Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 8.7s

Alternatives: 15

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: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1}\\ t_1 := \frac{\left(y + x\right) \cdot \frac{x}{x + 1}}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ x 1.0)))
        (t_1 (/ (* (+ y x) (/ x (+ x 1.0))) y)))
   (if (<= t_0 -2e+277)
     t_1
     (if (<= t_0 1e+41) (/ (fma (/ x y) x x) (+ x 1.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (x + 1.0);
	double t_1 = ((y + x) * (x / (x + 1.0))) / y;
	double tmp;
	if (t_0 <= -2e+277) {
		tmp = t_1;
	} else if (t_0 <= 1e+41) {
		tmp = fma((x / y), x, x) / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(x + 1.0))
	t_1 = Float64(Float64(Float64(y + x) * Float64(x / Float64(x + 1.0))) / y)
	tmp = 0.0
	if (t_0 <= -2e+277)
		tmp = t_1;
	elseif (t_0 <= 1e+41)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+277], t$95$1, If[LessEqual[t$95$0, 1e+41], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

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))) < -2.00000000000000001e277 or 1.00000000000000001e41 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 61.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000001e277 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000001e41

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1} \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{x + 1}}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1} \leq 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{x + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1}\\ \mathbf{if}\;t\_0 \leq -10000000000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ x 1.0))))
   (if (<= t_0 -10000000000.0)
     (/ (- x 1.0) y)
     (if (<= t_0 0.5) (fma (- x) x x) (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (x + 1.0);
	double tmp;
	if (t_0 <= -10000000000.0) {
		tmp = (x - 1.0) / y;
	} else if (t_0 <= 0.5) {
		tmp = fma(-x, x, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -10000000000.0)
		tmp = Float64(Float64(x - 1.0) / y);
	elseif (t_0 <= 0.5)
		tmp = fma(Float64(-x), x, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000000.0], N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[((-x) * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e10

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
   4. 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. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.3%

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

   if -1e10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.5

   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%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot x, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.6%

       \[\leadsto \mathsf{fma}\left(-x, x, x\right) \]

   if 0.5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
   4. 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. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around inf

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 92.7%

      \[\leadsto 1 \]

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

   1. Initial program 72.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1} \leq -10000000000:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. 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 2024230 
(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)))