Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.8%

Time: 8.4s

Alternatives: 8

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: 99.8% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 35000:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -1.55e+43)
     (/ x y)
     (if (<= x 1e-30)
       t_0
       (if (<= x 35000.0) (* x (/ x y)) (if (<= x 8e+35) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.55e+43) {
		tmp = x / y;
	} else if (x <= 1e-30) {
		tmp = t_0;
	} else if (x <= 35000.0) {
		tmp = x * (x / y);
	} else if (x <= 8e+35) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + 1.0d0)
    if (x <= (-1.55d+43)) then
        tmp = x / y
    else if (x <= 1d-30) then
        tmp = t_0
    else if (x <= 35000.0d0) then
        tmp = x * (x / y)
    else if (x <= 8d+35) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -1.55e+43) {
		tmp = x / y;
	} else if (x <= 1e-30) {
		tmp = t_0;
	} else if (x <= 35000.0) {
		tmp = x * (x / y);
	} else if (x <= 8e+35) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -1.55e+43:
		tmp = x / y
	elif x <= 1e-30:
		tmp = t_0
	elif x <= 35000.0:
		tmp = x * (x / y)
	elif x <= 8e+35:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.55e+43)
		tmp = Float64(x / y);
	elseif (x <= 1e-30)
		tmp = t_0;
	elseif (x <= 35000.0)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 8e+35)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.55e+43)
		tmp = x / y;
	elseif (x <= 1e-30)
		tmp = t_0;
	elseif (x <= 35000.0)
		tmp = x * (x / y);
	elseif (x <= 8e+35)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55e+43], N[(x / y), $MachinePrecision], If[LessEqual[x, 1e-30], t$95$0, If[LessEqual[x, 35000.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e+35], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5500000000000001e43 or 7.9999999999999997e35 < x

   1. Initial program 66.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 82.5%

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

   if -1.5500000000000001e43 < x < 1e-30 or 35000 < x < 7.9999999999999997e35

   1. Initial program 99.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 80.4%

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

   if 1e-30 < x < 35000

   1. Initial program 99.4%

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

      1. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 84.6%

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

   6. Taylor expanded in x around 0 50.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 10^{-30}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 35000:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.84\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.84)))
   (+ (/ x y) 1.0)
   (* x (+ 1.0 (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.84)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.84d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x * (1.0d0 + ((x / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.84)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x * (1.0 + ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.84):
		tmp = (x / y) + 1.0
	else:
		tmp = x * (1.0 + ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.84))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.84)))
		tmp = (x / y) + 1.0;
	else
		tmp = x * (1.0 + ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.84]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.839999999999999969 < x

   1. Initial program 71.7%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 98.0%

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

   if -1 < x < 0.839999999999999969

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.8%

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

   6. Taylor expanded in x around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. distribute-rgt-in 98.0%

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

      4. neg-mul-1 98.0%

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

      5. unsub-neg 98.0%

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

      6. associate-*l/ 98.0%

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

      7. *-lft-identity 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.84\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(\frac{x}{y} - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.84\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.84)))
   (+ (/ x y) 1.0)
   (+ x (* x (- (/ x y) x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.84)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x + (x * ((x / y) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.84d0))) then
        tmp = (x / y) + 1.0d0
    else
        tmp = x + (x * ((x / y) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.84)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x + (x * ((x / y) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.84):
		tmp = (x / y) + 1.0
	else:
		tmp = x + (x * ((x / y) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.84))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x + Float64(x * Float64(Float64(x / y) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.84)))
		tmp = (x / y) + 1.0;
	else
		tmp = x + (x * ((x / y) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.84]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x + N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.839999999999999969 < x

   1. Initial program 71.7%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 98.0%

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

   if -1 < x < 0.839999999999999969

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.8%

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

   6. Taylor expanded in x around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. distribute-rgt-in 98.0%

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

      4. neg-mul-1 98.0%

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

      5. unsub-neg 98.0%

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

      6. associate-*l/ 98.0%

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

      7. *-lft-identity 98.0%

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

   8. Simplified 98.0%

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

      1. +-commutative 98.0%

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

      2. distribute-lft-in 98.0%

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

      3. *-rgt-identity 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.84\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) 1.0)))
   (if (or (<= x -1.0) (not (<= x 1.0))) t_0 (* x t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + 1.0;
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = t_0;
	} else {
		tmp = x * 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)) .or. (.not. (x <= 1.0d0))) then
        tmp = t_0
    else
        tmp = x * 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) || !(x <= 1.0)) {
		tmp = t_0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) + 1.0
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = t_0
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + 1.0)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = t_0;
	else
		tmp = Float64(x * 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) || ~((x <= 1.0)))
		tmp = t_0;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], t$95$0, N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 71.7%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 98.0%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. div-inv 99.8%

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

      4. associate-*l* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.2%

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

4. Final simplification 97.6%

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

Alternative 6: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+14} \lor \neg \left(x \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5e+14) (not (<= x 1.05e-8)))
   (+ (/ x y) 1.0)
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e+14) || !(x <= 1.05e-8)) {
		tmp = (x / y) + 1.0;
	} 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 <= (-3.5d+14)) .or. (.not. (x <= 1.05d-8))) then
        tmp = (x / y) + 1.0d0
    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 <= -3.5e+14) || !(x <= 1.05e-8)) {
		tmp = (x / y) + 1.0;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.5e+14) or not (x <= 1.05e-8):
		tmp = (x / y) + 1.0
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.5e+14) || !(x <= 1.05e-8))
		tmp = Float64(Float64(x / y) + 1.0);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.5e+14) || ~((x <= 1.05e-8)))
		tmp = (x / y) + 1.0;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.5e+14], N[Not[LessEqual[x, 1.05e-8]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e14 or 1.04999999999999997e-8 < x

   1. Initial program 71.9%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf 96.1%

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

   if -3.5e14 < x < 1.04999999999999997e-8

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 80.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+14} \lor \neg \left(x \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.05e-8))) (/ x y) x))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.05e-8)) {
		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 <= 1.05d-8))) 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 <= 1.05e-8)) {
		tmp = x / y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.05e-8):
		tmp = x / y
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.05e-8))
		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 <= 1.05e-8)))
		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, 1.05e-8]], $MachinePrecision]], N[(x / y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.04999999999999997e-8 < 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* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 73.6%

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

   if -1 < x < 1.04999999999999997e-8

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 79.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.5% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 86.8%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 43.3%

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

6. Final simplification 43.3%

   \[\leadsto x \]
7. Add Preprocessing

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 2024067 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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