Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 7.7s

Alternatives: 13

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. associate-/r/ 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-rgt-in 99.9%

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

   4. *-un-lft-identity 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 39000:\\ \;\;\;\;t_0 + x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1.0)
     (+ t_0 (/ x y))
     (if (<= x 39000.0)
       (+ t_0 (* x (/ x y)))
       (+ (+ 1.0 (/ x y)) (/ -1.0 y))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0 + (x / y);
	} else if (x <= 39000.0) {
		tmp = t_0 + (x * (x / y));
	} else {
		tmp = (1.0 + (x / y)) + (-1.0 / 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 <= (-1.0d0)) then
        tmp = t_0 + (x / y)
    else if (x <= 39000.0d0) then
        tmp = t_0 + (x * (x / y))
    else
        tmp = (1.0d0 + (x / y)) + ((-1.0d0) / 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 <= -1.0) {
		tmp = t_0 + (x / y);
	} else if (x <= 39000.0) {
		tmp = t_0 + (x * (x / y));
	} else {
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1.0:
		tmp = t_0 + (x / y)
	elif x <= 39000.0:
		tmp = t_0 + (x * (x / y))
	else:
		tmp = (1.0 + (x / y)) + (-1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(t_0 + Float64(x / y));
	elseif (x <= 39000.0)
		tmp = Float64(t_0 + Float64(x * Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0 + (x / y);
	elseif (x <= 39000.0)
		tmp = t_0 + (x * (x / y));
	else
		tmp = (1.0 + (x / y)) + (-1.0 / 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, -1.0], N[(t$95$0 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 39000.0], N[(t$95$0 + N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 72.8%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.5%

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

   if -1 < x < 39000

   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. Step-by-step derivation

      1. associate-/r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 39000 < x

   1. Initial program 74.1%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{elif}\;x \leq 39000:\\ \;\;\;\;\frac{x}{x + 1} + x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Alternative 3: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -390000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1260000000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -390000.0)
     t_0
     (if (<= x 7.7e-75)
       (/ x (+ x 1.0))
       (if (<= x 1260000000.0) (/ x (+ y (/ y x))) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -390000.0) {
		tmp = t_0;
	} else if (x <= 7.7e-75) {
		tmp = x / (x + 1.0);
	} else if (x <= 1260000000.0) {
		tmp = x / (y + (y / x));
	} 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 <= (-390000.0d0)) then
        tmp = t_0
    else if (x <= 7.7d-75) then
        tmp = x / (x + 1.0d0)
    else if (x <= 1260000000.0d0) then
        tmp = x / (y + (y / x))
    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 <= -390000.0) {
		tmp = t_0;
	} else if (x <= 7.7e-75) {
		tmp = x / (x + 1.0);
	} else if (x <= 1260000000.0) {
		tmp = x / (y + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -390000.0:
		tmp = t_0
	elif x <= 7.7e-75:
		tmp = x / (x + 1.0)
	elif x <= 1260000000.0:
		tmp = x / (y + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -390000.0)
		tmp = t_0;
	elseif (x <= 7.7e-75)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 1260000000.0)
		tmp = Float64(x / Float64(y + Float64(y / x)));
	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 <= -390000.0)
		tmp = t_0;
	elseif (x <= 7.7e-75)
		tmp = x / (x + 1.0);
	elseif (x <= 1260000000.0)
		tmp = x / (y + (y / x));
	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, -390000.0], t$95$0, If[LessEqual[x, 7.7e-75], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1260000000.0], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9e5 or 1.26e9 < x

   1. Initial program 72.4%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.9%

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

   7. Taylor expanded in x around inf 99.8%

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

      1. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   if -3.9e5 < x < 7.69999999999999958e-75

