Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.1% → 99.4%

Time: 6.5s

Alternatives: 12

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: 88.1% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ t_1 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)))
        (t_1 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= t_0 -2e+93)
     t_1
     (if (<= t_0 5e-128) (/ (+ (* (/ x y) x) x) (+ 1.0 x)) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double t_1 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (t_0 <= -2e+93) {
		tmp = t_1;
	} else if (t_0 <= 5e-128) {
		tmp = (((x / y) * x) + x) / (1.0 + x);
	} 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) / (1.0d0 + x)
    t_1 = ((y + x) * (x / (1.0d0 + x))) / y
    if (t_0 <= (-2d+93)) then
        tmp = t_1
    else if (t_0 <= 5d-128) then
        tmp = (((x / y) * x) + x) / (1.0d0 + x)
    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) / (1.0 + x);
	double t_1 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (t_0 <= -2e+93) {
		tmp = t_1;
	} else if (t_0 <= 5e-128) {
		tmp = (((x / y) * x) + x) / (1.0 + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x)
	t_1 = ((y + x) * (x / (1.0 + x))) / y
	tmp = 0
	if t_0 <= -2e+93:
		tmp = t_1
	elif t_0 <= 5e-128:
		tmp = (((x / y) * x) + x) / (1.0 + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	t_1 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (t_0 <= -2e+93)
		tmp = t_1;
	elseif (t_0 <= 5e-128)
		tmp = Float64(Float64(Float64(Float64(x / y) * x) + x) / Float64(1.0 + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	t_1 = ((y + x) * (x / (1.0 + x))) / y;
	tmp = 0.0;
	if (t_0 <= -2e+93)
		tmp = t_1;
	elseif (t_0 <= 5e-128)
		tmp = (((x / y) * x) + x) / (1.0 + x);
	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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+93], t$95$1, If[LessEqual[t$95$0, 5e-128], N[(N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -2.00000000000000009e93 or 5.0000000000000001e-128 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 79.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 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 -2.00000000000000009e93 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 5.0000000000000001e-128

   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-+.f64 N/A

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

      6. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot x + x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -5e-5)
     (/ x y)
     (if (<= t_0 1e-6)
       (* (fma (- x 1.0) x 1.0) x)
       (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = x / y;
	} else if (t_0 <= 1e-6) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -5e-5)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-6)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-5], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 33.0

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

   5. Applied rewrites 33.0%

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 81.1

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

   10. Applied rewrites 81.1%

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

   if -5.00000000000000024e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999955e-7

   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 83.4

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

   5. Applied rewrites 83.4%

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

   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 83.5%

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

   if 9.99999999999999955e-7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 94.7

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

   5. Applied rewrites 94.7%

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

   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 0.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 94.7%

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

4. Final simplification 83.4%

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

Alternative 3: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-x, x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -5e-5)
     (/ x y)
     (if (<= t_0 1e-6) (fma (- x) x x) (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = x / y;
	} else if (t_0 <= 1e-6) {
		tmp = fma(-x, x, x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x))
	tmp = 0.0
	if (t_0 <= -5e-5)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-6)
		tmp = fma(Float64(-x), x, x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-5], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-6], N[((-x) * x + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 33.0

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

   5. Applied rewrites 33.0%

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 81.1

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

   10. Applied rewrites 81.1%

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

   if -5.00000000000000024e-5 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999955e-7

   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 83.4

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

   5. Applied rewrites 83.4%

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

   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 83.3%

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

   if 9.99999999999999955e-7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 94.7

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

   5. Applied rewrites 94.7%

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

   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 0.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 94.7%

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

4. Final simplification 83.3%

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

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ t_1 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, \frac{-1}{y}, -x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)))
        (t_1 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= t_0 -2000.0)
     t_1
     (if (<= t_0 1e-95) (fma (fma (- x) (/ -1.0 y) (- x)) x x) t_1))))

