Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 89.1% → 99.9%

Time: 7.6s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-13} \lor \neg \left(x \leq 5 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot \left(y + x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{y}, -x, -x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.6e-13) (not (<= x 5e-150)))
   (/ (* (/ x (+ 1.0 x)) (+ y x)) y)
   (fma (fma (/ -1.0 y) (- x) (- x)) x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.6e-13) || !(x <= 5e-150)) {
		tmp = ((x / (1.0 + x)) * (y + x)) / y;
	} else {
		tmp = fma(fma((-1.0 / y), -x, -x), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.6e-13) || !(x <= 5e-150))
		tmp = Float64(Float64(Float64(x / Float64(1.0 + x)) * Float64(y + x)) / y);
	else
		tmp = fma(fma(Float64(-1.0 / y), Float64(-x), Float64(-x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.6e-13], N[Not[LessEqual[x, 5e-150]], $MachinePrecision]], N[(N[(N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(-1.0 / y), $MachinePrecision] * (-x) + (-x)), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e-13 or 4.9999999999999999e-150 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 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 -1.6e-13 < x < 4.9999999999999999e-150

   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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-13} \lor \neg \left(x \leq 5 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\frac{x}{1 + x} \cdot \left(y + x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{y}, -x, -x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -500000000000 \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -500000000000.0) (not (<= t_0 2.0)))
     (* (pow y -1.0) x)
     (/ x (+ 1.0 x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -500000000000.0) || !(t_0 <= 2.0)) {
		tmp = pow(y, -1.0) * x;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
    if ((t_0 <= (-500000000000.0d0)) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = (y ** (-1.0d0)) * x
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -500000000000.0) || !(t_0 <= 2.0)) {
		tmp = Math.pow(y, -1.0) * x;
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0)
	tmp = 0
	if (t_0 <= -500000000000.0) or not (t_0 <= 2.0):
		tmp = math.pow(y, -1.0) * x
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -500000000000.0) || !(t_0 <= 2.0))
		tmp = Float64((y ^ -1.0) * x);
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	tmp = 0.0;
	if ((t_0 <= -500000000000.0) || ~((t_0 <= 2.0)))
		tmp = (y ^ -1.0) * x;
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -500000000000.0], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(N[Power[y, -1.0], $MachinePrecision] * x), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $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))) < -5e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.8%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.4%

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

   if -5e11 < (/.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 87.8

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

   5. Applied rewrites 87.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -500000000000 \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 2\right):\\ \;\;\;\;{y}^{-1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 10^{+185}:\\ \;\;\;\;1 - {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 0.01)
     (- x (* x x))
     (if (<= t_0 1e+185) (- 1.0 (pow x -1.0)) (* (fma (- x 1.0) x 1.0) x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = x - (x * x);
	} else if (t_0 <= 1e+185) {
		tmp = 1.0 - pow(x, -1.0);
	} else {
		tmp = fma((x - 1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 1e+185)
		tmp = Float64(1.0 - (x ^ -1.0));
	else
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+185], N[(1.0 - N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $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))) < 0.0100000000000000002

   1. Initial program 91.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.4%

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

   10. Applied rewrites 66.4%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.9999999999999998e184

   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 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.7%

