Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.5% → 99.9%

Time: 7.8s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{y}{x}} + 1\right) \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= x -1e-60)
     t_0
     (if (<= x 2e-7) (/ (* (+ (/ 1.0 (/ y x)) 1.0) x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (x <= -1e-60) {
		tmp = t_0;
	} else if (x <= 2e-7) {
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) * (x / (1.0d0 + x))) / y
    if (x <= (-1d-60)) then
        tmp = t_0
    else if (x <= 2d-7) then
        tmp = (((1.0d0 / (y / x)) + 1.0d0) * x) / (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (x <= -1e-60) {
		tmp = t_0;
	} else if (x <= 2e-7) {
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((y + x) * (x / (1.0 + x))) / y
	tmp = 0
	if x <= -1e-60:
		tmp = t_0
	elif x <= 2e-7:
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (x <= -1e-60)
		tmp = t_0;
	elseif (x <= 2e-7)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(y / x)) + 1.0) * x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((y + x) * (x / (1.0 + x))) / y;
	tmp = 0.0;
	if (x <= -1e-60)
		tmp = t_0;
	elseif (x <= 2e-7)
		tmp = (((1.0 / (y / x)) + 1.0) * x) / (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1e-60], t$95$0, If[LessEqual[x, 2e-7], N[(N[(N[(N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999997e-61 or 1.9999999999999999e-7 < x

   1. Initial program 78.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 -9.9999999999999997e-61 < x < 1.9999999999999999e-7

   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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\frac{1}{\frac{y}{x}} + 1\right) \cdot x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -2e-31)
     (/ x y)
     (if (<= t_0 1e-11) (- 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 <= -2e-31) {
		tmp = x / y;
	} else if (t_0 <= 1e-11) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x / y) + 1.0d0) * x) / (1.0d0 + x)
    if (t_0 <= (-2d-31)) then
        tmp = x / y
    else if (t_0 <= 1d-11) then
        tmp = x - (x * x)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	double tmp;
	if (t_0 <= -2e-31) {
		tmp = x / y;
	} else if (t_0 <= 1e-11) {
		tmp = x - (x * x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} 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 <= -2e-31:
		tmp = x / y
	elif t_0 <= 1e-11:
		tmp = x - (x * x)
	elif 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 <= -2e-31)
		tmp = Float64(x / y);
	elseif (t_0 <= 1e-11)
		tmp = Float64(x - Float64(x * x));
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (((x / y) + 1.0) * x) / (1.0 + x);
	tmp = 0.0;
	if (t_0 <= -2e-31)
		tmp = x / y;
	elseif (t_0 <= 1e-11)
		tmp = x - (x * x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-31], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 1e-11], N[(x - N[(x * x), $MachinePrecision]), $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))) < -2e-31 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -2e-31 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 9.99999999999999939e-12

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 91.8%

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

   if 9.99999999999999939e-12 < (/.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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 10^{-11}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x))))
   (if (<= t_0 -2e-31) (/ 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 <= -2e-31) {
		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 <= (-2d-31)) 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 <= -2e-31) {
		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 <= -2e-31:
		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 <= -2e-31)
		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 <= -2e-31)
		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, -2e-31], 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))) < -2e-31 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 73.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -2e-31 < (/.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 93.6

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

   5. Applied rewrites 93.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 10^{-11}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 1e-11) (- x (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 1e-11) {
		tmp = x - (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)) <= 1d-11) then
        tmp = x - (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)) <= 1e-11) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((((x / y) + 1.0) * x) / (1.0 + x)) <= 1e-11:
		tmp = 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-11)
		tmp = Float64(x - 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)) <= 1e-11)
		tmp = x - (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], 1e-11], N[(x - N[(x * x), $MachinePrecision]), $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))) < 9.99999999999999939e-12

   1. Initial program 91.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.2%

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

   if 9.99999999999999939e-12 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 44.7

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

   7. Applied rewrites 44.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 44.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 10^{-11}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 20.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (+ (/ x y) 1.0) x) (+ 1.0 x)) 2e-159) (* (- x) x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((((x / y) + 1.0) * x) / (1.0 + x)) <= 2e-159) {
		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-159) 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-159) {
		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-159:
		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-159)
		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-159)
		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-159], 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))) < 1.99999999999999998e-159

   1. Initial program 90.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 12.1%

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

   if 1.99999999999999998e-159 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 86.1%

