Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.2% → 99.9%

Time: 6.6s

Alternatives: 10

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: 88.2% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e-20)
   (/ (* (+ y x) (/ x (+ 1.0 x))) y)
   (if (<= x 1.6e+16)
     (* (- x 1.0) (/ (fma (/ x y) x x) (fma x x -1.0)))
     (+ (/ (- x 1.0) y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-20) {
		tmp = ((y + x) * (x / (1.0 + x))) / y;
	} else if (x <= 1.6e+16) {
		tmp = (x - 1.0) * (fma((x / y), x, x) / fma(x, x, -1.0));
	} else {
		tmp = ((x - 1.0) / y) + 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e-20)
		tmp = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y);
	elseif (x <= 1.6e+16)
		tmp = Float64(Float64(x - 1.0) * Float64(fma(Float64(x / y), x, x) / fma(x, x, -1.0)));
	else
		tmp = Float64(Float64(Float64(x - 1.0) / y) + 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e-20], N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.6e+16], N[(N[(x - 1.0), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-20

   1. Initial program 80.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 -1.45e-20 < x < 1.6e16

   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

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 1.6e16 < x

   1. Initial program 67.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+16}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -100000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -100000.0)
     (/ x y)
     (if (<= t_0 0.002) (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) / (1.0 + x);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.002) {
		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(1.0 + x))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.002)
		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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.002], N[((-x) * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

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))) < -1e5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -1e5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 85.8%

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

   if 2e-3 < (/.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 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 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 90.4%

      \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -100000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -100000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -100000.0) (/ x y) (if (<= t_0 2.0) (/ x (+ 1.0 x)) (/ x y)))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} 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 / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-100000.0d0)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -100000.0) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	tmp = 0
	if t_0 <= -100000.0:
		tmp = x / y
	elif t_0 <= 2.0:
		tmp = x / (1.0 + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -100000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -100000.0)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -1e5 < (/.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. lower-+.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -100000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 0.002) (fma (- x) x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 0.002) {
		tmp = fma(-x, x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x)) <= 0.002)
		tmp = fma(Float64(-x), x, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 0.002], N[((-x) * x + x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2e-3

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.7%

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

   if 2e-3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 34.8%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= x -1.45e-20) t_0 (if (<= x 5e-83) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (x <= -1.45e-20) {
		tmp = t_0;
	} else if (x <= 5e-83) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (x <= -1.45e-20)
		tmp = t_0;
	elseif (x <= 5e-83)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.45e-20], t$95$0, If[LessEqual[x, 5e-83], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e-20 or 5e-83 < x

   1. Initial program 78.0%

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

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

   5. Applied rewrites 99.9%

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

   if -1.45e-20 < x < 5e-83

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -2e+38)
     t_0
     (if (<= x 2e+16) (/ (fma (/ x y) x x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -2e+38) {
		tmp = t_0;
	} else if (x <= 2e+16) {
		tmp = fma((x / y), x, x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -2e+38)
		tmp = t_0;
	elseif (x <= 2e+16)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -2e+38], t$95$0, If[LessEqual[x, 2e+16], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999995e38 or 2e16 < x

   1. Initial program 71.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 100.0%

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

   if -1.99999999999999995e38 < x < 2e16

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 73.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.9%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. 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

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

      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. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 8: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0) t_0 (if (<= x 1.25) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.25) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.25)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.25], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25 < x

   1. Initial program 73.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.9%

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

   if -1 < x < 1.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -960000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -960000.0) t_0 (if (<= x 7000.0) (/ x (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -960000.0) {
		tmp = t_0;
	} else if (x <= 7000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) / y) + 1.0d0
    if (x <= (-960000.0d0)) then
        tmp = t_0
    else if (x <= 7000.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -960000.0) {
		tmp = t_0;
	} else if (x <= 7000.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) / y) + 1.0
	tmp = 0
	if x <= -960000.0:
		tmp = t_0
	elif x <= 7000.0:
		tmp = x / (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -960000.0)
		tmp = t_0;
	elseif (x <= 7000.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) / y) + 1.0;
	tmp = 0.0;
	if (x <= -960000.0)
		tmp = t_0;
	elseif (x <= 7000.0)
		tmp = x / (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -960000.0], t$95$0, If[LessEqual[x, 7000.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6e5 or 7e3 < x

   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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{x} + \frac{1}{y}\right) - \frac{1}{x \cdot y}\right)} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 100.0%

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

   if -9.6e5 < x < 7e3

   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. lower-+.f64 73.4

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

   5. Applied rewrites 73.4%

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

Alternative 10: 14.3% accurate, 34.0× 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 87.5%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 47.4

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

5. Applied rewrites 47.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 13.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024248 
(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)))