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

   if 9.9999999999999998e184 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 62.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.1%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 0.01:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 10^{+185}:\\ \;\;\;\;1 - {x}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot {y}^{-1}\\ t_1 := \left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-182}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (pow y -1.0))) (t_1 (* (+ y x) (/ x y))))
   (if (<= x -1.0)
     t_0
     (if (<= x -2.4e-97)
       t_1
       (if (<= x 5.7e-182) (* (- 1.0 x) x) (if (<= x 1.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (y + x) * pow(y, -1.0);
	double t_1 = (y + x) * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.4e-97) {
		tmp = t_1;
	} else if (x <= 5.7e-182) {
		tmp = (1.0 - x) * x;
	} else if (x <= 1.0) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (y + x) * (y ** (-1.0d0))
    t_1 = (y + x) * (x / y)
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-2.4d-97)) then
        tmp = t_1
    else if (x <= 5.7d-182) then
        tmp = (1.0d0 - x) * x
    else if (x <= 1.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) * Math.pow(y, -1.0);
	double t_1 = (y + x) * (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -2.4e-97) {
		tmp = t_1;
	} else if (x <= 5.7e-182) {
		tmp = (1.0 - x) * x;
	} else if (x <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) * math.pow(y, -1.0)
	t_1 = (y + x) * (x / y)
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -2.4e-97:
		tmp = t_1
	elif x <= 5.7e-182:
		tmp = (1.0 - x) * x
	elif x <= 1.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * (y ^ -1.0))
	t_1 = Float64(Float64(y + x) * Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.4e-97)
		tmp = t_1;
	elseif (x <= 5.7e-182)
		tmp = Float64(Float64(1.0 - x) * x);
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) * (y ^ -1.0);
	t_1 = (y + x) * (x / y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -2.4e-97)
		tmp = t_1;
	elseif (x <= 5.7e-182)
		tmp = (1.0 - x) * x;
	elseif (x <= 1.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -2.4e-97], t$95$1, If[LessEqual[x, 5.7e-182], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.0%

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

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.7%

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

   if -1 < x < -2.4e-97 or 5.6999999999999998e-182 < x < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{1 + x} + \frac{{x}^{2}}{1 + x}}{y}} \]
   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 93.3

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

   5. Applied rewrites 93.3%

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 82.8%

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

   if -2.4e-97 < x < 5.6999999999999998e-182

   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 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.4%

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

   10. Applied rewrites 94.4%

       \[\leadsto \left(1 - x\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-182}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(y + x\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = fma((y ^ -1.0), Float64(x - 1.0), 1.0);
	else
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 79.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

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

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. rgt-mult-inverse N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.2%

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

Alternative 6: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.85)))
   (* (+ y x) (pow y -1.0))
   (fma (- (/ x y) x) x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.85)) {
		tmp = (y + x) * pow(y, -1.0);
	} else {
		tmp = fma(((x / y) - x), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.85))
		tmp = Float64(Float64(y + x) * (y ^ -1.0));
	else
		tmp = fma(Float64(Float64(x / y) - x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(N[(y + x), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.849999999999999978 < x

   1. Initial program 79.0%

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

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.7%

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

   if -1 < x < 0.849999999999999978

   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

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      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-lft-out-- N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3200000000 \lor \neg \left(x \leq 6000\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3200000000.0) (not (<= x 6000.0)))
   (* (+ y x) (pow y -1.0))
   (/ x (+ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3200000000.0) || !(x <= 6000.0)) {
		tmp = (y + x) * pow(y, -1.0);
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3200000000.0d0)) .or. (.not. (x <= 6000.0d0))) then
        tmp = (y + x) * (y ** (-1.0d0))
    else
        tmp = x / (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3200000000.0) || !(x <= 6000.0)) {
		tmp = (y + x) * Math.pow(y, -1.0);
	} else {
		tmp = x / (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3200000000.0) or not (x <= 6000.0):
		tmp = (y + x) * math.pow(y, -1.0)
	else:
		tmp = x / (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3200000000.0) || !(x <= 6000.0))
		tmp = Float64(Float64(y + x) * (y ^ -1.0));
	else
		tmp = Float64(x / Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3200000000.0) || ~((x <= 6000.0)))
		tmp = (y + x) * (y ^ -1.0);
	else
		tmp = x / (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3200000000.0], N[Not[LessEqual[x, 6000.0]], $MachinePrecision]], N[(N[(y + x), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e9 or 6e3 < x

