Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.8%

Time: 1.2s

Alternatives: 8

Speedup: 1.1×

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x \cdot t\_0}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (<= x -6.2e+37) t_0 (if (<= x 5e+14) (/ (* x t_0) (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -6.2e+37) {
		tmp = t_0;
	} else if (x <= 5e+14) {
		tmp = (x * t_0) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) + 1.0d0
    if (x <= (-6.2d+37)) then
        tmp = t_0
    else if (x <= 5d+14) then
        tmp = (x * t_0) / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -6.2e+37) {
		tmp = t_0;
	} else if (x <= 5e+14) {
		tmp = (x * t_0) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if x <= -6.2e+37:
		tmp = t_0
	elif x <= 5e+14:
		tmp = (x * t_0) / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if (x <= -6.2e+37)
		tmp = t_0;
	elseif (x <= 5e+14)
		tmp = Float64(Float64(x * t_0) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if (x <= -6.2e+37)
		tmp = t_0;
	elseif (x <= 5e+14)
		tmp = (x * t_0) / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e+37], t$95$0, If[LessEqual[x, 5e+14], N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2000000000000004e37 or 5e14 < x

   1. Initial program 69.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   if -6.2000000000000004e37 < x < 5e14

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 56.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -5000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(\left(x + 1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -5000000000000.0)
     t_0
     (if (<= t_1 5e-5) (* x (+ (+ x 1.0) 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-5) {
		tmp = x * ((x + 1.0) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) + 1.0d0
    t_1 = (x * t_0) / (x + 1.0d0)
    if (t_1 <= (-5000000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5d-5) then
        tmp = x * ((x + 1.0d0) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -5000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-5) {
		tmp = x * ((x + 1.0) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	t_1 = (x * t_0) / (x + 1.0)
	tmp = 0
	if t_1 <= -5000000000000.0:
		tmp = t_0
	elif t_1 <= 5e-5:
		tmp = x * ((x + 1.0) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -5000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-5)
		tmp = Float64(x * Float64(Float64(x + 1.0) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	t_1 = (x * t_0) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -5000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 5e-5)
		tmp = x * ((x + 1.0) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000000.0], t$95$0, If[LessEqual[t$95$1, 5e-5], N[(x * N[(N[(x + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e12 or 5.00000000000000024e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.7%

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

   if -5e12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.00000000000000024e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 17.0%

      \[\leadsto x \cdot \left(\left(x + 1\right) + 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 44.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -5000000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 1.0000005:\\ \;\;\;\;x \cdot \left(\left(x + 1\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -5000000000000.0)
     (/ x y)
     (if (<= t_0 1.0000005) (* x (+ (+ x 1.0) 1.0)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 1.0000005) {
		tmp = x * ((x + 1.0) + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if (t_0 <= (-5000000000000.0d0)) then
        tmp = x / y
    else if (t_0 <= 1.0000005d0) then
        tmp = x * ((x + 1.0d0) + 1.0d0)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -5000000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 1.0000005) {
		tmp = x * ((x + 1.0) + 1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if t_0 <= -5000000000000.0:
		tmp = x / y
	elif t_0 <= 1.0000005:
		tmp = x * ((x + 1.0) + 1.0)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -5000000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 1.0000005)
		tmp = Float64(x * Float64(Float64(x + 1.0) + 1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -5000000000000.0)
		tmp = x / y;
	elseif (t_0 <= 1.0000005)
		tmp = x * ((x + 1.0) + 1.0);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5000000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1.0000005], N[(x * N[(N[(x + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5e12 or 1.0000005000000001 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.2%

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

   if -5e12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.0000005000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 14.4%

      \[\leadsto x \cdot \left(\left(x + 1\right) + 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.7%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 96.6%

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

Alternative 5: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 4.6e-5) (* x (/ x y)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 4.6e-5) {
		tmp = x * (x / y);
	} 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 = (x / y) + 1.0d0
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 4.6d-5) then
        tmp = x * (x / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 4.6e-5) {
		tmp = x * (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 4.6e-5:
		tmp = x * (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 4.6e-5)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) + 1.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 4.6e-5)
		tmp = x * (x / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 4.6e-5], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.6e-5 < x

   1. Initial program 72.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.0%

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

   if -1 < x < 4.6e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot 1 \]

   8. Applied rewrites 26.3%

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

Alternative 6: 14.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 13.2%

   \[\leadsto x \cdot \left(\left(x + 1\right) + 1\right) \]
9. Add Preprocessing

Alternative 7: 3.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.1%

   \[\leadsto \left(x + 1\right) + 1 \]
9. Add Preprocessing

Alternative 8: 3.7% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 39.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.1%

   \[\leadsto x + \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 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 2024321 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (* (/ x 1) (/ (+ (/ x y) 1) (+ x 1))))

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