Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 8.1s

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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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 87.8%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.7%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.8)))
   (+ 1.0 (/ x y))
   (* x (+ 1.0 (* x (+ (/ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -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)) .or. (.not. (x <= 0.8d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.8)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.8):
		tmp = 1.0 + (x / y)
	else:
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.8))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.8)))
		tmp = 1.0 + (x / y);
	else
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.8]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in y around 0 56.4%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 56.4%

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

      2. distribute-lft-out 56.7%

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

   5. Simplified 56.7%

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

      1. associate-/r* 62.6%

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

      2. *-commutative 62.6%

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

      3. +-commutative 62.6%

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

      4. associate-/l* 73.2%

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

      5. +-commutative 73.2%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 96.2%

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

   9. Taylor expanded in x around 0 96.5%

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

       1. +-commutative 96.5%

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

   11. Simplified 96.5%

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

   if -1 < x < 0.80000000000000004

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.8\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\left(x + 1\right) \cdot \frac{y}{x}}\\ \mathbf{elif}\;x \leq 920000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1500000000.0)
     t_0
     (if (<= x -8.5e-41)
       (/ x (* (+ x 1.0) (/ y x)))
       (if (<= x 920000.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1500000000.0) {
		tmp = t_0;
	} else if (x <= -8.5e-41) {
		tmp = x / ((x + 1.0) * (y / x));
	} else if (x <= 920000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (x <= (-1500000000.0d0)) then
        tmp = t_0
    else if (x <= (-8.5d-41)) then
        tmp = x / ((x + 1.0d0) * (y / x))
    else if (x <= 920000.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1500000000.0) {
		tmp = t_0;
	} else if (x <= -8.5e-41) {
		tmp = x / ((x + 1.0) * (y / x));
	} else if (x <= 920000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1500000000.0:
		tmp = t_0
	elif x <= -8.5e-41:
		tmp = x / ((x + 1.0) * (y / x))
	elif x <= 920000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1500000000.0)
		tmp = t_0;
	elseif (x <= -8.5e-41)
		tmp = Float64(x / Float64(Float64(x + 1.0) * Float64(y / x)));
	elseif (x <= 920000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -1500000000.0)
		tmp = t_0;
	elseif (x <= -8.5e-41)
		tmp = x / ((x + 1.0) * (y / x));
	elseif (x <= 920000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1500000000.0], t$95$0, If[LessEqual[x, -8.5e-41], N[(x / N[(N[(x + 1.0), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 920000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e9 or 9.2e5 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0 54.7%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 54.7%

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

      2. distribute-lft-out 55.0%

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

   5. Simplified 55.0%

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

      1. associate-/r* 61.2%

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

      2. *-commutative 61.2%

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

      3. +-commutative 61.2%

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

      4. associate-/l* 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 98.1%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. +-commutative 98.4%

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

   11. Simplified 98.4%

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

   if -1.5e9 < x < -8.4999999999999996e-41

   1. Initial program 99.5%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 74.7%

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

      1. clear-num 74.7%

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

      2. un-div-inv 74.7%

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

      3. +-commutative 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/l* 74.9%

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

   7. Applied egg-rr 74.9%

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

   if -8.4999999999999996e-41 < x < 9.2e5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.2%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 84.2%

