Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.5% → 99.9%

Time: 6.0s

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: 87.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * (1.0 + ((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 <= 1.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x * (1.0d0 + ((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 <= 1.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.6%

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

   5. Taylor expanded in y around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. neg-mul-1 97.6%

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

      4. sub-neg 97.6%

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

   7. Simplified 97.6%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in y around 0 93.5%

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

      1. +-commutative 93.5%

         \[\leadsto x + \color{blue}{\left(\frac{{x}^{2}}{y} + -1 \cdot {x}^{2}\right)} \]

      2. unpow2 93.5%

         \[\leadsto x + \left(\frac{\color{blue}{x \cdot x}}{y} + -1 \cdot {x}^{2}\right) \]

      3. unpow2 93.5%

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

      4. neg-mul-1 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 99.2%

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

      7. associate-/r/ 99.2%

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

      8. distribute-rgt-out-- 99.2%

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

   9. Simplified 99.2%

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

       1. *-commutative 99.2%

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

       2. distribute-rgt1-in 99.2%

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

   11. Applied egg-rr 99.2%

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

4. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 75.5%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in y around 0 93.5%

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

      1. +-commutative 93.5%

         \[\leadsto x + \color{blue}{\left(\frac{{x}^{2}}{y} + -1 \cdot {x}^{2}\right)} \]

      2. unpow2 93.5%

         \[\leadsto x + \left(\frac{\color{blue}{x \cdot x}}{y} + -1 \cdot {x}^{2}\right) \]

      3. unpow2 93.5%

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

      4. neg-mul-1 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 99.2%

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

      7. associate-/r/ 99.2%

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

      8. distribute-rgt-out-- 99.2%

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

   9. Simplified 99.2%

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

       1. *-commutative 99.2%

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

       2. distribute-rgt1-in 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 1 < x

   1. Initial program 81.2%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.4%

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

   5. Taylor expanded in y around -inf 97.4%

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

      1. mul-1-neg 97.4%

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

      2. unsub-neg 97.4%

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

      3. neg-mul-1 97.4%

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

      4. sub-neg 97.4%

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

   7. Simplified 97.4%

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

4. Final simplification 98.3%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = t_0
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 78.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.6%

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

   5. Taylor expanded in y around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. neg-mul-1 97.6%

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

      4. sub-neg 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 96.8%

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

      1. neg-mul-1 96.8%

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

      2. distribute-neg-frac 96.8%

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

   10. Simplified 96.8%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in y around 0 93.5%

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

      1. +-commutative 93.5%

         \[\leadsto x + \color{blue}{\left(\frac{{x}^{2}}{y} + -1 \cdot {x}^{2}\right)} \]

      2. unpow2 93.5%

         \[\leadsto x + \left(\frac{\color{blue}{x \cdot x}}{y} + -1 \cdot {x}^{2}\right) \]

      3. unpow2 93.5%

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

      4. neg-mul-1 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 99.2%

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

      7. associate-/r/ 99.2%

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

      8. distribute-rgt-out-- 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in y around 0 92.5%

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

       1. unpow2 92.5%

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

       2. associate-*r/ 98.2%

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

   12. Simplified 98.2%

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

       1. *-commutative 98.2%

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

       2. distribute-rgt1-in 98.2%

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

   14. Applied egg-rr 98.2%

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

4. Final simplification 97.4%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.1)))
   (+ 1.0 (/ (+ x -1.0) y))
   (* x (+ 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.1)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * (1.0 + (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)) .or. (.not. (x <= 1.1d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x * (1.0d0 + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.1)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x * (1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.1):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x * (1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.1))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.1)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x * (1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.1]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.1000000000000001 < x

   1. Initial program 78.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.6%

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

   5. Taylor expanded in y around -inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. neg-mul-1 97.6%

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

      4. sub-neg 97.6%

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

   7. Simplified 97.6%

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

   if -1 < x < 1.1000000000000001

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. unpow2 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in y around 0 93.5%

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

      1. +-commutative 93.5%

         \[\leadsto x + \color{blue}{\left(\frac{{x}^{2}}{y} + -1 \cdot {x}^{2}\right)} \]

      2. unpow2 93.5%

         \[\leadsto x + \left(\frac{\color{blue}{x \cdot x}}{y} + -1 \cdot {x}^{2}\right) \]

      3. unpow2 93.5%

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

      4. neg-mul-1 93.5%

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

      5. unsub-neg 93.5%

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

      6. associate-/l* 99.2%

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

      7. associate-/r/ 99.2%

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

      8. distribute-rgt-out-- 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in y around 0 92.5%

