Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.8%

Time: 6.3s

Alternatives: 10

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: 88.4% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.8%

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

   3. clear-num 99.9%

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

   4. +-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.05)
     (/ x y)
     (if (<= x 1.05e-144)
       t_0
       (if (<= x 5.5e-107) (/ x (/ y x)) (if (<= x 3.1e+16) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.05) {
		tmp = x / y;
	} else if (x <= 1.05e-144) {
		tmp = t_0;
	} else if (x <= 5.5e-107) {
		tmp = x / (y / x);
	} else if (x <= 3.1e+16) {
		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 / (1.0d0 + x)
    if (x <= (-1.05d0)) then
        tmp = x / y
    else if (x <= 1.05d-144) then
        tmp = t_0
    else if (x <= 5.5d-107) then
        tmp = x / (y / x)
    else if (x <= 3.1d+16) 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 / (1.0 + x);
	double tmp;
	if (x <= -1.05) {
		tmp = x / y;
	} else if (x <= 1.05e-144) {
		tmp = t_0;
	} else if (x <= 5.5e-107) {
		tmp = x / (y / x);
	} else if (x <= 3.1e+16) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.05:
		tmp = x / y
	elif x <= 1.05e-144:
		tmp = t_0
	elif x <= 5.5e-107:
		tmp = x / (y / x)
	elif x <= 3.1e+16:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x / y);
	elseif (x <= 1.05e-144)
		tmp = t_0;
	elseif (x <= 5.5e-107)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 3.1e+16)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x / y;
	elseif (x <= 1.05e-144)
		tmp = t_0;
	elseif (x <= 5.5e-107)
		tmp = x / (y / x);
	elseif (x <= 3.1e+16)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.05e-144], t$95$0, If[LessEqual[x, 5.5e-107], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+16], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004 or 3.1e16 < x

   1. Initial program 72.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 71.6%

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

   if -1.05000000000000004 < x < 1.0500000000000001e-144 or 5.49999999999999986e-107 < x < 3.1e16

   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 78.0%

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

   if 1.0500000000000001e-144 < x < 5.49999999999999986e-107

   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* 99.7%

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

      3. +-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. div-sub 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. 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. Taylor expanded in y around 0 78.7%

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

      1. distribute-lft-in 78.7%

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

      2. *-rgt-identity 78.7%

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

      3. associate-*r/ 78.5%

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

      4. *-rgt-identity 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in x around 0 78.5%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.05)
     (/ (+ x -1.0) y)
     (if (<= x 1.05e-144)
       t_0
       (if (<= x 5.5e-107) (/ x (/ y x)) (if (<= x 7.5e+15) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.05) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1.05e-144) {
		tmp = t_0;
	} else if (x <= 5.5e-107) {
		tmp = x / (y / x);
	} else if (x <= 7.5e+15) {
		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 / (1.0d0 + x)
    if (x <= (-1.05d0)) then
        tmp = (x + (-1.0d0)) / y
    else if (x <= 1.05d-144) then
        tmp = t_0
    else if (x <= 5.5d-107) then
        tmp = x / (y / x)
    else if (x <= 7.5d+15) 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 / (1.0 + x);
	double tmp;
	if (x <= -1.05) {
		tmp = (x + -1.0) / y;
	} else if (x <= 1.05e-144) {
		tmp = t_0;
	} else if (x <= 5.5e-107) {
		tmp = x / (y / x);
	} else if (x <= 7.5e+15) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.05:
		tmp = (x + -1.0) / y
	elif x <= 1.05e-144:
		tmp = t_0
	elif x <= 5.5e-107:
		tmp = x / (y / x)
	elif x <= 7.5e+15:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(x + -1.0) / y);
	elseif (x <= 1.05e-144)
		tmp = t_0;
	elseif (x <= 5.5e-107)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 7.5e+15)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (x + -1.0) / y;
	elseif (x <= 1.05e-144)
		tmp = t_0;
	elseif (x <= 5.5e-107)
		tmp = x / (y / x);
	elseif (x <= 7.5e+15)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.05e-144], t$95$0, If[LessEqual[x, 5.5e-107], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+15], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05000000000000004

   1. Initial program 80.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 98.5%

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

   5. Taylor expanded in y around 0 66.3%

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

   if -1.05000000000000004 < x < 1.0500000000000001e-144 or 5.49999999999999986e-107 < x < 7.5e15

