Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.8%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := y + \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (+ y (/ y x))))
   (if (<= t_0 -1e+100) (/ x t_1) (if (<= t_0 2e-34) t_0 (/ (+ x y) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = y + (y / x);
	double tmp;
	if (t_0 <= -1e+100) {
		tmp = x / t_1;
	} else if (t_0 <= 2e-34) {
		tmp = t_0;
	} else {
		tmp = (x + y) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = y + (y / x)
    if (t_0 <= (-1d+100)) then
        tmp = x / t_1
    else if (t_0 <= 2d-34) then
        tmp = t_0
    else
        tmp = (x + y) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = y + (y / x);
	double tmp;
	if (t_0 <= -1e+100) {
		tmp = x / t_1;
	} else if (t_0 <= 2e-34) {
		tmp = t_0;
	} else {
		tmp = (x + y) / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = y + (y / x)
	tmp = 0
	if t_0 <= -1e+100:
		tmp = x / t_1
	elif t_0 <= 2e-34:
		tmp = t_0
	else:
		tmp = (x + y) / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(y + Float64(y / x))
	tmp = 0.0
	if (t_0 <= -1e+100)
		tmp = Float64(x / t_1);
	elseif (t_0 <= 2e-34)
		tmp = t_0;
	else
		tmp = Float64(Float64(x + y) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = y + (y / x);
	tmp = 0.0;
	if (t_0 <= -1e+100)
		tmp = x / t_1;
	elseif (t_0 <= 2e-34)
		tmp = t_0;
	else
		tmp = (x + y) / t_1;
	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]}, Block[{t$95$1 = N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+100], N[(x / t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2e-34], t$95$0, N[(N[(x + y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1)) < -1.00000000000000002e100

   1. Initial program 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      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 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if -1.00000000000000002e100 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1)) < 1.99999999999999986e-34

   1. Initial program 99.9%

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

   if 1.99999999999999986e-34 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1))

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-/l* 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. +-commutative 99.9%

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

      8. remove-double-neg 99.9%

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

      9. unsub-neg 99.9%

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

      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 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l/ 99.9%

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

      3. frac-add 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/r* 99.9%

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

       2. div-inv 99.6%

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

       3. *-commutative 99.6%

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

       4. distribute-lft-out 99.6%

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

       5. distribute-lft-in 99.6%

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

       6. *-rgt-identity 99.6%

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

       7. un-div-inv 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. associate-/l* 99.7%

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

       4. associate-/r/ 100.0%

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

       5. *-inverses 100.0%

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

       6. *-lft-identity 100.0%

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

       7. +-commutative 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-43} \lor \neg \left(x \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x + y}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{x}{y}}{1 + \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.1e-43) (not (<= x 6e-23)))
   (/ (+ x y) (+ y (/ y x)))
   (+ x (/ (/ x y) (+ 1.0 (/ 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e-43) || !(x <= 6e-23)) {
		tmp = (x + y) / (y + (y / x));
	} else {
		tmp = x + ((x / y) / (1.0 + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.1d-43)) .or. (.not. (x <= 6d-23))) then
        tmp = (x + y) / (y + (y / x))
    else
        tmp = x + ((x / y) / (1.0d0 + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.1e-43) || !(x <= 6e-23)) {
		tmp = (x + y) / (y + (y / x));
	} else {
		tmp = x + ((x / y) / (1.0 + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.1e-43) or not (x <= 6e-23):
		tmp = (x + y) / (y + (y / x))
	else:
		tmp = x + ((x / y) / (1.0 + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.1e-43) || !(x <= 6e-23))
		tmp = Float64(Float64(x + y) / Float64(y + Float64(y / x)));
	else
		tmp = Float64(x + Float64(Float64(x / y) / Float64(1.0 + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.1e-43) || ~((x <= 6e-23)))
		tmp = (x + y) / (y + (y / x));
	else
		tmp = x + ((x / y) / (1.0 + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.1e-43], N[Not[LessEqual[x, 6e-23]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(x / y), $MachinePrecision] / N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000001e-43 or 6.00000000000000006e-23 < 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 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-/l/ 100.0%

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

      3. frac-add 99.9%

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

      4. *-un-lft-identity 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-/r* 99.9%

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

       2. div-inv 99.7%

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

       3. *-commutative 99.7%

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

       4. distribute-lft-out 99.7%

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

       5. distribute-lft-in 99.7%

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

       6. *-rgt-identity 99.7%

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

       7. un-div-inv 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

       3. associate-/l* 99.8%

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

       4. associate-/r/ 100.0%

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

       5. *-inverses 100.0%

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

       6. *-lft-identity 100.0%

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

       7. +-commutative 100.0%

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

   13. Simplified 100.0%

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

   if -2.1000000000000001e-43 < x < 6.00000000000000006e-23

   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* 99.6%

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

      3. remove-double-neg 99.6%

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

      4. neg-mul-1 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. +-commutative 99.6%

