Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 88.4% → 99.9%

Time: 9.3s

Alternatives: 16

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: 88.4% 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, 1.0× speedup

Math

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

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(Float64(x / 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[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

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

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

   8. /-lowering-/.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.8:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 0.8)
       (* x (fma x (+ x -1.0) 1.0))
       (if (<= t_0 2.0) (+ 1.0 (/ -1.0 x)) (/ 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 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.8) {
		tmp = x * fma(x, (x + -1.0), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + (-1.0 / x);
	} 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 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.8)
		tmp = Float64(x * fma(x, Float64(x + -1.0), 1.0));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(x / y);
	end
	return 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, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.8], N[(x * N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 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))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004

   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

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

   5. Simplified 86.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 85.3

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

   8. Simplified 85.3%

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

   if 0.80000000000000004 < (/.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 0

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 98.0

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

   8. Simplified 98.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.8:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 0.02)
       (* x (fma x (+ x -1.0) 1.0))
       (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 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = x * fma(x, (x + -1.0), 1.0);
	} else if (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 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.02)
		tmp = Float64(x * fma(x, Float64(x + -1.0), 1.0));
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(x * N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $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))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   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

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

   5. Simplified 87.9%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 86.7

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

   8. Simplified 86.7%

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

   if 0.0200000000000000004 < (/.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 0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 90.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -100000000000.0)
     (/ x y)
     (if (<= t_0 0.02) (fma x (- 0.0 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 <= -100000000000.0) {
		tmp = x / y;
	} else if (t_0 <= 0.02) {
		tmp = fma(x, (0.0 - x), x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(1.0 + Float64(x / y))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(x / y);
	elseif (t_0 <= 0.02)
		tmp = fma(x, Float64(0.0 - x), x);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100000000000.0], N[(x / y), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(x * N[(0.0 - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < -1e11 or 2 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 74.6%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.4

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

   5. Simplified 81.4%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   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

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

      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 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

      3. --lowering--.f64 86.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   8. Simplified 86.3%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   if 0.0200000000000000004 < (/.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 0

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

   5. Simplified 93.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 90.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1.000000000000002:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -100000000000.0)
     t_0
     (if (<= t_1 1.000000000000002) (/ 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 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1.000000000000002) {
		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 <= (-100000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 1.000000000000002d0) 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 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1.000000000000002) {
		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 <= -100000000000.0:
		tmp = t_0
	elif t_1 <= 1.000000000000002:
		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 <= -100000000000.0)
		tmp = t_0;
	elseif (t_1 <= 1.000000000000002)
		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 <= -100000000000.0)
		tmp = t_0;
	elseif (t_1 <= 1.000000000000002)
		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, -100000000000.0], t$95$0, If[LessEqual[t$95$1, 1.000000000000002], 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))) < -1e11 or 1.000000000000002 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 75.3%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 84.8

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

   8. Simplified 84.8%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 1.000000000000002

   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

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

   5. Simplified 89.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 1.000000000000002:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x \cdot t\_0}{x + 1}\\ \mathbf{if}\;t\_1 \leq -100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.8:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ (* x t_0) (+ x 1.0))))
   (if (<= t_1 -100000000000.0)
     t_0
     (if (<= t_1 0.8) (* x (fma x (+ x -1.0) 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 <= -100000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.8) {
		tmp = x * fma(x, (x + -1.0), 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 <= -100000000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.8)
		tmp = Float64(x * fma(x, Float64(x + -1.0), 1.0));
	else
		tmp = t_0;
	end
	return 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, -100000000000.0], t$95$0, If[LessEqual[t$95$1, 0.8], N[(x * N[(x * N[(x + -1.0), $MachinePrecision] + 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))) < -1e11 or 0.80000000000000004 < (/.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 67.3%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 87.2

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

   8. Simplified 87.2%

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

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.80000000000000004

   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

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

   5. Simplified 86.4%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 85.3

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

   8. Simplified 85.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.8:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+242}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 0.02)
     (fma x (- 0.0 x) x)
     (if (<= t_0 2e+242) 1.0 (* x (* x x))))))

C

double code(double x, double y) {
	double t_0 = (x * (1.0 + (x / y))) / (x + 1.0);
	double tmp;
	if (t_0 <= 0.02) {
		tmp = fma(x, (0.0 - x), x);
	} else if (t_0 <= 2e+242) {
		tmp = 1.0;
	} else {
		tmp = x * (x * x);
	}
	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 <= 0.02)
		tmp = fma(x, Float64(0.0 - x), x);
	elseif (t_0 <= 2e+242)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * x));
	end
	return 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, 0.02], N[(x * N[(0.0 - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 2e+242], 1.0, N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   1. Initial program 88.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

      3. --lowering--.f64 64.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   8. Simplified 64.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 2.0000000000000001e242

   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

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 48.5%

      \[\leadsto \color{blue}{1} \]

   if 2.0000000000000001e242 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 3.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 17.1

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

   8. Simplified 17.1%

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

   9. Taylor expanded in x around inf

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

       1. +-rgt-identity N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} + 0\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{x \cdot x} + 0\right) \]

       3. accelerator-lowering-fma.f64 17.1

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   11. Simplified 17.1%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity N/A