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 75.7%

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

   if 7.69999999999999958e-75 < x < 1.26e9

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/l* 99.3%

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

      3. +-commutative 99.3%

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

      4. remove-double-neg 99.3%

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

      5. unsub-neg 99.3%

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

      6. div-sub 99.4%

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

      7. distribute-frac-neg 99.4%

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

      8. *-inverses 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 78.9%

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

      1. distribute-rgt-in 78.9%

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

      2. *-lft-identity 78.9%

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

      3. associate-*l/ 78.8%

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

      4. *-lft-identity 78.8%

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

   6. Simplified 78.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -390000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 7.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 1260000000:\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 4: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-12} \lor \neg \left(x \leq 4.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot x}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.1e-12) (not (<= x 4.8e-12)))
   (+ (/ x (+ x 1.0)) (/ x y))
   (- x (/ (* x x) (- y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.1e-12) || !(x <= 4.8e-12)) {
		tmp = (x / (x + 1.0)) + (x / 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 <= (-3.1d-12)) .or. (.not. (x <= 4.8d-12))) then
        tmp = (x / (x + 1.0d0)) + (x / 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 <= -3.1e-12) || !(x <= 4.8e-12)) {
		tmp = (x / (x + 1.0)) + (x / y);
	} else {
		tmp = x - ((x * x) / -y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.1e-12) or not (x <= 4.8e-12):
		tmp = (x / (x + 1.0)) + (x / y)
	else:
		tmp = x - ((x * x) / -y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.1e-12) || !(x <= 4.8e-12))
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(x / y));
	else
		tmp = Float64(x - Float64(Float64(x * x) / Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.1e-12) || ~((x <= 4.8e-12)))
		tmp = (x / (x + 1.0)) + (x / y);
	else
		tmp = x - ((x * x) / -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.1e-12], N[Not[LessEqual[x, 4.8e-12]], $MachinePrecision]], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x * x), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1000000000000001e-12 or 4.79999999999999974e-12 < x

   1. Initial program 74.4%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.0%

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

   if -3.1000000000000001e-12 < x < 4.79999999999999974e-12

   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. Step-by-step derivation

      1. associate-/r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. frac-2neg 99.8%

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

      3. associate-*r/ 90.8%

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

   7. Applied egg-rr 90.8%

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

   8. Taylor expanded in x around 0 90.8%

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

   9. Taylor expanded in x around 0 90.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-12} \lor \neg \left(x \leq 4.8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot x}{-y}\\ \end{array} \]

Alternative 5: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.28:\\ \;\;\;\;x - \frac{x \cdot x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-12)
   (+ (/ x (+ x 1.0)) (/ x y))
   (if (<= x 1.28) (- x (/ (* x x) (- y))) (+ (+ 1.0 (/ x y)) (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-12) {
		tmp = (x / (x + 1.0)) + (x / y);
	} else if (x <= 1.28) {
		tmp = x - ((x * x) / -y);
	} else {
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.1d-12)) then
        tmp = (x / (x + 1.0d0)) + (x / y)
    else if (x <= 1.28d0) then
        tmp = x - ((x * x) / -y)
    else
        tmp = (1.0d0 + (x / y)) + ((-1.0d0) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-12) {
		tmp = (x / (x + 1.0)) + (x / y);
	} else if (x <= 1.28) {
		tmp = x - ((x * x) / -y);
	} else {
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e-12:
		tmp = (x / (x + 1.0)) + (x / y)
	elif x <= 1.28:
		tmp = x - ((x * x) / -y)
	else:
		tmp = (1.0 + (x / y)) + (-1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e-12)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(x / y));
	elseif (x <= 1.28)
		tmp = Float64(x - Float64(Float64(x * x) / Float64(-y)));
	else
		tmp = Float64(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.1e-12)
		tmp = (x / (x + 1.0)) + (x / y);
	elseif (x <= 1.28)
		tmp = x - ((x * x) / -y);
	else
		tmp = (1.0 + (x / y)) + (-1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e-12], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.28], N[(x - N[(N[(x * x), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1000000000000001e-12

   1. Initial program 74.1%

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

      1. associate-/r/ 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. *-un-lft-identity 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 98.5%

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

   if -3.1000000000000001e-12 < x < 1.28000000000000003

   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. Step-by-step derivation

      1. associate-/r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. frac-2neg 99.8%

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

      3. associate-*r/ 90.9%

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

   7. Applied egg-rr 90.9%

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

   8. Taylor expanded in x around 0 90.4%

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

   9. Taylor expanded in x around 0 90.2%

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

   if 1.28000000000000003 < x

   1. Initial program 74.1%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.28:\\ \;\;\;\;x - \frac{x \cdot x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Alternative 6: 95.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.0)))
		tmp = 1.0 + (x / 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, 1.0]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x * x), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.5%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.7%

