Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.9%

Time: 8.8s

Alternatives: 13

Speedup: 1.1×

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \left(x + y\right)}{y}\\ \mathbf{elif}\;x \leq 5800000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-62)
   (/ (* (/ x (+ x 1.0)) (+ x y)) y)
   (if (<= x 5800000000000.0)
     (/ (fma (/ x y) x x) (+ x 1.0))
     (+ 1.0 (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-62) {
		tmp = ((x / (x + 1.0)) * (x + y)) / y;
	} else if (x <= 5800000000000.0) {
		tmp = fma((x / y), x, x) / (x + 1.0);
	} else {
		tmp = 1.0 + ((x + -1.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-62)
		tmp = Float64(Float64(Float64(x / Float64(x + 1.0)) * Float64(x + y)) / y);
	elseif (x <= 5800000000000.0)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-62], N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 5800000000000.0], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000002e-62

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.0000000000000002e-62 < x < 5.8e12

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 5.8e12 < x

   1. Initial program 75.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{x}{y} - \color{blue}{x}, x\right) \]

      9. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{x}{y} - x}, x\right) \]

      10. lower-/.f64 28.1

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{x}{y}} - x, x\right) \]

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{y} - x, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{-1 \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in x around inf

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

       1. associate--l+ N/A

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

       2. distribute-rgt-in N/A

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

       3. lft-mult-inverse N/A

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

       4. *-commutative N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

       7. associate-*r/ N/A

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

       8. *-rgt-identity N/A

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

       9. distribute-rgt-neg-out N/A

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

       10. associate-/r* N/A

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

       11. associate-*r/ N/A

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

       12. rgt-mult-inverse N/A

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

       13. sub-neg N/A

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

       14. lower-+.f64 N/A

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

       15. div-sub N/A

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

       16. lower-/.f64 N/A

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

       17. sub-neg N/A

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

       18. metadata-eval N/A

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

       19. lower-+.f64 100.0

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

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{1 + \frac{x + -1}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ t_1 := \frac{x + -1}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+111}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))) (t_1 (/ (+ x -1.0) y)))
   (if (<= t_0 -2e+25)
     t_1
     (if (<= t_0 2.0)
       (/ x (+ x 1.0))
       (if (<= t_0 5e+111) (* x (/ x y)) t_1)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double t_1 = (x + -1.0) / y;
	double tmp;
	if (t_0 <= -2e+25) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else if (t_0 <= 5e+111) {
		tmp = x * (x / y);
	} else {
		tmp = t_1;
	}
	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 * ((x / y) + 1.0d0)) / (x + 1.0d0)
    t_1 = (x + (-1.0d0)) / y
    if (t_0 <= (-2d+25)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = x / (x + 1.0d0)
    else if (t_0 <= 5d+111) then
        tmp = x * (x / y)
    else
        tmp = t_1
    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 t_1 = (x + -1.0) / y;
	double tmp;
	if (t_0 <= -2e+25) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = x / (x + 1.0);
	} else if (t_0 <= 5e+111) {
		tmp = x * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	t_1 = (x + -1.0) / y
	tmp = 0
	if t_0 <= -2e+25:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = x / (x + 1.0)
	elif t_0 <= 5e+111:
		tmp = x * (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	t_1 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (t_0 <= -2e+25)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (t_0 <= 5e+111)
		tmp = Float64(x * Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	t_1 = (x + -1.0) / y;
	tmp = 0.0;
	if (t_0 <= -2e+25)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = x / (x + 1.0);
	elseif (t_0 <= 5e+111)
		tmp = x * (x / y);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+25], t$95$1, If[LessEqual[t$95$0, 2.0], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+111], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000018e25 or 4.9999999999999997e111 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 67.4%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 87.8

         \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(x, y, y\right)}} \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(x, y, y\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.7%

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

   if -2.00000000000000018e25 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.9999999999999997e111

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. lower-fma.f64 81.6

         \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(x, y, y\right)}} \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left(x, y, y\right)}} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.3%

      \[\leadsto x \cdot \frac{x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. 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 2024228 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

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