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

      2. distribute-lft-out 84.3%

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

   5. Simplified 84.3%

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

      1. associate-/r* 84.2%

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

      2. *-commutative 84.2%

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

      3. +-commutative 84.2%

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

      4. associate-/l* 69.9%

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

      5. +-commutative 69.9%

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

      6. associate-/r* 69.9%

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

      7. +-commutative 69.9%

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

   7. Applied egg-rr 69.9%

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

   8. Taylor expanded in y around inf 78.9%

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

      1. +-commutative 78.9%

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

   10. Simplified 78.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{\left(x + 1\right) \cdot \frac{y}{x}}\\ \mathbf{elif}\;x \leq 920000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \frac{x}{\left(x + 1\right) \cdot y}\\ \mathbf{elif}\;x \leq 90000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1500000000.0)
     t_0
     (if (<= x -3e-43)
       (* x (/ x (* (+ x 1.0) y)))
       (if (<= x 90000.0) (/ x (+ x 1.0)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1500000000.0) {
		tmp = t_0;
	} else if (x <= -3e-43) {
		tmp = x * (x / ((x + 1.0) * y));
	} else if (x <= 90000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (x <= (-1500000000.0d0)) then
        tmp = t_0
    else if (x <= (-3d-43)) then
        tmp = x * (x / ((x + 1.0d0) * y))
    else if (x <= 90000.0d0) then
        tmp = x / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1500000000.0) {
		tmp = t_0;
	} else if (x <= -3e-43) {
		tmp = x * (x / ((x + 1.0) * y));
	} else if (x <= 90000.0) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -1500000000.0:
		tmp = t_0
	elif x <= -3e-43:
		tmp = x * (x / ((x + 1.0) * y))
	elif x <= 90000.0:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1500000000.0)
		tmp = t_0;
	elseif (x <= -3e-43)
		tmp = Float64(x * Float64(x / Float64(Float64(x + 1.0) * y)));
	elseif (x <= 90000.0)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -1500000000.0)
		tmp = t_0;
	elseif (x <= -3e-43)
		tmp = x * (x / ((x + 1.0) * y));
	elseif (x <= 90000.0)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1500000000.0], t$95$0, If[LessEqual[x, -3e-43], N[(x * N[(x / N[(N[(x + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 90000.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e9 or 9e4 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in y around 0 54.7%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 54.7%

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

      2. distribute-lft-out 55.0%

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

   5. Simplified 55.0%

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

      1. associate-/r* 61.2%

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

      2. *-commutative 61.2%

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

      3. +-commutative 61.2%

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

      4. associate-/l* 72.2%

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

      5. +-commutative 72.2%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 98.1%

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

   9. Taylor expanded in x around 0 98.4%

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

       1. +-commutative 98.4%

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

   11. Simplified 98.4%

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

   if -1.5e9 < x < -3.00000000000000003e-43

   1. Initial program 99.5%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 74.7%

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

   if -3.00000000000000003e-43 < x < 9e4

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 84.2%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 84.2%

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

      2. distribute-lft-out 84.3%

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

   5. Simplified 84.3%

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

      1. associate-/r* 84.2%

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

      2. *-commutative 84.2%

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

      3. +-commutative 84.2%

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

      4. associate-/l* 69.9%

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

      5. +-commutative 69.9%

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

      6. associate-/r* 69.9%

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

      7. +-commutative 69.9%

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

   7. Applied egg-rr 69.9%

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

   8. Taylor expanded in y around inf 78.9%

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

      1. +-commutative 78.9%

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

   10. Simplified 78.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1500000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \frac{x}{\left(x + 1\right) \cdot y}\\ \mathbf{elif}\;x \leq 90000:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -900 \lor \neg \left(x \leq 320000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -900.0) (not (<= x 320000.0))) (+ 1.0 (/ x y)) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -900.0) || !(x <= 320000.0)) {
		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 <= (-900.0d0)) .or. (.not. (x <= 320000.0d0))) 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 <= -900.0) || !(x <= 320000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -900.0) or not (x <= 320000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -900.0) || !(x <= 320000.0))
		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 <= -900.0) || ~((x <= 320000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -900.0], N[Not[LessEqual[x, 320000.0]], $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 < -900 or 3.2e5 < x

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0 56.1%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 56.1%

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

      2. distribute-lft-out 56.4%

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

   5. Simplified 56.4%

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

      1. associate-/r* 62.3%

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

      2. *-commutative 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-/l* 73.0%

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

      5. +-commutative 73.0%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 96.6%

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

   9. Taylor expanded in x around 0 96.9%

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

       1. +-commutative 96.9%

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

   11. Simplified 96.9%

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

   if -900 < x < 3.2e5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 85.7%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 85.7%

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

      2. distribute-lft-out 85.7%

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

   5. Simplified 85.7%

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

      1. associate-/r* 85.7%

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

      2. *-commutative 85.7%

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

      3. +-commutative 85.7%

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

      4. associate-/l* 72.0%

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

      5. +-commutative 72.0%

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

      6. associate-/r* 72.0%

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

      7. +-commutative 72.0%

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

   7. Applied egg-rr 72.0%

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

   8. Taylor expanded in y around inf 74.0%

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

      1. +-commutative 74.0%

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

   10. Simplified 74.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -900 \lor \neg \left(x \leq 320000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 320000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -600.0) (not (<= x 320000.0))) (+ 1.0 (/ x y)) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -600.0) || !(x <= 320000.0)) {
		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 <= (-600.0d0)) .or. (.not. (x <= 320000.0d0))) 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 <= -600.0) || !(x <= 320000.0)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -600.0) or not (x <= 320000.0):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -600.0) || !(x <= 320000.0))
		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 <= -600.0) || ~((x <= 320000.0)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -600.0], N[Not[LessEqual[x, 320000.0]], $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 < -600 or 3.2e5 < x

