Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 7.5s

Alternatives: 12

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: 88.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{\left(y + x\right) \cdot \frac{x}{x - -1}}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-6) t_1 (if (<= t_0 1e-12) (fma (- (/ x y) x) x x) 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-6) {
		tmp = t_1;
	} else if (t_0 <= 1e-12) {
		tmp = fma(((x / y) - x), x, x);
	} 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-6)
		tmp = t_1;
	elseif (t_0 <= 1e-12)
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	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-6], t$95$1, If[LessEqual[t$95$0, 1e-12], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $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))) < -1.99999999999999991e-6 or 9.9999999999999998e-13 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 78.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 -1.99999999999999991e-6 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e-13

   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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{x - -1}}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{x - -1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))))
   (if (<= t_0 -10.0)
     (/ x y)
     (if (<= t_0 2e-5) (fma (- x) x x) (if (<= t_0 50000000.0) 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 <= -10.0) {
		tmp = x / y;
	} else if (t_0 <= 2e-5) {
		tmp = fma(-x, x, x);
	} else if (t_0 <= 50000000.0) {
		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 <= -10.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 2e-5)
		tmp = fma(Float64(-x), x, x);
	elseif (t_0 <= 50000000.0)
		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, -10.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2e-5], N[((-x) * x + x), $MachinePrecision], If[LessEqual[t$95$0, 50000000.0], 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))) < -10 or 5e7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.00000000000000016e-5

   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 99.1

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

   5. Applied rewrites 99.1%

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

   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 88.0%

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

   if 2.00000000000000016e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5e7

   1. Initial program 100.0%

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

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

      5. associate-*l/ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

      14. lower-pow.f64 99.6

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

   4. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{x}{y} - 1}{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}}}}{x + 1} \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 88.0

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

   7. Applied rewrites 88.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 81.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -10:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ t_1 := \frac{x - 1}{y}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 50000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))) (t_1 (/ (- x 1.0) y)))
   (if (<= t_0 -10.0) t_1 (if (<= t_0 50000000.0) (/ 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 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000.0) {
		tmp = x / (x - -1.0);
	} 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 / y) - (-1.0d0)) * x) / (x - (-1.0d0))
    t_1 = (x - 1.0d0) / y
    if (t_0 <= (-10.0d0)) then
        tmp = t_1
    else if (t_0 <= 50000000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	double t_1 = (x - 1.0) / y;
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 50000000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0)
	t_1 = (x - 1.0) / y
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 50000000.0:
		tmp = 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(x - 1.0) / y)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 50000000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	t_1 = (x - 1.0) / y;
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 50000000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_1;
	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[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 50000000.0], N[(x / 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))) < -10 or 5e7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.5%

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 84.9%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 83.9%

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

   if -10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5e7

   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 88.6

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

   5. Applied rewrites 88.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -10:\\ \;\;\;\;\frac{x - 1}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 50000000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- (/ x y) -1.0) x) (- x -1.0))))
   (if (<= t_0 -10.0) (/ x y) (if (<= t_0 4.0) (/ x (- x -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 <= -10.0) {
		tmp = x / y;
	} else if (t_0 <= 4.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) - (-1.0d0)) * x) / (x - (-1.0d0))
    if (t_0 <= (-10.0d0)) then
        tmp = x / y
    else if (t_0 <= 4.0d0) then
        tmp = x / (x - (-1.0d0))
    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) / (x - -1.0);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = x / y;
	} else if (t_0 <= 4.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0)
	tmp = 0
	if t_0 <= -10.0:
		tmp = x / y
	elif t_0 <= 4.0:
		tmp = x / (x - -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 <= -10.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 4.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) - -1.0) * x) / (x - -1.0);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = x / y;
	elseif (t_0 <= 4.0)
		tmp = x / (x - -1.0);
	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[(x - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(x / N[(x - -1.0), $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))) < -10 or 4 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -10 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4

