Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.9% → 99.9%

Time: 9.6s

Alternatives: 15

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: 87.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{x + 1} \cdot \left(x + y\right)}{y}\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{y}, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (/ x (+ x 1.0)) (+ x y)) y)))
   (if (<= x -1.95e-16)
     t_0
     (if (<= x 5.1e-15) (fma x (fma (/ 1.0 y) x (- x)) x) t_0))))

C

double code(double x, double y) {
	double t_0 = ((x / (x + 1.0)) * (x + y)) / y;
	double tmp;
	if (x <= -1.95e-16) {
		tmp = t_0;
	} else if (x <= 5.1e-15) {
		tmp = fma(x, fma((1.0 / y), x, -x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x / Float64(x + 1.0)) * Float64(x + y)) / y)
	tmp = 0.0
	if (x <= -1.95e-16)
		tmp = t_0;
	elseif (x <= 5.1e-15)
		tmp = fma(x, fma(Float64(1.0 / y), x, Float64(-x)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.95e-16], t$95$0, If[LessEqual[x, 5.1e-15], N[(x * N[(N[(1.0 / y), $MachinePrecision] * x + (-x)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999989e-16 or 5.1e-15 < x

   1. Initial program 78.7%

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

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

      9. /-lowering-/.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   5. Simplified 99.9%

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

   if -1.94999999999999989e-16 < x < 5.1e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 99.9

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

   5. Simplified 99.9%

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

      1. sub-neg N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

Alternative 2: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -400.0)
     (/ x y)
     (if (<= t_0 4e-8) (- x (* x x)) (if (<= t_0 2.0) 1.0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = x / y;
	} else if (t_0 <= 4e-8) {
		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 * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-400.0d0)) then
        tmp = x / y
    else if (t_0 <= 4d-8) 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 * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = x / y;
	} else if (t_0 <= 4e-8) {
		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 * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -400.0:
		tmp = x / y
	elif t_0 <= 4e-8:
		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(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 4e-8)
		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 * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -400.0)
		tmp = x / y;
	elseif (t_0 <= 4e-8)
		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[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 4e-8], 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))) < -400 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 76.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 85.6

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

   5. Simplified 85.6%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.8

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

   8. Simplified 87.8%

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

   if 4.0000000000000001e-8 < (/.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 x around inf

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 98.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -400.0) t_0 (if (<= t_1 4e-8) (/ x (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = (x * t_0) / (x + 1.0d0)
    if (t_1 <= (-400.0d0)) then
        tmp = t_0
    else if (t_1 <= 4d-8) then
        tmp = x / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = x / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = (x * t_0) / (x + 1.0)
	tmp = 0
	if t_1 <= -400.0:
		tmp = t_0
	elif t_1 <= 4e-8:
		tmp = x / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -400.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = (x * t_0) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -400.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = x / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400.0], t$95$0, If[LessEqual[t$95$1, 4e-8], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$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))) < -400 or 4.0000000000000001e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e-8

   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

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

   5. Simplified 88.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -400.0) t_0 (if (<= t_1 4e-8) (- x (* x x)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = (x * t_0) / (x + 1.0d0)
    if (t_1 <= (-400.0d0)) then
        tmp = t_0
    else if (t_1 <= 4d-8) then
        tmp = 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 = 1.0 + (x / y);
	double t_1 = (x * t_0) / (x + 1.0);
	double tmp;
	if (t_1 <= -400.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-8) {
		tmp = x - (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = (x * t_0) / (x + 1.0)
	tmp = 0
	if t_1 <= -400.0:
		tmp = t_0
	elif t_1 <= 4e-8:
		tmp = x - (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(Float64(x * t_0) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= -400.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = (x * t_0) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= -400.0)
		tmp = t_0;
	elseif (t_1 <= 4e-8)
		tmp = x - (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -400.0], t$95$0, If[LessEqual[t$95$1, 4e-8], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$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))) < -400 or 4.0000000000000001e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 69.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 89.5

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

   7. Applied egg-rr 89.5%

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

   if -400 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 87.8

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

   8. Simplified 87.8%

      \[\leadsto \color{blue}{x - x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -400:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -8000000000:\\ \;\;\;\;-x \cdot x\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -8000000000.0) (- (* x x)) (if (<= t_0 4e-8) x 1.0))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -8000000000.0) {
		tmp = -(x * x);
	} else if (t_0 <= 4e-8) {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (1.0d0 + (x / y))) / (x + 1.0d0)
    if (t_0 <= (-8000000000.0d0)) then
        tmp = -(x * x)
    else if (t_0 <= 4d-8) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= -8000000000.0) {
		tmp = -(x * x);
	} else if (t_0 <= 4e-8) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0)
	tmp = 0
	if t_0 <= -8000000000.0:
		tmp = -(x * x)
	elif t_0 <= 4e-8:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -8000000000.0)
		tmp = Float64(-Float64(x * x));
	elseif (t_0 <= 4e-8)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	tmp = 0.0;
	if (t_0 <= -8000000000.0)
		tmp = -(x * x);
	elseif (t_0 <= 4e-8)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -8000000000.0], (-N[(x * x), $MachinePrecision]), If[LessEqual[t$95$0, 4e-8], x, 1.0]]]

