Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.7% → 99.9%

Time: 7.2s

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: 88.7% 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 92.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -200000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -200000.0)
     (/ x y)
     (if (<= x 5e-57)
       t_0
       (if (<= x 6e-27) (/ x (/ y x)) (if (<= x 3e+56) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -200000.0) {
		tmp = x / y;
	} else if (x <= 5e-57) {
		tmp = t_0;
	} else if (x <= 6e-27) {
		tmp = x / (y / x);
	} else if (x <= 3e+56) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-200000.0d0)) then
        tmp = x / y
    else if (x <= 5d-57) then
        tmp = t_0
    else if (x <= 6d-27) then
        tmp = x / (y / x)
    else if (x <= 3d+56) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -200000.0) {
		tmp = x / y;
	} else if (x <= 5e-57) {
		tmp = t_0;
	} else if (x <= 6e-27) {
		tmp = x / (y / x);
	} else if (x <= 3e+56) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -200000.0:
		tmp = x / y
	elif x <= 5e-57:
		tmp = t_0
	elif x <= 6e-27:
		tmp = x / (y / x)
	elif x <= 3e+56:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -200000.0)
		tmp = Float64(x / y);
	elseif (x <= 5e-57)
		tmp = t_0;
	elseif (x <= 6e-27)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 3e+56)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -200000.0)
		tmp = x / y;
	elseif (x <= 5e-57)
		tmp = t_0;
	elseif (x <= 6e-27)
		tmp = x / (y / x);
	elseif (x <= 3e+56)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -200000.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 5e-57], t$95$0, If[LessEqual[x, 6e-27], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+56], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e5 or 3.00000000000000006e56 < x

   1. Initial program 82.3%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 76.5%

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

   if -2e5 < x < 5.0000000000000002e-57 or 6.0000000000000002e-27 < x < 3.00000000000000006e56

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 78.7%

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

   if 5.0000000000000002e-57 < x < 6.0000000000000002e-27

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.3%

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

      3. remove-double-neg 99.3%

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

      4. neg-mul-1 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. +-commutative 99.3%

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

      8. remove-double-neg 99.3%

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

      9. unsub-neg 99.3%

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

      10. div-sub 99.3%

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

      11. *-inverses 99.3%

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

      12. div-sub 99.3%

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

      13. associate-/r* 99.3%

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

      14. *-commutative 99.3%

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

      15. neg-mul-1 99.3%

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

      16. remove-double-neg 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 74.1%

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

      1. distribute-rgt-in 74.1%

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

      2. *-lft-identity 74.1%

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

      3. associate-*l/ 74.1%

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

      4. *-lft-identity 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in x around 0 74.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -44:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -44.0)
     (/ (+ x -1.0) y)
     (if (<= x 6.2e-57)
       t_0
       (if (<= x 1.42e-30) (/ x (/ y x)) (if (<= x 3e+56) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -44.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e-57) {
		tmp = t_0;
	} else if (x <= 1.42e-30) {
		tmp = x / (y / x);
	} else if (x <= 3e+56) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-44.0d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 6.2d-57) then
        tmp = t_0
    else if (x <= 1.42d-30) then
        tmp = x / (y / x)
    else if (x <= 3d+56) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -44.0) {
		tmp = (x + -1.0) / y;
	} else if (x <= 6.2e-57) {
		tmp = t_0;
	} else if (x <= 1.42e-30) {
		tmp = x / (y / x);
	} else if (x <= 3e+56) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -44.0:
		tmp = (x + -1.0) / y
	elif x <= 6.2e-57:
		tmp = t_0
	elif x <= 1.42e-30:
		tmp = x / (y / x)
	elif x <= 3e+56:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -44.0)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 6.2e-57)
		tmp = t_0;
	elseif (x <= 1.42e-30)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 3e+56)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -44.0)
		tmp = (x + -1.0) / y;
	elseif (x <= 6.2e-57)
		tmp = t_0;
	elseif (x <= 1.42e-30)
		tmp = x / (y / x);
	elseif (x <= 3e+56)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -44.0], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 6.2e-57], t$95$0, If[LessEqual[x, 1.42e-30], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+56], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -44