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

       2. *-lowering-*.f64 17.1

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

   13. Applied egg-rr 17.1%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 2 \cdot 10^{+242}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1}\\ \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;0 - x \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (+ 1.0 (/ x y))) (+ x 1.0))))
   (if (<= t_0 -100000000000.0) (- 0.0 (* x x)) (if (<= t_0 0.02) 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 <= -100000000000.0) {
		tmp = 0.0 - (x * x);
	} else if (t_0 <= 0.02) {
		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 <= (-100000000000.0d0)) then
        tmp = 0.0d0 - (x * x)
    else if (t_0 <= 0.02d0) 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 <= -100000000000.0) {
		tmp = 0.0 - (x * x);
	} else if (t_0 <= 0.02) {
		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 <= -100000000000.0:
		tmp = 0.0 - (x * x)
	elif t_0 <= 0.02:
		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 <= -100000000000.0)
		tmp = Float64(0.0 - Float64(x * x));
	elseif (t_0 <= 0.02)
		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 <= -100000000000.0)
		tmp = 0.0 - (x * x);
	elseif (t_0 <= 0.02)
		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, -100000000000.0], N[(0.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.02], 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))) < -1e11

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 1.2%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. lft-mult-inverse N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 20.1

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

   8. Simplified 20.1%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - x\right)} \]

       3. --lowering--.f64 20.3

          \[\leadsto x \cdot \color{blue}{\left(0 - x\right)} \]

   11. Simplified 20.3%

       \[\leadsto x \cdot \color{blue}{\left(0 - x\right)} \]

   if -1e11 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64))) < 0.0200000000000000004

   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

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

   5. Simplified 85.3%

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

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 34.0%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq -100000000000:\\ \;\;\;\;0 - x \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 0.02) (fma x (- 0.0 x) x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02) {
		tmp = fma(x, (0.0 - 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)) <= 0.02)
		tmp = fma(x, Float64(0.0 - x), x);
	else
		tmp = 1.0;
	end
	return 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], 0.02], N[(x * N[(0.0 - x), $MachinePrecision] + 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))) < 0.0200000000000000004

   1. Initial program 88.3%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 71.6

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

   5. Simplified 71.6%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

      3. --lowering--.f64 64.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   8. Simplified 64.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - x}, x\right) \]

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 34.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(x, 0 - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 0.02) (* x (- 1.0 x)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02) {
		tmp = x * (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * (1.0d0 + (x / y))) / (x + 1.0d0)) <= 0.02d0) then
        tmp = x * (1.0d0 - x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02) {
		tmp = x * (1.0 - x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02:
		tmp = x * (1.0 - 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)) <= 0.02)
		tmp = Float64(x * Float64(1.0 - 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)) <= 0.02)
		tmp = x * (1.0 - 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], 0.02], N[(x * N[(1.0 - 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))) < 0.0200000000000000004

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. lft-mult-inverse N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 64.4

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

   8. Simplified 64.4%

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

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 34.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{y}, x, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -5e+21)
     t_0
     (if (<= x 2e+14) (/ (fma (/ x y) x x) (+ x 1.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double tmp;
	if (x <= -5e+21) {
		tmp = t_0;
	} else if (x <= 2e+14) {
		tmp = fma((x / y), x, x) / (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 <= -5e+21)
		tmp = t_0;
	elseif (x <= 2e+14)
		tmp = Float64(fma(Float64(x / y), x, x) / Float64(x + 1.0));
	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, -5e+21], t$95$0, If[LessEqual[x, 2e+14], N[(N[(N[(x / y), $MachinePrecision] * x + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e21 or 2e14 < x

   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 75.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 100.0

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

   8. Simplified 100.0%

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

   if -5e21 < x < 2e14

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. distribute-lft1-in N/A

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

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

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

      4. /-lowering-/.f64 99.8

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

   4. Applied egg-rr 99.8%

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

Alternative 12: 50.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x (+ 1.0 (/ x y))) (+ x 1.0)) 0.02) x 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02) {
		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)) <= 0.02d0) 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)) <= 0.02) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * (1.0 + (x / y))) / (x + 1.0)) <= 0.02:
		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)) <= 0.02)
		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)) <= 0.02)
		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], 0.02], 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))) < 0.0200000000000000004

   1. Initial program 88.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 58.4%

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

   if 0.0200000000000000004 < (/.f64 (*.f64 x (+.f64 (/.f64 x y) #s(literal 1 binary64))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 34.5%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 34.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(1 + \frac{x}{y}\right)}{x + 1} \leq 0.02:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = fma(Float64(1.0 / y), Float64(x + -1.0), 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		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[(N[(1.0 / y), $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], 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 1 < x

   1. Initial program 77.4%

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

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

   5. Simplified 98.6%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-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 97.4

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

   5. Simplified 97.4%

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

Alternative 14: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.82:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y} - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))))
   (if (<= x -1.0) t_0 (if (<= x 0.82) (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 <= 0.82) {
		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 <= 0.82)
		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, 0.82], 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 0.819999999999999951 < x

   1. Initial program 77.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 75.5%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 98.1

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

   8. Simplified 98.1%

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

   if -1 < x < 0.819999999999999951

   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 97.4

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

   5. Simplified 97.4%

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

Alternative 15: 97.8% 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 77.4%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   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 75.5%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 98.1

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

   8. Simplified 98.1%

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

   if -1 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 97.4

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

   5. Simplified 97.4%

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

   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 96.5

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

   8. Simplified 96.5%

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

Alternative 16: 14.1% accurate, 34.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 88.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.3%

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

6. Taylor expanded in x around inf

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

8. Simplified 15.4%

   \[\leadsto \color{blue}{1} \]
9. 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 2024196 
(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)))