   1. Initial program 76.3%

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

   3. Taylor expanded in y around 0 56.1%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 56.1%

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

      2. distribute-lft-out 56.4%

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

   5. Simplified 56.4%

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

      1. associate-/r* 62.3%

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

      2. *-commutative 62.3%

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

      3. +-commutative 62.3%

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

      4. associate-/l* 73.0%

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

      5. +-commutative 73.0%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 96.6%

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

   9. Taylor expanded in x around 0 96.9%

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

       1. +-commutative 96.9%

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

   11. Simplified 96.9%

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

   if -600 < x < 3.2e5

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 74.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -600 \lor \neg \left(x \leq 320000\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.054\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.054))) (+ 1.0 (/ x y)) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.054)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.054d0))) then
        tmp = 1.0d0 + (x / y)
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.054)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.054):
		tmp = 1.0 + (x / y)
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.054))
		tmp = Float64(1.0 + Float64(x / y));
	else
		tmp = Float64(x * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.054)))
		tmp = 1.0 + (x / y);
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.054]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.0539999999999999994 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in y around 0 56.4%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 56.4%

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

      2. distribute-lft-out 56.7%

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

   5. Simplified 56.7%

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

      1. associate-/r* 62.6%

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

      2. *-commutative 62.6%

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

      3. +-commutative 62.6%

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

      4. associate-/l* 73.2%

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

      5. +-commutative 73.2%

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

      6. associate-/r* 99.8%

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

      7. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x around inf 96.2%

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

   9. Taylor expanded in x around 0 96.5%

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

       1. +-commutative 96.5%

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

   11. Simplified 96.5%

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

   if -1 < x < 0.0539999999999999994

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 85.6%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 85.6%

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

      2. distribute-lft-out 85.6%

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

   5. Simplified 85.6%

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

      1. associate-/r* 85.6%

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

      2. *-commutative 85.6%

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

      3. +-commutative 85.6%

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

      4. associate-/l* 71.8%

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

      5. +-commutative 71.8%

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

      6. associate-/r* 71.8%

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

      7. +-commutative 71.8%

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

   7. Applied egg-rr 71.8%

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

   8. Taylor expanded in y around inf 74.0%

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

      1. +-commutative 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in x around 0 74.0%

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

       1. neg-mul-1 74.0%

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

       2. sub-neg 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.054\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.62))) (/ x y) (* x (- 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.62)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.62d0))) then
        tmp = x / y
    else
        tmp = x * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.62)) {
		tmp = x / y;
	} else {
		tmp = x * (1.0 - x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.62):
		tmp = x / y
	else:
		tmp = x * (1.0 - x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.62))
		tmp = Float64(x / y);
	else
		tmp = Float64(x * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.62)))
		tmp = x / y;
	else
		tmp = x * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.62]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.619999999999999996 < x

   1. Initial program 76.5%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 73.3%

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

   if -1 < x < 0.619999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 85.6%

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y + {x}^{2}}{y}}}{x + 1} \]
   4. Step-by-step derivation

      1. unpow2 85.6%

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

      2. distribute-lft-out 85.6%

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

   5. Simplified 85.6%

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

      1. associate-/r* 85.6%

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

      2. *-commutative 85.6%

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

      3. +-commutative 85.6%

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

      4. associate-/l* 71.8%

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

      5. +-commutative 71.8%

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

      6. associate-/r* 71.8%

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

      7. +-commutative 71.8%

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

   7. Applied egg-rr 71.8%

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

   8. Taylor expanded in y around inf 74.0%

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

      1. +-commutative 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in x around 0 74.0%

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

       1. neg-mul-1 74.0%

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

       2. sub-neg 74.0%

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

   13. Simplified 74.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 21000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 21000.0))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 21000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 21000.0d0))) then
        tmp = x / y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 21000.0)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 21000.0):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 21000.0))
		tmp = Float64(x / y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 21000.0)))
		tmp = x / y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 21000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 21000 < x

   1. Initial program 76.3%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 73.8%

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

   if -1 < x < 21000

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 73.4%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 21000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $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. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 38.7% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 87.8%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 37.5%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))