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))) < -8e9

   1. Initial program 87.3%

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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 32.4

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

   5. Simplified 32.4%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 32.8

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

   8. Simplified 32.8%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 33.1

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

   11. Simplified 33.1%

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

   if -8e9 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 4.0000000000000001e-8

   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

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

   5. Simplified 85.6%

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

   if 4.0000000000000001e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 66.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 27.1%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -8000000000:\\ \;\;\;\;-x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 4e-8) (- x (* x x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8) {
		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 * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 4d-8) 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 * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8:
		tmp = x - (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0)) <= 4e-8)
		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 * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8)
		tmp = x - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 4e-8], 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))) < 4.0000000000000001e-8

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 81.3

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

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 72.3

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

   8. Simplified 72.3%

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

   if 4.0000000000000001e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 66.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 27.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot t\_0}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -2e+65) t_0 (if (<= x 9e+15) (/ (* x t_0) (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2e+65) {
		tmp = t_0;
	} else if (x <= 9e+15) {
		tmp = (x * t_0) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    if (x <= (-2d+65)) then
        tmp = t_0
    else if (x <= 9d+15) then
        tmp = (x * t_0) / (x + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -2e+65) {
		tmp = t_0;
	} else if (x <= 9e+15) {
		tmp = (x * t_0) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	tmp = 0
	if x <= -2e+65:
		tmp = t_0
	elif x <= 9e+15:
		tmp = (x * t_0) / (x + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -2e+65)
		tmp = t_0;
	elseif (x <= 9e+15)
		tmp = Float64(Float64(x * t_0) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	tmp = 0.0;
	if (x <= -2e+65)
		tmp = t_0;
	elseif (x <= 9e+15)
		tmp = (x * t_0) / (x + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+65], t$95$0, If[LessEqual[x, 9e+15], N[(N[(x * t$95$0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e65 or 9e15 < x

   1. Initial program 75.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 75.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -2e65 < x < 9e15

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+65}:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} + \left(1 + \frac{-1}{y}\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.98:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{y}, x, -x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ x y) (+ 1.0 (/ -1.0 y)))))
   (if (<= x -1.0)
     t_0
     (if (<= x 0.98) (fma x (fma (/ 1.0 y) x (- x)) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (x / y) + (1.0 + (-1.0 / y));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.98) {
		tmp = fma(x, fma((1.0 / y), x, -x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) + Float64(1.0 + Float64(-1.0 / y)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.98)
		tmp = fma(x, fma(Float64(1.0 / y), x, Float64(-x)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] + N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.98], N[(x * N[(N[(1.0 / y), $MachinePrecision] * x + (-x)), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.97999999999999998 < x

   1. Initial program 77.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

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

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

      11. associate-+l+ N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. associate-/r* N/A

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

      15. associate-*r/ N/A

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

      16. rgt-mult-inverse N/A

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

      17. sub-neg N/A

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

      18. +-lowering-+.f64 N/A

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

      19. /-lowering-/.f64 N/A

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

   7. Simplified 99.8%

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

   if -1 < x < 0.97999999999999998

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

      1. sub-neg N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 98.3

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

   7. Applied egg-rr 98.3%

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

Alternative 9: 49.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 4e-8) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8) {
		tmp = 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 * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 4d-8) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0)) <= 4e-8)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 4e-8)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 4e-8], x, 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))) < 4.0000000000000001e-8

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 64.0%

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

   if 4.0000000000000001e-8 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 66.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 27.1%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. /-lowering-/.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 11: 98.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

      1. sub-neg N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. neg-lowering-neg.f64 98.3

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

   7. Applied egg-rr 98.3%

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

   if 1 < x

   1. Initial program 81.4%

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

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

      14. /-lowering-/.f64 N/A

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

      15. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.9%

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

Alternative 12: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   if 1 < x

   1. Initial program 81.4%

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

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

      14. /-lowering-/.f64 N/A

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

      15. +-lowering-+.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 98.9%

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

Alternative 13: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y} - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0) t_0 (if (<= x 0.8) (fma x (- (/ x y) x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 0.8) {
		tmp = fma(x, ((x / y) - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 0.8)
		tmp = fma(x, Float64(Float64(x / y) - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 0.8], N[(x * N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.80000000000000004 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   if -1 < x < 0.80000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y} - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 77.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. /-rgt-identity N/A

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

      7. +-lowering-+.f64 N/A

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

      8. /-lowering-/.f64 99.3

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

   7. Applied egg-rr 99.3%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

      6. associate-*l/ N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. --lowering--.f64 N/A

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

      10. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 97.5

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

   8. Simplified 97.5%

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

4. Final simplification 98.3%

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

Alternative 15: 14.8% 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 89.6%

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

3. Taylor expanded in x around inf

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

5. Simplified 37.7%

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

6. Taylor expanded in x around 0

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

8. Simplified 11.0%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(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)))