   1. Initial program 83.3%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.5%

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

   6. Taylor expanded in y around 0 75.0%

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

   if -44 < x < 6.19999999999999952e-57 or 1.42e-30 < x < 3.00000000000000006e56

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 78.7%

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

   if 6.19999999999999952e-57 < x < 1.42e-30

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.3%

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

      3. remove-double-neg 99.3%

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

      4. neg-mul-1 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. +-commutative 99.3%

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

      8. remove-double-neg 99.3%

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

      9. unsub-neg 99.3%

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

      10. div-sub 99.3%

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

      11. *-inverses 99.3%

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

      12. div-sub 99.3%

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

      13. associate-/r* 99.3%

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

      14. *-commutative 99.3%

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

      15. neg-mul-1 99.3%

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

      16. remove-double-neg 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 74.1%

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

      1. distribute-rgt-in 74.1%

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

      2. *-lft-identity 74.1%

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

      3. associate-*l/ 74.1%

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

      4. *-lft-identity 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in x around 0 74.1%

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

   if 3.00000000000000006e56 < x

   1. Initial program 81.5%

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

   5. Taylor expanded in x around inf 79.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6200000000 \lor \neg \left(x \leq 380000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6200000000.0) (not (<= x 380000.0)))
   (+ 1.0 (/ (+ x -1.0) y))
   (+ x (/ x (+ y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6200000000.0) || !(x <= 380000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (y + (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-6200000000.0d0)) .or. (.not. (x <= 380000.0d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + (x / (y + (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6200000000.0) || !(x <= 380000.0)) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else {
		tmp = x + (x / (y + (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6200000000.0) or not (x <= 380000.0):
		tmp = 1.0 + ((x + -1.0) / y)
	else:
		tmp = x + (x / (y + (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6200000000.0) || !(x <= 380000.0))
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	else
		tmp = Float64(x + Float64(x / Float64(y + Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6200000000.0) || ~((x <= 380000.0)))
		tmp = 1.0 + ((x + -1.0) / y);
	else
		tmp = x + (x / (y + (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6200000000.0], N[Not[LessEqual[x, 380000.0]], $MachinePrecision]], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e9 or 3.8e5 < x

   1. Initial program 83.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. sub-div 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -6.2e9 < x < 3.8e5

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. *-lft-identity 99.6%

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

      4. associate-*l/ 99.6%

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

      5. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6200000000 \lor \neg \left(x \leq 380000\right):\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{y + \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 8.5))) (/ (+ x -1.0) y) (+ x (* x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 8.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x + (x * (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 <= 8.5d0))) then
        tmp = (x + (-1.0d0)) / y
    else
        tmp = x + (x * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 8.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x + (x * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 8.5):
		tmp = (x + -1.0) / y
	else:
		tmp = x + (x * (x / y))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 8.5 < x

   1. Initial program 83.6%

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

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in y around 0 74.5%

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

   if -1 < x < 8.5

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-rgt-in 99.5%

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

      3. *-lft-identity 99.5%

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

      4. associate-*l/ 99.6%

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

      5. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

   9. Taylor expanded in x around 0 97.8%

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

       1. associate-/r/ 97.7%

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

   11. Applied egg-rr 97.7%

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

4. Final simplification 87.4%

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

Alternative 6: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 7.5\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 7.5))) (/ (+ x -1.0) y) (+ x (/ x (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 7.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x + (x / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 7.5d0))) then
        tmp = (x + (-1.0d0)) / y
    else
        tmp = x + (x / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 7.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x + (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 7.5):
		tmp = (x + -1.0) / y
	else:
		tmp = x + (x / (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 7.5))
		tmp = Float64(Float64(x + -1.0) / y);
	else
		tmp = Float64(x + Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 7.5)))
		tmp = (x + -1.0) / y;
	else
		tmp = x + (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 7.5]], $MachinePrecision]], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], N[(x + N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 7.5 < x

