Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.0% → 99.9%

Time: 4.8s

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: 88.0% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 10^{+16}\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4e+17) (not (<= x 1e+16)))
   (+ 1.0 (/ (+ x -1.0) y))
   (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4e+17) || !(x <= 1e+16)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = (x * (1.0 + (x / y))) / (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 <= (-4d+17)) .or. (.not. (x <= 1d+16))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4e+17) || !(x <= 1e+16)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = (x * (1.0 + (x / y))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4e+17) or not (x <= 1e+16):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = (x * (1.0 + (x / y))) / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4e+17) || !(x <= 1e+16))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4e+17) || ~((x <= 1e+16)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = (x * (1.0 + (x / y))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4e+17], N[Not[LessEqual[x, 1e+16]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e17 or 1e16 < x

   1. Initial program 76.3%

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

   3. Taylor expanded in x around inf 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   if -4e17 < x < 1e16

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 10^{+16}\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   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. distribute-lft-in 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-rgt-identity 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 235\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 235.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (+ x (/ x (+ y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.2) || !(x <= 235.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (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 ((x <= (-4.2d0)) .or. (.not. (x <= 235.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + (x / (y + (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.2) || !(x <= 235.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (y + (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.2) or not (x <= 235.0):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x + (x / (y + (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 235.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x + Float64(x / Float64(y + Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.2) || ~((x <= 235.0)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x + (x / (y + (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 235.0]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 235 < x

   1. Initial program 77.8%

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

   3. Taylor expanded in x around inf 98.7%

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

   4. Taylor expanded in y around -inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. sub-neg 98.8%

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

   6. Simplified 98.8%

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

   if -4.20000000000000018 < x < 235

   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 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. *-rgt-identity 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 235\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ x -1.0) y)
   (if (<= x 5e-6)
     (+ x (/ x (/ y x)))
     (if (<= x 9.5e+82) (/ x (+ x 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 5e-6) {
		tmp = x + (x / (y / x));
	} else if (x <= 9.5e+82) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 5d-6) then
        tmp = x + (x / (y / x))
    else if (x <= 9.5d+82) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 5e-6) {
		tmp = x + (x / (y / x));
	} else if (x <= 9.5e+82) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (x + -1.0) / y
	elif x <= 5e-6:
		tmp = x + (x / (y / x))
	elif x <= 9.5e+82:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 5e-6)
		tmp = Float64(x + Float64(x / Float64(y / x)));
	elseif (x <= 9.5e+82)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 5e-6)
		tmp = x + (x / (y / x));
	elseif (x <= 9.5e+82)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 5e-6], N[(x + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+82], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 76.8%

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

   3. Taylor expanded in x around inf 98.6%

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

   4. Taylor expanded in y around 0 75.2%

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

   if -1 < x < 5.00000000000000041e-6

   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 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. *-rgt-identity 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 99.4%

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

   9. Taylor expanded in x around 0 97.9%

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

   if 5.00000000000000041e-6 < x < 9.50000000000000049e82

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf 75.6%

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

      1. +-commutative 75.6%

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

   5. Simplified 75.6%

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

   if 9.50000000000000049e82 < x

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf 76.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.25)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (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 ((x <= (-1.0d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + (x / (y / x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 77.8%

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

   3. Taylor expanded in x around inf 98.7%

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

   4. Taylor expanded in y around -inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

      3. neg-mul-1 98.8%

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

      4. sub-neg 98.8%

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

   6. Simplified 98.8%

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

   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* 99.6%

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

      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 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. *-rgt-identity 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 99.4%

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

   9. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.4%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (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 = (1.0d0 + (x / y)) / ((1.0d0 / x) - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   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%

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

Alternative 7: 73.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+84}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 4.8e-6) x (if (<= x 1.3e+84) 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 4.8e-6) {
		tmp = x;
	} else if (x <= 1.3e+84) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 4.8d-6) then
        tmp = x
    else if (x <= 1.3d+84) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 4.8e-6) {
		tmp = x;
	} else if (x <= 1.3e+84) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 4.8e-6:
		tmp = x
	elif x <= 1.3e+84:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 4.8e-6)
		tmp = x;
	elseif (x <= 1.3e+84)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 4.8e-6)
		tmp = x;
	elseif (x <= 1.3e+84)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.8e-6], x, If[LessEqual[x, 1.3e+84], 1.0, N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1.3000000000000001e84 < x

   1. Initial program 75.7%

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

   3. Taylor expanded in x around inf 75.1%

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

   if -1 < x < 4.7999999999999998e-6

   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 79.5%

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

   if 4.7999999999999998e-6 < x < 1.3000000000000001e84

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 71.4%

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

      1. +-commutative 71.4%

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

   5. Simplified 71.4%

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

   6. Taylor expanded in x around inf 67.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+84}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -245000000 \lor \neg \left(x \leq 6.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -245000000.0) (not (<= x 6.5e+82))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -245000000.0) || !(x <= 6.5e+82)) {
		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 <= (-245000000.0d0)) .or. (.not. (x <= 6.5d+82))) 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 <= -245000000.0) || !(x <= 6.5e+82)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -245000000.0) or not (x <= 6.5e+82):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -245000000.0) || !(x <= 6.5e+82))
		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 <= -245000000.0) || ~((x <= 6.5e+82)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -245000000.0], N[Not[LessEqual[x, 6.5e+82]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.45e8 or 6.5000000000000003e82 < x

   1. Initial program 75.3%

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

   3. Taylor expanded in x around inf 76.2%

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

   if -2.45e8 < x < 6.5000000000000003e82

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 79.3%

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

      1. +-commutative 79.3%

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

   5. Simplified 79.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -245000000 \lor \neg \left(x \leq 6.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -54000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -54000000.0)
   (/ (+ x -1.0) y)
   (if (<= x 7.2e+83) (/ x (+ x 1.0)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -54000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 7.2e+83) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-54000000.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 7.2d+83) then
        tmp = x / (x + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -54000000.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 7.2e+83) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -54000000.0:
		tmp = (x + -1.0) / y
	elif x <= 7.2e+83:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -54000000.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 7.2e+83)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -54000000.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 7.2e+83)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -54000000.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 7.2e+83], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4e7

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 100.0%

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

   4. Taylor expanded in y around 0 77.1%

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

   if -5.4e7 < x < 7.1999999999999995e83

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 79.3%

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

      1. +-commutative 79.3%

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

   5. Simplified 79.3%

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

   if 7.1999999999999995e83 < x

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf 76.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -54000000:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) 1.0 (if (<= x 4.8e-6) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 4.8e-6) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 4.8d-6) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 4.8e-6) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 4.8e-6:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 4.8e-6)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 4.8e-6)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 4.8e-6], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.7999999999999998e-6 < x

   1. Initial program 78.0%

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

   3. Taylor expanded in y around inf 31.2%

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

      1. +-commutative 31.2%

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

   5. Simplified 31.2%

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

   6. Taylor expanded in x around inf 30.0%

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

   if -1 < x < 4.7999999999999998e-6

   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 79.5%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 14.5% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around inf 53.2%

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

   1. +-commutative 53.2%

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

5. Simplified 53.2%

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

6. Taylor expanded in x around inf 18.1%

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

7. Final simplification 18.1%

   \[\leadsto 1 \]
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 2024010 
(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)))