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

      8. remove-double-neg 99.6%

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

      9. unsub-neg 99.6%

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

      10. div-sub 99.6%

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

      11. *-inverses 99.6%

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

      12. div-sub 99.6%

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

      13. associate-/r* 99.6%

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

      14. *-commutative 99.6%

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

      15. neg-mul-1 99.6%

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

      16. remove-double-neg 99.6%

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

      17. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-43} \lor \neg \left(x \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{x + y}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{x}{y}}{1 + \frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+168} \lor \neg \left(y \leq 3.2 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.5e+168) (not (<= y 3.2e+149)))
   (/ x (+ x 1.0))
   (/ (+ x y) (+ y (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+168) || !(y <= 3.2e+149)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / (y + (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.5d+168)) .or. (.not. (y <= 3.2d+149))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = (x + y) / (y + (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.5e+168) || !(y <= 3.2e+149)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = (x + y) / (y + (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.5e+168) or not (y <= 3.2e+149):
		tmp = x / (x + 1.0)
	else:
		tmp = (x + y) / (y + (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.5e+168) || !(y <= 3.2e+149))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + y) / Float64(y + Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.5e+168) || ~((y <= 3.2e+149)))
		tmp = x / (x + 1.0);
	else
		tmp = (x + y) / (y + (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.5e+168], N[Not[LessEqual[y, 3.2e+149]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000001e168 or 3.2000000000000002e149 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 96.6%

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

      1. +-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -5.5000000000000001e168 < y < 3.2000000000000002e149

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-/l* 99.8%

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

      3. remove-double-neg 99.8%

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

      4. neg-mul-1 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

      8. remove-double-neg 99.8%

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

      9. unsub-neg 99.8%

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

      10. div-sub 99.8%

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

      11. *-inverses 99.8%

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

      12. div-sub 99.8%

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

      13. associate-/r* 99.8%

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

      14. *-commutative 99.8%

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

      15. neg-mul-1 99.8%

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

      16. remove-double-neg 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/l/ 99.8%

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

      3. frac-add 83.2%

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

      4. *-un-lft-identity 83.2%

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

      5. *-commutative 83.2%

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

      6. *-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

       1. associate-/r* 96.2%

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

       2. div-inv 95.9%

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

       3. *-commutative 95.9%

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

       4. distribute-lft-out 95.9%

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

       5. distribute-lft-in 95.9%

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

       6. *-rgt-identity 95.9%

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

       7. un-div-inv 95.9%

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

   11. Applied egg-rr 95.9%

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

       1. associate-*r/ 96.2%

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

       2. *-rgt-identity 96.2%

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

       3. associate-/l* 86.0%

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

       4. associate-/r/ 96.7%

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

       5. *-inverses 96.7%

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

       6. *-lft-identity 96.7%

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

       7. +-commutative 96.7%

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

   13. Simplified 96.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+168} \lor \neg \left(y \leq 3.2 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 3500\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -58000000000.0) (not (<= x 3500.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 3500.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-58000000000.0d0)) .or. (.not. (x <= 3500.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -58000000000.0) || !(x <= 3500.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -58000000000.0) or not (x <= 3500.0):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -58000000000.0) || !(x <= 3500.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -58000000000.0) || ~((x <= 3500.0)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -58000000000.0], N[Not[LessEqual[x, 3500.0]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e10 or 3500 < x

   1. Initial program 71.6%

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

   3. Taylor expanded in x around inf 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-div 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -5.8e10 < x < 3500

   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 77.5%

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

      1. +-commutative 77.5%

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

   5. Simplified 77.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58000000000 \lor \neg \left(x \leq 3500\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. *-commutative 85.6%

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

   2. associate-/l* 99.8%

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

   3. remove-double-neg 99.8%

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

   4. neg-mul-1 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-/r* 99.8%

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

   7. +-commutative 99.8%

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

   8. remove-double-neg 99.8%

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

   9. unsub-neg 99.8%

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

   10. div-sub 99.8%

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

   11. *-inverses 99.8%

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

   12. div-sub 99.8%

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

   13. associate-/r* 99.8%

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

   14. *-commutative 99.8%

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

   15. neg-mul-1 99.8%

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

   16. remove-double-neg 99.8%

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

   17. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-/r* 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+42} \lor \neg \left(x \leq 4.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.5e+42) (not (<= x 4.5e+95))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.5e+42) || !(x <= 4.5e+95)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.5d+42)) .or. (.not. (x <= 4.5d+95))) then
        tmp = x / y
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.5e+42) || !(x <= 4.5e+95)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.5e+42) or not (x <= 4.5e+95):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.5e+42) || !(x <= 4.5e+95))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.5e+42) || ~((x <= 4.5e+95)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.5e+42], N[Not[LessEqual[x, 4.5e+95]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000003e42 or 4.50000000000000017e95 < x

   1. Initial program 66.5%

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

   3. Taylor expanded in x around inf 82.7%

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

   if -2.50000000000000003e42 < x < 4.50000000000000017e95

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf 74.2%

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

      1. +-commutative 74.2%

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

   5. Simplified 74.2%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+42} \lor \neg \left(x \leq 4.5 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.6e-5))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.6e-5)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 3.6d-5))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.6e-5)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 3.6e-5):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 3.6e-5))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 3.6e-5)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 3.6e-5]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.60000000000000009e-5 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf 73.5%

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

   if -1 < x < 3.60000000000000009e-5

   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 76.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.6 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. *-commutative 85.6%

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

   2. associate-/l* 99.8%

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

   3. remove-double-neg 99.8%

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

   4. neg-mul-1 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-/r* 99.8%

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

   7. +-commutative 99.8%

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

   8. remove-double-neg 99.8%

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

   9. unsub-neg 99.8%

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

   10. div-sub 99.8%

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

   11. *-inverses 99.8%

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

   12. div-sub 99.8%

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

   13. associate-/r* 99.8%

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

   14. *-commutative 99.8%

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

   15. neg-mul-1 99.8%

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

   16. remove-double-neg 99.8%

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

   17. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 38.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 39.3%

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

4. Final simplification 39.3%

   \[\leadsto x \]
5. 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 2024026 
(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)))