   1. Initial program 78.5%

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

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   if -3.2e9 < x < 6e3

   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 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3200000000 \lor \neg \left(x \leq 6000\right):\\ \;\;\;\;\left(y + x\right) \cdot {y}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+19} \lor \neg \left(t\_0 \leq 10^{+41}\right):\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (or (<= t_0 -1e+19) (not (<= t_0 1e+41)))
     (/ x (+ (/ y x) y))
     (/ (fma (/ x y) x x) (+ x 1.0)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if ((t_0 <= -1e+19) || !(t_0 <= 1e+41)) {
		tmp = x / ((y / x) + y);
	} else {
		tmp = fma((x / y), x, x) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_0 <= -1e+19) || !(t_0 <= 1e+41))
		tmp = Float64(x / Float64(Float64(y / x) + y));
	else
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+19], N[Not[LessEqual[t$95$0, 1e+41]], $MachinePrecision]], N[(x / N[(N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $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))) < -1e19 or 1.00000000000000001e41 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.4%

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 83.8

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

   7. Applied rewrites 83.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   if -1e19 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.00000000000000001e41

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq -1 \cdot 10^{+19} \lor \neg \left(\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \leq 10^{+41}\right):\\ \;\;\;\;\frac{x}{\frac{y}{x} + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -500000000000:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0))))
   (if (<= t_0 -500000000000.0)
     (* (- x) x)
     (if (<= t_0 5e+181) (/ x (+ 1.0 x)) (* (fma (- x 1.0) x 1.0) x)))))

C

double code(double x, double y) {
	double t_0 = (x * ((x / y) + 1.0)) / (x + 1.0);
	double tmp;
	if (t_0 <= -500000000000.0) {
		tmp = -x * x;
	} else if (t_0 <= 5e+181) {
		tmp = x / (1.0 + x);
	} else {
		tmp = fma((x - 1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -500000000000.0)
		tmp = Float64(Float64(-x) * x);
	elseif (t_0 <= 5e+181)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500000000000.0], N[((-x) * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+181], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $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))) < -5e11

   1. Initial program 72.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 1.3

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

   5. Applied rewrites 1.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 26.1%

       \[\leadsto \left(-x\right) \cdot x \]

   if -5e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000003e181

   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 75.1

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

   5. Applied rewrites 75.1%

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

   if 5.0000000000000003e181 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 3.7

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

   5. Applied rewrites 3.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 23.5%

      \[\leadsto \mathsf{fma}\left(x - 1, x, 1\right) \cdot \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.9

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 99.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 11: 48.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(1 - x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-310) (* (- 1.0 x) x) (* (fma (- x 1.0) x 1.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (1.0 - x) * x;
	} else {
		tmp = fma((x - 1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(1.0 - x) * x);
	else
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-310], N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 87.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.4%

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

   10. Applied rewrites 41.4%

       \[\leadsto \left(1 - x\right) \cdot x \]

   if -4.999999999999985e-310 < y

   1. Initial program 91.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.3%

      \[\leadsto \mathsf{fma}\left(x - 1, x, 1\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x - x \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* x x)))

C

double code(double x, double y) {
	return x - (x * x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - (x * x)
end function

Java

public static double code(double x, double y) {
	return x - (x * x);
}

Python

def code(x, y):
	return x - (x * x)

Julia

function code(x, y)
	return Float64(x - Float64(x * x))
end

MATLAB

function tmp = code(x, y)
	tmp = x - (x * x);
end

Wolfram

code[x_, y_] := N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 42.1%

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

10. Applied rewrites 42.1%

    \[\leadsto x - x \cdot \color{blue}{x} \]
11. Add Preprocessing

Alternative 13: 44.4% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (- 1.0 x) x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (1.0 - x) * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 42.1%

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

10. Applied rewrites 42.1%

    \[\leadsto \left(1 - x\right) \cdot x \]
11. Add Preprocessing

Alternative 14: 8.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- x) x))

C

double code(double x, double y) {
	return -x * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x * x
end function

Java

public static double code(double x, double y) {
	return -x * x;
}

Python

def code(x, y):
	return -x * x

Julia

function code(x, y)
	return Float64(Float64(-x) * x)
end

MATLAB

function tmp = code(x, y)
	tmp = -x * x;
end

Wolfram

code[x_, y_] := N[((-x) * x), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 50.4

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

5. Applied rewrites 50.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 42.1%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 8.1%

    \[\leadsto \left(-x\right) \cdot x \]
12. 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 2024318 
(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)))