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double t_1 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-95) {
		tmp = fma(fma(-x, (-1.0 / 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(1.0 + x))
	t_1 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-95)
		tmp = fma(fma(Float64(-x), Float64(-1.0 / y), Float64(-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[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000.0], t$95$1, If[LessEqual[t$95$0, 1e-95], N[(N[((-x) * N[(-1.0 / 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))) < -2e3 or 9.99999999999999989e-96 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.7%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 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 -2e3 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999989e-96

   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 99.9

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

   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 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -2000:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-x, \frac{-1}{y}, -x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-5d-5)) then
        tmp = x / y
    else if (t_0 <= 2.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -5e-5) {
		tmp = x / y;
	} else if (t_0 <= 2.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -5e-5)
		tmp = x / y;
	elseif (t_0 <= 2.0)
		tmp = x / (1.0 + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-5], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -5.00000000000000024e-5 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 33.0

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

   5. Applied rewrites 33.0%

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

   7. Applied rewrites 33.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 81.1

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

   10. Applied rewrites 81.1%

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

   if -5.00000000000000024e-5 < (/.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 85.5

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

   5. Applied rewrites 85.5%

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

4. Final simplification 83.4%

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

Alternative 6: 54.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 1e-6) (fma (- x) x x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 1e-6) {
		tmp = fma(-x, x, x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x)) <= 1e-6)
		tmp = fma(Float64(-x), x, x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1e-6], 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))) < 9.99999999999999955e-7

   1. Initial program 92.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   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 64.0%

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

   if 9.99999999999999955e-7 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 28.8

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

   5. Applied rewrites 28.8%

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

   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 0.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 29.2%

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

4. Final simplification 51.9%

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

Alternative 7: 21.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 2e-155) (* (- x) x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 2e-155) {
		tmp = -x * 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) / (1.0d0 + x)) <= 2d-155) then
        tmp = -x * 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) / (1.0 + x)) <= 2e-155) {
		tmp = -x * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) + 1.0) * x) / (1.0 + x)) <= 2e-155:
		tmp = -x * x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x / y) + 1.0) * x) / Float64(1.0 + x)) <= 2e-155)
		tmp = Float64(Float64(-x) * 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) / (1.0 + x)) <= 2e-155)
		tmp = -x * 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[(1.0 + x), $MachinePrecision]), $MachinePrecision], 2e-155], N[((-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))) < 2.00000000000000003e-155

   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 52.4

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

   5. Applied rewrites 52.4%

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

   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 62.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 14.4%

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

   if 2.00000000000000003e-155 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 84.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 37.8

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

   5. Applied rewrites 37.8%

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

   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 14.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 25.3%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

   \[\leadsto \frac{x}{\frac{1 + x}{\frac{x}{y} + 1}} \]
6. Add Preprocessing

Alternative 9: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{y} \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x - 1, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.0)
     t_0
     (if (<= x -1.35e-47)
       (* (/ x y) x)
       (if (<= x 1.0) (* (fma (- x 1.0) x 1.0) x) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -1.35e-47) {
		tmp = (x / y) * x;
	} else if (x <= 1.0) {
		tmp = fma((x - 1.0), x, 1.0) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -1.35e-47)
		tmp = Float64(Float64(x / y) * x);
	elseif (x <= 1.0)
		tmp = Float64(fma(Float64(x - 1.0), x, 1.0) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -1.35e-47], N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 1 < x

   1. Initial program 76.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 100.0%

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

   if -1 < x < -1.3499999999999999e-47

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot \left(1 + x\right)}} \]
   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 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{{x}^{2}}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.5%

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

   if -1.3499999999999999e-47 < 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 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%

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

   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 75.2%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 76.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 100.0%

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

   if -1 < 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 98.2

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

Alternative 11: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 76.0%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 100.0%

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

   if -1 < 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 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y} - x, x, x\right)} \]
3. Recombined 2 regimes into one program.
4. 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 88.2%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
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 46.2

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

5. Applied rewrites 46.2%

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

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 42.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 11.9%

    \[\leadsto 1 \]
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 2024296 
(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)))