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

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 53.5

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

   7. Applied rewrites 53.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 34.3%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\frac{x}{y} + 1\right) \cdot x}{1 + x} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\left(-x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (/ x (+ 1.0 x))) y)))
   (if (<= x -1.15e-38) t_0 (if (<= x 1e-147) (fma (/ x y) x x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((y + x) * (x / (1.0 + x))) / y;
	double tmp;
	if (x <= -1.15e-38) {
		tmp = t_0;
	} else if (x <= 1e-147) {
		tmp = fma((x / y), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) * Float64(x / Float64(1.0 + x))) / y)
	tmp = 0.0
	if (x <= -1.15e-38)
		tmp = t_0;
	elseif (x <= 1e-147)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.15e-38], t$95$0, If[LessEqual[x, 1e-147], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15000000000000001e-38 or 9.9999999999999997e-148 < x

   1. Initial program 81.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 -1.15000000000000001e-38 < x < 9.9999999999999997e-148

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \mathbf{elif}\;x \leq 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \frac{x}{1 + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 1}{y} + 1\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ (- x 1.0) y) 1.0)))
   (if (<= x -1.32e+14)
     t_0
     (if (<= x 4.2e+14) (/ (fma (/ x y) x x) (+ 1.0 x)) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) / y) + 1.0;
	double tmp;
	if (x <= -1.32e+14) {
		tmp = t_0;
	} else if (x <= 4.2e+14) {
		tmp = fma((x / y), x, x) / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.32e+14)
		tmp = t_0;
	elseif (x <= 4.2e+14)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.32e+14], t$95$0, If[LessEqual[x, 4.2e+14], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32e14 or 4.2e14 < x

   1. Initial program 75.3%

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

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 28.7

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-rgt-in N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -1.32e14 < x < 4.2e14

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 1}{y} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - 1\right) + y}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{x \cdot x}{y}\\ \mathbf{elif}\;x \leq 680:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ (- x 1.0) y) y)))
   (if (<= x -1.0)
     t_0
     (if (<= x -5.8e-64)
       (/ (* x x) y)
       (if (<= x 680.0) (/ x (+ 1.0 x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = ((x - 1.0) + y) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -5.8e-64) {
		tmp = (x * x) / y;
	} else if (x <= 680.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - 1.0d0) + y) / y
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= (-5.8d-64)) then
        tmp = (x * x) / y
    else if (x <= 680.0d0) then
        tmp = x / (1.0d0 + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x - 1.0) + y) / y;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= -5.8e-64) {
		tmp = (x * x) / y;
	} else if (x <= 680.0) {
		tmp = x / (1.0 + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - 1.0) + y) / y
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= -5.8e-64:
		tmp = (x * x) / y
	elif x <= 680.0:
		tmp = x / (1.0 + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) + y) / y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -5.8e-64)
		tmp = Float64(Float64(x * x) / y);
	elseif (x <= 680.0)
		tmp = Float64(x / Float64(1.0 + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - 1.0) + y) / y;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= -5.8e-64)
		tmp = (x * x) / y;
	elseif (x <= 680.0)
		tmp = x / (1.0 + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, -5.8e-64], N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 680.0], N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 680 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.1%

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

   if -1 < x < -5.7999999999999998e-64

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 11.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 76.0%

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

   if -5.7999999999999998e-64 < x < 680

   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 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 9: 98.5% 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 76.1%

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

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 65.8

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-rgt-in N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) / y) + 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.3], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.30000000000000004 < x

   1. Initial program 76.1%

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

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*l/ N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. associate-/r* N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 65.8

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.4

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

   7. Applied rewrites 29.4%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-rgt-in N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. associate-+l+ N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 99.2

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

   10. Applied rewrites 99.2%

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

   if -1 < x < 1.30000000000000004

   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.3%

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

Alternative 11: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - 1.0) + y) / y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.3)
		tmp = fma(Float64(x / y), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.3], N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.30000000000000004 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.1%

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

   if -1 < x < 1.30000000000000004

   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.3%

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

Alternative 12: 14.3% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.5%

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

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

   3. distribute-rgt-in N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. associate-*l/ N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. neg-mul-1 N/A

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

   13. associate-/r* N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 82.0

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 54.5

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

7. Applied rewrites 54.5%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 15.8%

    \[\leadsto 1 \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024249 
(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)))