   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 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq -10:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 4:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- (/ x y) -1.0) x) (- x -1.0)) 2e-5) (fma (- x) x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= 2e-5) {
		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(x - -1.0)) <= 2e-5)
		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[(x - -1.0), $MachinePrecision]), $MachinePrecision], 2e-5], 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))) < 2.00000000000000016e-5

   1. Initial program 90.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

   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 65.2%

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

   if 2.00000000000000016e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.6%

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

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

      5. associate-*l/ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

      14. lower-pow.f64 53.6

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

   4. Applied rewrites 53.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{x}{y} - 1}{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}}}}{x + 1} \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 40.3

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 37.6%

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

4. Final simplification 56.5%

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

Alternative 6: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} - -1\right) \cdot x}{x - -1} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- (/ x y) -1.0) x) (- x -1.0)) 2e-5) (* 1.0 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= 2e-5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((((x / y) - (-1.0d0)) * x) / (x - (-1.0d0))) <= 2d-5) then
        tmp = 1.0d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= 2e-5) {
		tmp = 1.0 * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) - -1.0) * x) / (x - -1.0)) <= 2e-5:
		tmp = 1.0 * 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(x - -1.0)) <= 2e-5)
		tmp = Float64(1.0 * x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((((x / y) - -1.0) * x) / (x - -1.0)) <= 2e-5)
		tmp = 1.0 * x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 2e-5], N[(1.0 * 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))) < 2.00000000000000016e-5

   1. Initial program 90.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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. 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 87.1

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

   5. Applied rewrites 87.1%

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

   7. Applied rewrites 77.0%

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

   9. Applied rewrites 93.7%

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

   10. Taylor expanded in x around 0

       \[\leadsto x \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 57.4%

       \[\leadsto x \cdot 1 \]

   if 2.00000000000000016e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.6%

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

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

      5. associate-*l/ N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

      14. lower-pow.f64 53.6

          \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

   4. Applied rewrites 53.6%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{x}{y} - 1}{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}}}}{x + 1} \]

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 40.3

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 37.6%

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

4. Final simplification 51.2%

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

Alternative 7: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - \left(-x\right)}{y}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+16}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y (- x)) y)))
   (if (<= x -1e+16)
     t_0
     (if (<= x 1.72e+16) (* (/ (+ y x) (fma y x y)) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - -x) / y;
	double tmp;
	if (x <= -1e+16) {
		tmp = t_0;
	} else if (x <= 1.72e+16) {
		tmp = ((y + x) / fma(y, x, y)) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - Float64(-x)) / y)
	tmp = 0.0
	if (x <= -1e+16)
		tmp = t_0;
	elseif (x <= 1.72e+16)
		tmp = Float64(Float64(Float64(y + x) / fma(y, x, y)) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - (-x)), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1e+16], t$95$0, If[LessEqual[x, 1.72e+16], N[(N[(N[(y + x), $MachinePrecision] / N[(y * x + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e16 or 1.72e16 < x

   1. Initial program 73.9%

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{y - -1 \cdot x}{y} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{y - \left(-x\right)}{y} \]

   if -1e16 < x < 1.72e16

   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 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 83.6

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

   5. Applied rewrites 83.6%

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

   7. Applied rewrites 83.6%

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{y - \left(-x\right)}{y}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+16}:\\ \;\;\;\;\frac{y + x}{\mathsf{fma}\left(y, x, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(-x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y (- 1.0 x)) y)))
   (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 = (y - (1.0 - x)) / y;
	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(y - Float64(1.0 - x)) / y)
	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[(y - N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $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 75.4%

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

   3. Taylor expanded in y around 0

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 97.6%

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

   if -1 < x < 1

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 9: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - \left(1 - x\right)}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y (- 1.0 x)) y)))
   (if (<= x -1.0) t_0 (if (<= x 1.3) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - (1.0 - x)) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.3) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y - Float64(1.0 - x)) / y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3)
		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[(y - N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.3], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.30000000000000004 < x

   1. Initial program 75.4%

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

   3. Taylor expanded in y around 0

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 97.6%

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

   if -1 < x < 1.30000000000000004

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   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 97.4%