   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 78.0%

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

   if 1.0500000000000001e-144 < x < 5.49999999999999986e-107

   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* 99.7%

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

      3. +-commutative 99.7%

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

      4. remove-double-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. div-sub 99.7%

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

      7. distribute-frac-neg 99.7%

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

      8. *-inverses 99.7%

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

      9. 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. Taylor expanded in y around 0 78.7%

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

      1. distribute-lft-in 78.7%

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

      2. *-rgt-identity 78.7%

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

      3. associate-*r/ 78.5%

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

      4. *-rgt-identity 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in x around 0 78.5%

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

   if 7.5e15 < x

   1. Initial program 64.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 78.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-144}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 80.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. sub-div 98.5%

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

      4. sub-neg 98.5%

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

      5. metadata-eval 98.5%

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

   6. Applied egg-rr 98.5%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-/r/ 98.8%

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

      3. /-rgt-identity 98.8%

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

   6. Applied egg-rr 98.8%

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

   if 1 < x

   1. Initial program 67.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 97.8%

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

4. Final simplification 98.5%

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

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 15.5)))
   (/ (+ x -1.0) y)
   (* x (+ 1.0 (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 15.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x * (1.0 + (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 15.5d0))) then
        tmp = (x + (-1.0d0)) / y
    else
        tmp = x * (1.0d0 + (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 15.5)) {
		tmp = (x + -1.0) / y;
	} else {
		tmp = x * (1.0 + (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 15.5):
		tmp = (x + -1.0) / y
	else:
		tmp = x * (1.0 + (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 15.5))
		tmp = Float64(Float64(x + -1.0) / y);
	else
		tmp = Float64(x * Float64(1.0 + Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 15.5)))
		tmp = (x + -1.0) / y;
	else
		tmp = x * (1.0 + (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 15.5]], $MachinePrecision]], N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 15.5 < x

   1. Initial program 73.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 98.8%

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

   5. Taylor expanded in y around 0 71.4%

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

   if -1 < x < 15.5

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. associate-/r/ 98.2%

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

      3. /-rgt-identity 98.2%

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

   6. Applied egg-rr 98.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{x + -1}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.6%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.1%

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

      1. associate--l+ 98.1%

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

      2. +-commutative 98.1%

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

      3. sub-div 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

   6. Applied egg-rr 98.1%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.6%

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

      3. +-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. div-sub 99.6%

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

      7. distribute-frac-neg 99.6%

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

      8. *-inverses 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. associate-/r/ 98.8%

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

      3. /-rgt-identity 98.8%

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

   6. Applied egg-rr 98.8%

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

4. Final simplification 98.5%

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

Alternative 7: 75.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.1e+15))) (/ x y) (/ x (+ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.1e+15)) {
		tmp = x / y;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 2.1d+15))) then
        tmp = x / y
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.1e+15)) {
		tmp = x / y;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.1e+15):
		tmp = x / y
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.1e+15))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.1e+15)))
		tmp = x / y;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.1e+15]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.1e15 < x

   1. Initial program 72.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 71.6%

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

   if -1.05000000000000004 < x < 2.1e15

   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 74.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.1 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]

Alternative 8: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.15))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15)) {
		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 <= 1.15d0))) 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 <= 1.15)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.15):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.15))
		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 <= 1.15)))
		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, 1.15]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.1499999999999999 < x

   1. Initial program 73.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. Taylor expanded in x around inf 70.4%

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

   if -1 < x < 1.1499999999999999

   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 x around 0 74.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 49.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -960000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 400000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -960000000000.0) 1.0 (if (<= x 400000000.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -960000000000.0) {
		tmp = 1.0;
	} else if (x <= 400000000.0) {
		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 <= (-960000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 400000000.0d0) 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 <= -960000000000.0) {
		tmp = 1.0;
	} else if (x <= 400000000.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -960000000000.0:
		tmp = 1.0
	elif x <= 400000000.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -960000000000.0)
		tmp = 1.0;
	elseif (x <= 400000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -960000000000.0)
		tmp = 1.0;
	elseif (x <= 400000000.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -960000000000.0], 1.0, If[LessEqual[x, 400000000.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6e11 or 4e8 < x

   1. Initial program 72.2%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 27.6%

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

   5. Taylor expanded in x around inf 27.1%

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

   if -9.6e11 < x < 4e8

   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 71.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -960000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 400000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 13.9% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 87.9%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around inf 53.0%

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

5. Taylor expanded in x around inf 13.8%

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

6. Final simplification 13.8%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023336 
(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)))