Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.9%

Time: 6.7s

Alternatives: 9

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: 87.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 74.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0))))
   (if (<= x -5.6e+14)
     (/ x y)
     (if (<= x 4.3e-62)
       t_0
       (if (<= x 2.3e-24) (* x (/ x y)) (if (<= x 2.4e+21) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	double tmp;
	if (x <= -5.6e+14) {
		tmp = x / y;
	} else if (x <= 4.3e-62) {
		tmp = t_0;
	} else if (x <= 2.3e-24) {
		tmp = x * (x / y);
	} else if (x <= 2.4e+21) {
		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 <= (-5.6d+14)) then
        tmp = x / y
    else if (x <= 4.3d-62) then
        tmp = t_0
    else if (x <= 2.3d-24) then
        tmp = x * (x / y)
    else if (x <= 2.4d+21) 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 <= -5.6e+14) {
		tmp = x / y;
	} else if (x <= 4.3e-62) {
		tmp = t_0;
	} else if (x <= 2.3e-24) {
		tmp = x * (x / y);
	} else if (x <= 2.4e+21) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + 1.0)
	tmp = 0
	if x <= -5.6e+14:
		tmp = x / y
	elif x <= 4.3e-62:
		tmp = t_0
	elif x <= 2.3e-24:
		tmp = x * (x / y)
	elif x <= 2.4e+21:
		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 <= -5.6e+14)
		tmp = Float64(x / y);
	elseif (x <= 4.3e-62)
		tmp = t_0;
	elseif (x <= 2.3e-24)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 2.4e+21)
		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 <= -5.6e+14)
		tmp = x / y;
	elseif (x <= 4.3e-62)
		tmp = t_0;
	elseif (x <= 2.3e-24)
		tmp = x * (x / y);
	elseif (x <= 2.4e+21)
		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, -5.6e+14], N[(x / y), $MachinePrecision], If[LessEqual[x, 4.3e-62], t$95$0, If[LessEqual[x, 2.3e-24], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e+21], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6e14 or 2.4e21 < x

   1. Initial program 71.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 76.4%

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

   if -5.6e14 < x < 4.2999999999999997e-62 or 2.3000000000000001e-24 < x < 2.4e21

   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 81.9%

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

   if 4.2999999999999997e-62 < x < 2.3000000000000001e-24

   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%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 67.0%

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

      1. clear-num 66.7%

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

      2. un-div-inv 67.0%

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

      3. +-commutative 67.0%

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

      4. *-commutative 67.0%

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

      5. associate-/l* 67.0%

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

   7. Applied egg-rr 67.0%

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

   8. Taylor expanded in x around 0 67.0%

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

      1. associate-/r/ 67.0%

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

   10. Applied egg-rr 67.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-6} \lor \neg \left(x \leq 0.07\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.4e-6) (not (<= x 0.07)))
   (/ x (+ y (/ y x)))
   (* x (+ 1.0 (* x (+ (/ 1.0 y) -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -6.4e-6) || !(x <= 0.07)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -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 <= (-6.4d-6)) .or. (.not. (x <= 0.07d0))) then
        tmp = x / (y + (y / x))
    else
        tmp = x * (1.0d0 + (x * ((1.0d0 / y) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -6.4e-6) || !(x <= 0.07)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -6.4e-6) or not (x <= 0.07):
		tmp = x / (y + (y / x))
	else:
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -6.4e-6) || !(x <= 0.07))
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(1.0 / y) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -6.4e-6) || ~((x <= 0.07)))
		tmp = x / (y + (y / x));
	else
		tmp = x * (1.0 + (x * ((1.0 / y) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -6.4e-6], N[Not[LessEqual[x, 0.07]], $MachinePrecision]], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(N[(1.0 / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999997e-6 or 0.070000000000000007 < x

   1. Initial program 74.1%

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

      1. associate-/l* 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 y around 0 65.7%