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

       1. unpow2 92.5%

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

       2. associate-*r/ 98.2%

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

   12. Simplified 98.2%

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

       1. *-commutative 98.2%

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

       2. distribute-rgt1-in 98.2%

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

   14. Applied egg-rr 98.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3150 \lor \neg \left(x \leq 195000000\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3150.0) (not (<= x 195000000.0)))
   (/ (+ x -1.0) y)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3150.0) || !(x <= 195000000.0)) {
		tmp = (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 <= (-3150.0d0)) .or. (.not. (x <= 195000000.0d0))) then
        tmp = (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 <= -3150.0) || !(x <= 195000000.0)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3150.0) or not (x <= 195000000.0):
		tmp = (x + -1.0) / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3150.0) || !(x <= 195000000.0))
		tmp = 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 <= -3150.0) || ~((x <= 195000000.0)))
		tmp = (x + -1.0) / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3150.0], N[Not[LessEqual[x, 195000000.0]], $MachinePrecision]], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3150 or 1.95e8 < x

   1. Initial program 78.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

   5. Taylor expanded in y around 0 80.1%

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

   if -3150 < x < 1.95e8

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 81.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3150 \lor \neg \left(x \leq 195000000\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 7: 87.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7000 \lor \neg \left(x \leq 86000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -7000.0) (not (<= x 86000000.0)))
   (+ 1.0 (/ x y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -7000.0) || !(x <= 86000000.0)) {
		tmp = 1.0 + (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 <= (-7000.0d0)) .or. (.not. (x <= 86000000.0d0))) then
        tmp = 1.0d0 + (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 <= -7000.0) || !(x <= 86000000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -7000.0) or not (x <= 86000000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -7000.0) || !(x <= 86000000.0))
		tmp = Float64(1.0 + 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 <= -7000.0) || ~((x <= 86000000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -7000.0], N[Not[LessEqual[x, 86000000.0]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7e3 or 8.6e7 < x

   1. Initial program 78.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.6%

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

   5. Taylor expanded in y around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. neg-mul-1 99.6%

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

      4. sub-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. distribute-neg-frac 98.8%

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

   10. Simplified 98.8%

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

   if -7e3 < x < 8.6e7

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 81.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7000 \lor \neg \left(x \leq 86000000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 8: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 16500000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8000.0) (/ x y) (if (<= x 16500000000.0) (/ x (+ x 1.0)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8000.0) {
		tmp = x / y;
	} else if (x <= 16500000000.0) {
		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 <= (-8000.0d0)) then
        tmp = x / y
    else if (x <= 16500000000.0d0) 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 <= -8000.0) {
		tmp = x / y;
	} else if (x <= 16500000000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8000.0:
		tmp = x / y
	elif x <= 16500000000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8000.0)
		tmp = Float64(x / y);
	elseif (x <= 16500000000.0)
		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 <= -8000.0)
		tmp = x / y;
	elseif (x <= 16500000000.0)
		tmp = x / (x + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 16500000000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8e3 or 1.65e10 < x

   1. Initial program 78.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 79.3%

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

   if -8e3 < x < 1.65e10

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 81.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 16500000000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 1.9) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.9) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 1.9d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.9) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 1.9:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 1.9)
		tmp = x;
	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 <= 1.9)
		tmp = x;
	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, 1.9], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8999999999999999 < x

   1. Initial program 78.5%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.7%

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

   if -1 < x < 1.8999999999999999

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 79.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.9:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 10: 49.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e-13) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e-13) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.5d-13)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.5e-13) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.5e-13:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e-13)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.5e-13)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e-13], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999995e-13 or 1 < x

   1. Initial program 78.8%

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

      1. distribute-lft-in 78.8%

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

      2. *-rgt-identity 78.8%

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

   3. Applied egg-rr 78.8%

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

   4. Taylor expanded in y around inf 21.8%

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

      1. +-commutative 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in x around inf 20.3%

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

   if -2.49999999999999995e-13 < x < 1

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 80.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 13.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.5%

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

   1. distribute-lft-in 88.5%

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

   2. *-rgt-identity 88.5%

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

3. Applied egg-rr 88.5%

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

4. Taylor expanded in y around inf 49.6%

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

   1. +-commutative 49.6%

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

6. Simplified 49.6%

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

7. Taylor expanded in x around inf 12.7%

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

8. Final simplification 12.7%

   \[\leadsto 1 \]

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 2023293 
(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)))