1 < x < 2.60000000000000003e-17

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.0%

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

   if 5.49999999999999999e90 < x < 1.5500000000000001e95

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 26:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 26.0)
     (* x (+ 1.0 (/ x y)))
     (if (or (<= x 8.5e+90) (not (<= x 1.55e+95))) (/ x y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 26.0) {
		tmp = x * (1.0 + (x / y));
	} else if ((x <= 8.5e+90) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} 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 = x / y
    else if (x <= 26.0d0) then
        tmp = x * (1.0d0 + (x / y))
    else if ((x <= 8.5d+90) .or. (.not. (x <= 1.55d+95))) then
        tmp = x / y
    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 = x / y;
	} else if (x <= 26.0) {
		tmp = x * (1.0 + (x / y));
	} else if ((x <= 8.5e+90) || !(x <= 1.55e+95)) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 26.0:
		tmp = x * (1.0 + (x / y))
	elif (x <= 8.5e+90) or not (x <= 1.55e+95):
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 26.0)
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	elseif ((x <= 8.5e+90) || !(x <= 1.55e+95))
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 26.0)
		tmp = x * (1.0 + (x / y));
	elseif ((x <= 8.5e+90) || ~((x <= 1.55e+95)))
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 26.0], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 8.5e+90], N[Not[LessEqual[x, 1.55e+95]], $MachinePrecision]], N[(x / y), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 26 < x < 8.5000000000000002e90 or 1.5500000000000001e95 < x

   1. Initial program 72.7%

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

   5. Taylor expanded in x around inf 78.0%

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

   if -1 < x < 26

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.4%

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

      11. *-inverses 99.4%

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

      12. div-sub 99.4%

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

      13. associate-/r* 99.4%

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

      14. *-commutative 99.4%

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

      15. neg-mul-1 99.4%

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

      16. remove-double-neg 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. associate-/r/ 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. +-commutative 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 8.5000000000000002e90 < x < 1.5500000000000001e95

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 26:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90} \lor \neg \left(x \leq 1.55 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + \frac{y}{x}}\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (/ y x)))))
   (if (<= x -6.4e-16)
     t_0
     (if (<= x 2.6e-17)
       (* x (+ 1.0 (/ x y)))
       (if (<= x 8.5e+90) t_0 (if (<= x 1.55e+95) 1.0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (y / x));
	double tmp;
	if (x <= -6.4e-16) {
		tmp = t_0;
	} else if (x <= 2.6e-17) {
		tmp = x * (1.0 + (x / y));
	} else if (x <= 8.5e+90) {
		tmp = t_0;
	} else if (x <= 1.55e+95) {
		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) :: t_0
    real(8) :: tmp
    t_0 = x / (y + (y / x))
    if (x <= (-6.4d-16)) then
        tmp = t_0
    else if (x <= 2.6d-17) then
        tmp = x * (1.0d0 + (x / y))
    else if (x <= 8.5d+90) then
        tmp = t_0
    else if (x <= 1.55d+95) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + (y / x));
	double tmp;
	if (x <= -6.4e-16) {
		tmp = t_0;
	} else if (x <= 2.6e-17) {
		tmp = x * (1.0 + (x / y));
	} else if (x <= 8.5e+90) {
		tmp = t_0;
	} else if (x <= 1.55e+95) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (y / x))
	tmp = 0
	if x <= -6.4e-16:
		tmp = t_0
	elif x <= 2.6e-17:
		tmp = x * (1.0 + (x / y))
	elif x <= 8.5e+90:
		tmp = t_0
	elif x <= 1.55e+95:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(y / x)))
	tmp = 0.0
	if (x <= -6.4e-16)
		tmp = t_0;
	elseif (x <= 2.6e-17)
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	elseif (x <= 8.5e+90)
		tmp = t_0;
	elseif (x <= 1.55e+95)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (y / x));
	tmp = 0.0;
	if (x <= -6.4e-16)
		tmp = t_0;
	elseif (x <= 2.6e-17)
		tmp = x * (1.0 + (x / y));
	elseif (x <= 8.5e+90)
		tmp = t_0;
	elseif (x <= 1.55e+95)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.4e-16], t$95$0, If[LessEqual[x, 2.6e-17], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+90], t$95$0, If[LessEqual[x, 1.55e+95], 1.0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.40000000000000046e-16 or 2.60000000000000003e-17 < x < 8.5000000000000002e90

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 99.9%

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

      11. *-inverses 99.9%

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

      12. div-sub 99.9%

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

      13. associate-/r* 99.9%

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

      14. *-commutative 99.9%

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

      15. neg-mul-1 99.9%

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

      16. remove-double-neg 99.9%

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

      17. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 73.9%

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

      1. distribute-lft-in 73.9%

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

      2. *-rgt-identity 73.9%

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

      3. associate-*r/ 73.9%

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

      4. *-rgt-identity 73.9%

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

   7. Simplified 73.9%

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

   if -6.40000000000000046e-16 < x < 2.60000000000000003e-17

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.4%

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

      3. remove-double-neg 99.4%

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

      4. neg-mul-1 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. +-commutative 99.4%