   1. Initial program 83.6%

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

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in y around 0 74.5%

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

   if -1 < x < 7.5

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-rgt-in 99.5%

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

      3. *-lft-identity 99.5%

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

      4. associate-*l/ 99.6%

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

      5. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

   9. Taylor expanded in x around 0 97.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 7.5\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else
        tmp = x + (x / (y / x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 83.6%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-div 99.1%

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

   7. Applied egg-rr 99.1%

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

   if -1 < x < 1.25

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.5%

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

      3. remove-double-neg 99.5%

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

      4. neg-mul-1 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. +-commutative 99.5%

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

      8. remove-double-neg 99.5%

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

      9. unsub-neg 99.5%

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

      10. div-sub 99.5%

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

      11. *-inverses 99.5%

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

      12. div-sub 99.5%

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

      13. associate-/r* 99.5%

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

      14. *-commutative 99.5%

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

      15. neg-mul-1 99.5%

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

      16. remove-double-neg 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-rgt-in 99.5%

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

      3. *-lft-identity 99.5%

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

      4. associate-*l/ 99.6%

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

      5. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 98.5%

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

   9. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.4%

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

Alternative 8: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.05 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -90000.0) (not (<= x 1.05e+57))) (/ x y) (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.05e+57)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-90000.0d0)) .or. (.not. (x <= 1.05d+57))) then
        tmp = x / y
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.05e+57)) {
		tmp = x / y;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -90000.0) or not (x <= 1.05e+57):
		tmp = x / y
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -90000.0) || !(x <= 1.05e+57))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -90000.0) || ~((x <= 1.05e+57)))
		tmp = x / y;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -90000.0], N[Not[LessEqual[x, 1.05e+57]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e4 or 1.04999999999999995e57 < x

   1. Initial program 82.3%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 76.5%

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

   if -9e4 < x < 1.04999999999999995e57

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 75.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.05 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.32\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.32))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.32)) {
		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 <= 0.32d0))) 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 <= 0.32)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.32):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.32))
		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 <= 0.32)))
		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, 0.32]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.320000000000000007 < x

   1. Initial program 83.6%

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

   5. Taylor expanded in x around inf 73.2%

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

   if -1 < x < 0.320000000000000007

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.32\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -440000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -440000.0) 1.0 (if (<= x 3.1e+18) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -440000.0) {
		tmp = 1.0;
	} else if (x <= 3.1e+18) {
		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 <= (-440000.0d0)) then
        tmp = 1.0d0
    else if (x <= 3.1d+18) 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 <= -440000.0) {
		tmp = 1.0;
	} else if (x <= 3.1e+18) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -440000.0:
		tmp = 1.0
	elif x <= 3.1e+18:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -440000.0)
		tmp = 1.0;
	elseif (x <= 3.1e+18)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -440000.0)
		tmp = 1.0;
	elseif (x <= 3.1e+18)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -440000.0], 1.0, If[LessEqual[x, 3.1e+18], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4e5 or 3.1e18 < x

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/l* 100.0%

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

      3. remove-double-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r* 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. unsub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. *-inverses 100.0%

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

      12. div-sub 100.0%

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

      13. associate-/r* 100.0%

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

      14. *-commutative 100.0%

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

      15. neg-mul-1 100.0%

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

      16. remove-double-neg 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 26.8%

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

   6. Taylor expanded in x around inf 26.6%

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

   if -4.4e5 < x < 3.1e18

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 73.2%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -440000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 14.7% 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 92.6%

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

   1. *-commutative 92.6%

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

   2. associate-/l* 99.7%

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

   3. remove-double-neg 99.7%

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

   4. neg-mul-1 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-/r* 99.7%

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

   7. +-commutative 99.7%

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

   8. remove-double-neg 99.7%

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

   9. unsub-neg 99.7%

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

   10. div-sub 99.7%

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

   11. *-inverses 99.7%

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

   12. div-sub 99.7%

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

   13. associate-/r* 99.7%

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

   14. *-commutative 99.7%

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

   15. neg-mul-1 99.7%

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

   16. remove-double-neg 99.7%

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

   17. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around inf 54.0%

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

6. Taylor expanded in x around inf 13.3%

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

7. Final simplification 13.3%

   \[\leadsto 1 \]
8. 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 2024020 
(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)))