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

   7. Taylor expanded in x around inf 97.6%

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

      1. +-commutative 97.6%

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

   9. Simplified 97.6%

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

   if -1 < x < 1

   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. Step-by-step derivation

      1. associate-/r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. *-un-lft-identity 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. frac-2neg 99.8%

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

      3. associate-*r/ 91.1%

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

   7. Applied egg-rr 91.1%

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

   8. Taylor expanded in x around 0 90.6%

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

   9. Taylor expanded in x around 0 89.8%

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

4. Final simplification 93.7%

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

Alternative 7: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -11000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -11000000.0)
     t_0
     (if (<= x 1.12e-73)
       (/ x (+ x 1.0))
       (if (<= x 0.108) (/ x (/ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -11000000.0) {
		tmp = t_0;
	} else if (x <= 1.12e-73) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.108) {
		tmp = x / (y / x);
	} 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 <= (-11000000.0d0)) then
        tmp = t_0
    else if (x <= 1.12d-73) then
        tmp = x / (x + 1.0d0)
    else if (x <= 0.108d0) then
        tmp = x / (y / x)
    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 <= -11000000.0) {
		tmp = t_0;
	} else if (x <= 1.12e-73) {
		tmp = x / (x + 1.0);
	} else if (x <= 0.108) {
		tmp = x / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -11000000.0:
		tmp = t_0
	elif x <= 1.12e-73:
		tmp = x / (x + 1.0)
	elif x <= 0.108:
		tmp = x / (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -11000000.0)
		tmp = t_0;
	elseif (x <= 1.12e-73)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 0.108)
		tmp = Float64(x / Float64(y / x));
	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 <= -11000000.0)
		tmp = t_0;
	elseif (x <= 1.12e-73)
		tmp = x / (x + 1.0);
	elseif (x <= 0.108)
		tmp = x / (y / x);
	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, -11000000.0], t$95$0, If[LessEqual[x, 1.12e-73], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.108], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e7 or 0.107999999999999999 < x

   1. Initial program 72.8%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.2%

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

   7. Taylor expanded in x around inf 99.1%

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

      1. +-commutative 99.1%

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

   9. Simplified 99.1%

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

   if -1.1e7 < x < 1.11999999999999995e-73

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 75.7%

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

   if 1.11999999999999995e-73 < x < 0.107999999999999999

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-/l* 99.2%

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

      3. +-commutative 99.2%

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

      4. remove-double-neg 99.2%

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

      5. unsub-neg 99.2%

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

      6. div-sub 99.3%

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

      7. distribute-frac-neg 99.3%

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

      8. *-inverses 99.3%

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

      9. 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. Taylor expanded in y around 0 75.1%

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

      1. distribute-rgt-in 75.1%

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

      2. *-lft-identity 75.1%

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

      3. associate-*l/ 75.0%

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

      4. *-lft-identity 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in x around 0 68.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 0.108:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 8: 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 86.6%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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. Final simplification 99.9%

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

Alternative 9: 85.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.00155\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.00155))) (+ 1.0 (/ x y)) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.00155)) {
		tmp = 1.0 + (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 <= 0.00155d0))) then
        tmp = 1.0d0 + (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 <= 0.00155)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.00155):
		tmp = 1.0 + (x / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.00155))
		tmp = Float64(1.0 + 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 <= 0.00155)))
		tmp = 1.0 + (x / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.00155]], $MachinePrecision]], N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.00154999999999999995 < x

   1. Initial program 73.7%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.0%

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

   7. Taylor expanded in x around inf 96.9%

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

      1. +-commutative 96.9%

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

   9. Simplified 96.9%

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

   if -1 < x < 0.00154999999999999995

   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. Taylor expanded in x around 0 70.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.00155\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28000000 \lor \neg \left(x \leq 0.0027\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -28000000.0) (not (<= x 0.0027)))
   (+ 1.0 (/ x y))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -28000000.0) || !(x <= 0.0027)) {
		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 <= (-28000000.0d0)) .or. (.not. (x <= 0.0027d0))) 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 <= -28000000.0) || !(x <= 0.0027)) {
		tmp = 1.0 + (x / y);
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -28000000.0) or not (x <= 0.0027):
		tmp = 1.0 + (x / y)
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -28000000.0) || !(x <= 0.0027))
		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 <= -28000000.0) || ~((x <= 0.0027)))
		tmp = 1.0 + (x / y);
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -28000000.0], N[Not[LessEqual[x, 0.0027]], $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 < -2.8e7 or 0.0027000000000000001 < x