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

      8. remove-double-neg 99.4%

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

      9. unsub-neg 99.4%

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

      10. div-sub 99.4%

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

      11. *-inverses 99.4%

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

      12. div-sub 99.4%

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

      13. associate-/r* 99.4%

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

      14. *-commutative 99.4%

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

      15. neg-mul-1 99.4%

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

      16. remove-double-neg 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-/r/ 99.9%

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

      2. /-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 8.5000000000000002e90 < x < 1.5500000000000001e95

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

   if 1.5500000000000001e95 < x

   1. Initial program 54.0%

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

   5. Taylor expanded in x around inf 91.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+95}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35000 \lor \neg \left(x \leq 40\right):\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{\frac{x}{x + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -35000.0) (not (<= x 40.0)))
   (+ (+ 1.0 (/ x y)) (/ -1.0 y))
   (+ x (* x (/ (/ x (+ x 1.0)) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -35000.0) || !(x <= 40.0)) {
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	} else {
		tmp = x + (x * ((x / (x + 1.0)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-35000.0d0)) .or. (.not. (x <= 40.0d0))) then
        tmp = (1.0d0 + (x / y)) + ((-1.0d0) / y)
    else
        tmp = x + (x * ((x / (x + 1.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -35000.0) || !(x <= 40.0)) {
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	} else {
		tmp = x + (x * ((x / (x + 1.0)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -35000.0) or not (x <= 40.0):
		tmp = (1.0 + (x / y)) + (-1.0 / y)
	else:
		tmp = x + (x * ((x / (x + 1.0)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -35000.0) || !(x <= 40.0))
		tmp = Float64(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y));
	else
		tmp = Float64(x + Float64(x * Float64(Float64(x / Float64(x + 1.0)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -35000.0) || ~((x <= 40.0)))
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	else
		tmp = x + (x * ((x / (x + 1.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -35000.0], N[Not[LessEqual[x, 40.0]], $MachinePrecision]], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x * N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -35000 or 40 < x

   1. Initial program 73.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

   if -35000 < x < 40

   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. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. frac-2neg 99.9%

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

      3. associate-*r/ 97.6%

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

   8. Applied egg-rr 97.6%

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

   9. Taylor expanded in x around 0 97.6%

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

       1. associate-/l* 99.8%

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

       2. frac-2neg 99.8%

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

       3. associate-/r/ 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35000 \lor \neg \left(x \leq 40\right):\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{\frac{x}{x + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;t\_0 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (or (<= x -1.0) (not (<= x 1.25))) (+ t_0 (/ -1.0 y)) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if ((x <= -1.0) || !(x <= 1.25)) {
		tmp = t_0 + (-1.0 / y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = t_0 + ((-1.0d0) / y)
    else
        tmp = x * 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) || !(x <= 1.25)) {
		tmp = t_0 + (-1.0 / y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if (x <= -1.0) or not (x <= 1.25):
		tmp = t_0 + (-1.0 / y)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.25))
		tmp = Float64(t_0 + Float64(-1.0 / y));
	else
		tmp = Float64(x * 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) || ~((x <= 1.25)))
		tmp = t_0 + (-1.0 / y);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(t$95$0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 73.8%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.4%

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

   if -1 < x < 1.25

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.4%

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

      11. *-inverses 99.4%

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

      12. div-sub 99.4%

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

      13. associate-/r* 99.4%

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

      14. *-commutative 99.4%

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

      15. neg-mul-1 99.4%

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

      16. remove-double-neg 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.2%

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

      1. associate-/r/ 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. +-commutative 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.65) 1.0 (if (<= x 2.6e-17) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		tmp = 1.0;
	} else if (x <= 2.6e-17) {
		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 <= (-0.65d0)) then
        tmp = 1.0d0
    else if (x <= 2.6d-17) 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 <= -0.65) {
		tmp = 1.0;
	} else if (x <= 2.6e-17) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.65:
		tmp = 1.0
	elif x <= 2.6e-17:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.65)
		tmp = 1.0;
	elseif (x <= 2.6e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = 1.0;
	elseif (x <= 2.6e-17)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.65], 1.0, If[LessEqual[x, 2.6e-17], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022 or 2.60000000000000003e-17 < x

   1. Initial program 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 26.3%

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

   6. Taylor expanded in x around inf 25.7%

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

   if -0.650000000000000022 < x < 2.60000000000000003e-17

   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. Add Preprocessing

   5. Taylor expanded in x around 0 75.6%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 86.4%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 14.3% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 86.4%

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

   1. *-commutative 86.4%

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

   2. associate-/l* 99.7%

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

   3. remove-double-neg 99.7%

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

   4. neg-mul-1 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-/r* 99.7%

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

   7. +-commutative 99.7%

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

   8. remove-double-neg 99.7%

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

   9. unsub-neg 99.7%

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

   10. div-sub 99.7%

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

   11. *-inverses 99.7%

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

   12. div-sub 99.7%

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

   13. associate-/r* 99.7%

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

   14. *-commutative 99.7%

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

   15. neg-mul-1 99.7%

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

   16. remove-double-neg 99.7%

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around inf 49.2%

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

6. Taylor expanded in x around inf 15.5%

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

7. Final simplification 15.5%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(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)))