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

Alternative 10: 86.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - \left(1 - x\right)}{y}\\ \mathbf{if}\;x \leq -5500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2800:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y (- 1.0 x)) y)))
   (if (<= x -5500.0) t_0 (if (<= x 2800.0) (/ x (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - (1.0 - x)) / y;
	double tmp;
	if (x <= -5500.0) {
		tmp = t_0;
	} else if (x <= 2800.0) {
		tmp = x / (x - -1.0);
	} 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 = (y - (1.0d0 - x)) / y
    if (x <= (-5500.0d0)) then
        tmp = t_0
    else if (x <= 2800.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - (1.0 - x)) / y;
	double tmp;
	if (x <= -5500.0) {
		tmp = t_0;
	} else if (x <= 2800.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - (1.0 - x)) / y
	tmp = 0
	if x <= -5500.0:
		tmp = t_0
	elif x <= 2800.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - Float64(1.0 - x)) / y)
	tmp = 0.0
	if (x <= -5500.0)
		tmp = t_0;
	elseif (x <= 2800.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - (1.0 - x)) / y;
	tmp = 0.0;
	if (x <= -5500.0)
		tmp = t_0;
	elseif (x <= 2800.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -5500.0], t$95$0, If[LessEqual[x, 2800.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5500 or 2800 < x

   1. Initial program 74.8%

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.3%

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

   if -5500 < x < 2800

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5500:\\ \;\;\;\;\frac{y - \left(1 - x\right)}{y}\\ \mathbf{elif}\;x \leq 2800:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(1 - x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - \left(-x\right)}{y}\\ \mathbf{if}\;x \leq -15500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 46000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y (- x)) y)))
   (if (<= x -15500.0) t_0 (if (<= x 46000.0) (/ x (- x -1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = (y - -x) / y;
	double tmp;
	if (x <= -15500.0) {
		tmp = t_0;
	} else if (x <= 46000.0) {
		tmp = x / (x - -1.0);
	} 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 = (y - -x) / y
    if (x <= (-15500.0d0)) then
        tmp = t_0
    else if (x <= 46000.0d0) then
        tmp = x / (x - (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - -x) / y;
	double tmp;
	if (x <= -15500.0) {
		tmp = t_0;
	} else if (x <= 46000.0) {
		tmp = x / (x - -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - -x) / y
	tmp = 0
	if x <= -15500.0:
		tmp = t_0
	elif x <= 46000.0:
		tmp = x / (x - -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - Float64(-x)) / y)
	tmp = 0.0
	if (x <= -15500.0)
		tmp = t_0;
	elseif (x <= 46000.0)
		tmp = Float64(x / Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - -x) / y;
	tmp = 0.0;
	if (x <= -15500.0)
		tmp = t_0;
	elseif (x <= 46000.0)
		tmp = x / (x - -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - (-x)), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -15500.0], t$95$0, If[LessEqual[x, 46000.0], N[(x / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -15500 or 46000 < x

   1. Initial program 74.8%

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

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.3%

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

   12. Taylor expanded in x around inf

       \[\leadsto \frac{y - -1 \cdot x}{y} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.7%

       \[\leadsto \frac{y - \left(-x\right)}{y} \]

   if -15500 < x < 46000

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15500:\\ \;\;\;\;\frac{y - \left(-x\right)}{y}\\ \mathbf{elif}\;x \leq 46000:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \left(-x\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 14.4% 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 87.9%

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

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

   5. associate-*l/ N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. pow2 N/A

       \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

   14. lower-pow.f64 67.8

       \[\leadsto \frac{\frac{1}{\frac{\frac{x}{y} - 1}{\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}} - 1\right) \cdot x}}}{x + 1} \]

4. Applied rewrites 67.8%

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{x}{y} - 1}{\left({\left(\frac{x}{y}\right)}^{2} - 1\right) \cdot x}}}}{x + 1} \]

5. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 52.3

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

7. Applied rewrites 52.3%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 13.8%

    \[\leadsto 1 \]
11. Add Preprocessing

Developer Target 1: 99.9% 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 2024244 
(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)))