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

      1. clear-num 65.8%

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

      2. un-div-inv 65.9%

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

      3. +-commutative 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-/l* 74.8%

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

   7. Applied egg-rr 74.8%

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

      1. *-commutative 74.8%

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

      2. +-commutative 74.8%

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

      3. distribute-rgt-in 67.4%

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

      4. *-un-lft-identity 67.4%

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

      5. associate-/l* 65.9%

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

      6. *-commutative 65.9%

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

      7. associate-/l* 75.0%

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

      8. *-inverses 75.0%

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

      9. *-commutative 75.0%

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

      10. *-un-lft-identity 75.0%

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

   9. Applied egg-rr 75.0%

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

   if -6.3999999999999997e-6 < x < 0.070000000000000007

   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 99.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-6} \lor \neg \left(x \leq 0.07\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 1.35e-66) x (if (<= x 1.0) (* x (/ x y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.35e-66) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} 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) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 1.35d-66) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = x * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 1.35e-66) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = x * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 1.35e-66:
		tmp = x
	elif x <= 1.0:
		tmp = x * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 1.35e-66)
		tmp = x;
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 1.35e-66)
		tmp = x;
	elseif (x <= 1.0)
		tmp = x * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 1.35e-66], x, If[LessEqual[x, 1.0], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 72.5%

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

   if -1 < x < 1.34999999999999998e-66

   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 84.0%

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

   if 1.34999999999999998e-66 < x < 1

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 62.4%

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

      1. clear-num 62.2%

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

      2. un-div-inv 62.4%

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

      3. +-commutative 62.4%

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

      4. *-commutative 62.4%

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

      5. associate-/l* 62.4%

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

   7. Applied egg-rr 62.4%

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

   8. Taylor expanded in x around 0 62.4%

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

      1. associate-/r/ 62.4%

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

   10. Applied egg-rr 62.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-10} \lor \neg \left(x \leq 3.9 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1e-10) (not (<= x 3.9e-65)))
   (/ x (+ y (/ y x)))
   (/ x (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e-10) || !(x <= 3.9e-65)) {
		tmp = x / (y + (y / x));
	} 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 <= (-1.1d-10)) .or. (.not. (x <= 3.9d-65))) then
        tmp = x / (y + (y / x))
    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 <= -1.1e-10) || !(x <= 3.9e-65)) {
		tmp = x / (y + (y / x));
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1e-10) or not (x <= 3.9e-65):
		tmp = x / (y + (y / x))
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1e-10) || !(x <= 3.9e-65))
		tmp = Float64(x / Float64(y + Float64(y / x)));
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.1e-10) || ~((x <= 3.9e-65)))
		tmp = x / (y + (y / x));
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1e-10], N[Not[LessEqual[x, 3.9e-65]], $MachinePrecision]], N[(x / N[(y + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999995e-10 or 3.9000000000000004e-65 < x

   1. Initial program 77.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 y around 0 65.3%

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

      1. clear-num 65.4%

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

      2. un-div-inv 65.5%

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

      3. +-commutative 65.5%

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

      4. *-commutative 65.5%

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

      5. associate-/l* 73.4%

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

   7. Applied egg-rr 73.4%

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

      1. *-commutative 73.4%

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

      2. +-commutative 73.4%

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

      3. distribute-rgt-in 66.9%

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

      4. *-un-lft-identity 66.9%

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

      5. associate-/l* 65.5%

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

      6. *-commutative 65.5%

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

      7. associate-/l* 73.5%

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

      8. *-inverses 73.5%

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

      9. *-commutative 73.5%

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

      10. *-un-lft-identity 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -1.09999999999999995e-10 < x < 3.9000000000000004e-65

   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 84.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-10} \lor \neg \left(x \leq 3.9 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{x}{y + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 inf 72.5%

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

   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. 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 79.2%

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

4. Final simplification 76.4%

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

Alternative 7: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -740000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 44:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -740000000000.0) 1.0 (if (<= x 44.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -740000000000.0) {
		tmp = 1.0;
	} else if (x <= 44.0) {
		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 <= (-740000000000.0d0)) then
        tmp = 1.0d0
    else if (x <= 44.0d0) 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 <= -740000000000.0) {
		tmp = 1.0;
	} else if (x <= 44.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -740000000000.0:
		tmp = 1.0
	elif x <= 44.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -740000000000.0)
		tmp = 1.0;
	elseif (x <= 44.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -740000000000.0)
		tmp = 1.0;
	elseif (x <= 44.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -740000000000.0], 1.0, If[LessEqual[x, 44.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4e11 or 44 < x

   1. Initial program 73.1%

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

      1. associate-/l* 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 y around inf 28.2%

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

   6. Taylor expanded in x around inf 27.6%

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

   if -7.4e11 < x < 44

   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 77.7%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (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)) / (x + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

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

Alternative 9: 14.6% 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 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 57.7%

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

6. Taylor expanded in x around inf 13.4%

   \[\leadsto \color{blue}{1} \]
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 2024101 
(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)))