   1. Initial program 73.1%

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

      1. associate-/r/ 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.6%

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

   7. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

   9. Simplified 98.5%

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

   if -2.8e7 < x < 0.0027000000000000001

   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. Taylor expanded in y around inf 71.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28000000 \lor \neg \left(x \leq 0.0027\right):\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 11: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.00085\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.00085))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.00085)) {
		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 <= 0.00085d0))) 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 <= 0.00085)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.00085):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.00085))
		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 <= 0.00085)))
		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, 0.00085]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 8.49999999999999953e-4 < x

   1. Initial program 73.7%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 74.8%

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

   if -1 < x < 8.49999999999999953e-4

   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. Taylor expanded in x around 0 70.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.00085\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 49.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -38.0) 1.0 (if (<= x 33000000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -38.0) {
		tmp = 1.0;
	} else if (x <= 33000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-38.0d0)) then
        tmp = 1.0d0
    else if (x <= 33000000.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -38.0) {
		tmp = 1.0;
	} else if (x <= 33000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -38.0:
		tmp = 1.0
	elif x <= 33000000.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -38.0)
		tmp = 1.0;
	elseif (x <= 33000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -38.0)
		tmp = 1.0;
	elseif (x <= 33000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -38.0], 1.0, If[LessEqual[x, 33000000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -38 or 3.3e7 < x

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. *-inverses 100.0%

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

      9. 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. Taylor expanded in y around inf 24.6%

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

      1. flip-+ 24.6%

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

      2. associate-/r/ 24.6%

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

      3. metadata-eval 24.6%

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

      4. inv-pow 24.6%

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

      5. inv-pow 24.6%

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

      6. pow-prod-up 24.6%

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

      7. metadata-eval 24.6%

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

   6. Applied egg-rr 24.6%

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

      1. associate-*l/ 24.6%

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

      2. *-lft-identity 24.6%

         \[\leadsto \frac{\color{blue}{1 - \frac{1}{x}}}{1 - {x}^{-2}} \]

      3. sub-neg 24.6%

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

      4. distribute-neg-frac 24.6%

         \[\leadsto \frac{1 + \color{blue}{\frac{-1}{x}}}{1 - {x}^{-2}} \]

      5. metadata-eval 24.6%

         \[\leadsto \frac{1 + \frac{\color{blue}{-1}}{x}}{1 - {x}^{-2}} \]

   8. Simplified 24.6%

      \[\leadsto \color{blue}{\frac{1 + \frac{-1}{x}}{1 - {x}^{-2}}} \]

   9. Taylor expanded in x around inf 23.5%

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

   if -38 < x < 3.3e7

   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. Taylor expanded in x around 0 69.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -38:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 33000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 14.1% 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 86.6%

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

   1. *-commutative 86.6%

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

   2. associate-/l* 99.5%

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

   3. +-commutative 99.5%

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

   4. remove-double-neg 99.5%

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

   5. unsub-neg 99.5%

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

   6. div-sub 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. *-inverses 99.5%

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

   9. 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. Taylor expanded in y around inf 47.4%

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

   1. flip-+ 26.9%

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

   2. associate-/r/ 26.9%

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

   3. metadata-eval 26.9%

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

   4. inv-pow 26.9%

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

   5. inv-pow 26.9%

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

   6. pow-prod-up 26.9%

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

   7. metadata-eval 26.9%

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

6. Applied egg-rr 26.9%

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

   1. associate-*l/ 26.9%

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

   2. *-lft-identity 26.9%

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{x}}}{1 - {x}^{-2}} \]

   3. sub-neg 26.9%

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

   4. distribute-neg-frac 26.9%

      \[\leadsto \frac{1 + \color{blue}{\frac{-1}{x}}}{1 - {x}^{-2}} \]

   5. metadata-eval 26.9%

      \[\leadsto \frac{1 + \frac{\color{blue}{-1}}{x}}{1 - {x}^{-2}} \]

8. Simplified 26.9%

   \[\leadsto \color{blue}{\frac{1 + \frac{-1}{x}}{1 - {x}^{-2}}} \]

9. Taylor expanded in x around inf 13.2%

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

10. Final simplification 13.2%

    \[\leadsto 1 \]

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 2023318 
(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)))