Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 9

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 (fma y (- 0.5 x) x)))

C

double code(double x, double y) {
	return 0.918938533204673 - fma(y, (0.5 - x), x);
}

Julia

function code(x, y)
	return Float64(0.918938533204673 - fma(y, Float64(0.5 - x), x))
end

Wolfram

code[x_, y_] := N[(0.918938533204673 - N[(y * N[(0.5 - x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{0.918938533204673 - \mathsf{fma}\left(y, 0.5 - x, x\right)} \]
6. Add Preprocessing

Alternative 2: 49.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y + -1\right) - y \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x (+ y -1.0)) (* y 0.5))))
   (if (<= t_0 -2000.0)
     (- 0.0 x)
     (if (<= t_0 0.5) 0.918938533204673 (- 0.0 x)))))

C

double code(double x, double y) {
	double t_0 = (x * (y + -1.0)) - (y * 0.5);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = 0.0 - x;
	} else if (t_0 <= 0.5) {
		tmp = 0.918938533204673;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (y + (-1.0d0))) - (y * 0.5d0)
    if (t_0 <= (-2000.0d0)) then
        tmp = 0.0d0 - x
    else if (t_0 <= 0.5d0) then
        tmp = 0.918938533204673d0
    else
        tmp = 0.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * (y + -1.0)) - (y * 0.5);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = 0.0 - x;
	} else if (t_0 <= 0.5) {
		tmp = 0.918938533204673;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * (y + -1.0)) - (y * 0.5)
	tmp = 0
	if t_0 <= -2000.0:
		tmp = 0.0 - x
	elif t_0 <= 0.5:
		tmp = 0.918938533204673
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * Float64(y + -1.0)) - Float64(y * 0.5))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(0.0 - x);
	elseif (t_0 <= 0.5)
		tmp = 0.918938533204673;
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * (y + -1.0)) - (y * 0.5);
	tmp = 0.0;
	if (t_0 <= -2000.0)
		tmp = 0.0 - x;
	elseif (t_0 <= 0.5)
		tmp = 0.918938533204673;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000.0], N[(0.0 - x), $MachinePrecision], If[LessEqual[t$95$0, 0.5], 0.918938533204673, N[(0.0 - x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < -2e3 or 0.5 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64)))

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 35.9

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 35.9%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x} \]

      3. --lowering--.f64 34.9

         \[\leadsto \color{blue}{0 - x} \]

   8. Simplified 34.9%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 34.9

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

   10. Applied egg-rr 34.9%

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

   if -2e3 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < 0.5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 98.3

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
   7. Step-by-step derivation

   8. Simplified 95.0%

      \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(y + -1\right) - y \cdot 0.5 \leq -2000:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \cdot \left(y + -1\right) - y \cdot 0.5 \leq 0.5:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+252}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15)
   (* y x)
   (if (<= y 1.3e-7)
     (- 0.918938533204673 x)
     (if (<= y 6.4e+96)
       (fma -0.5 y 0.918938533204673)
       (if (<= y 5.4e+252) (* y x) (* y -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15) {
		tmp = y * x;
	} else if (y <= 1.3e-7) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 6.4e+96) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 5.4e+252) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15)
		tmp = Float64(y * x);
	elseif (y <= 1.3e-7)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 6.4e+96)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 5.4e+252)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.3e-7], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 6.4e+96], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 5.4e+252], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.14999999999999991 or 6.40000000000000013e96 < y < 5.40000000000000021e252

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 99.1

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]

   6. Taylor expanded in x around inf

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

   8. Simplified 66.4%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.4

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

   10. Applied egg-rr 66.4%

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

   if -2.14999999999999991 < y < 1.29999999999999999e-7

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 98.6

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 1.29999999999999999e-7 < y < 6.40000000000000013e96

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. accelerator-lowering-fma.f64 73.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   if 5.40000000000000021e252 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 100.0

         \[\leadsto \color{blue}{\left(x + -0.5\right)} \cdot y \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]
   9. Step-by-step derivation

   10. Simplified 68.0%

       \[\leadsto \color{blue}{-0.5} \cdot y \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+252}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.82:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+252}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15)
   (* y x)
   (if (<= y 1.82)
     (- 0.918938533204673 x)
     (if (<= y 4.5e+96) (* y -0.5) (if (<= y 3.1e+252) (* y x) (* y -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15) {
		tmp = y * x;
	} else if (y <= 1.82) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.5e+96) {
		tmp = y * -0.5;
	} else if (y <= 3.1e+252) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.15d0)) then
        tmp = y * x
    else if (y <= 1.82d0) then
        tmp = 0.918938533204673d0 - x
    else if (y <= 4.5d+96) then
        tmp = y * (-0.5d0)
    else if (y <= 3.1d+252) then
        tmp = y * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15) {
		tmp = y * x;
	} else if (y <= 1.82) {
		tmp = 0.918938533204673 - x;
	} else if (y <= 4.5e+96) {
		tmp = y * -0.5;
	} else if (y <= 3.1e+252) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15:
		tmp = y * x
	elif y <= 1.82:
		tmp = 0.918938533204673 - x
	elif y <= 4.5e+96:
		tmp = y * -0.5
	elif y <= 3.1e+252:
		tmp = y * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15)
		tmp = Float64(y * x);
	elseif (y <= 1.82)
		tmp = Float64(0.918938533204673 - x);
	elseif (y <= 4.5e+96)
		tmp = Float64(y * -0.5);
	elseif (y <= 3.1e+252)
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15)
		tmp = y * x;
	elseif (y <= 1.82)
		tmp = 0.918938533204673 - x;
	elseif (y <= 4.5e+96)
		tmp = y * -0.5;
	elseif (y <= 3.1e+252)
		tmp = y * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.82], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[y, 4.5e+96], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 3.1e+252], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.14999999999999991 or 4.49999999999999957e96 < y < 3.09999999999999982e252

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 99.1

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]

   6. Taylor expanded in x around inf

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

   8. Simplified 66.4%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 66.4

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

   10. Applied egg-rr 66.4%

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

   if -2.14999999999999991 < y < 1.82000000000000006

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 98.3

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 1.82000000000000006 < y < 4.49999999999999957e96 or 3.09999999999999982e252 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 95.3

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 95.3

         \[\leadsto \color{blue}{\left(x + -0.5\right)} \cdot y \]

   7. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]
   9. Step-by-step derivation

   10. Simplified 66.7%

       \[\leadsto \color{blue}{-0.5} \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.82:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+96}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+252}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.36:\\ \;\;\;\;\mathsf{fma}\left(y, x, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.36)
   (fma y x (* y -0.5))
   (if (<= y 1.65) (- 0.918938533204673 x) (* y (+ x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.36) {
		tmp = fma(y, x, (y * -0.5));
	} else if (y <= 1.65) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * (x + -0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.36)
		tmp = fma(y, x, Float64(y * -0.5));
	elseif (y <= 1.65)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * Float64(x + -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.36], N[(y * x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65], N[(0.918938533204673 - x), $MachinePrecision], N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3600000000000001

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 98.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \frac{-1}{2} \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{y \cdot \frac{-1}{2}}\right) \]

      7. *-lowering-*.f64 98.7

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{y \cdot -0.5}\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y \cdot -0.5\right)} \]

   if -1.3600000000000001 < y < 1.6499999999999999

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 98.3

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 1.6499999999999999 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 97.7

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 97.7

         \[\leadsto \color{blue}{\left(x + -0.5\right)} \cdot y \]

   7. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36:\\ \;\;\;\;\mathsf{fma}\left(y, x, y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.36) t_0 (if (<= y 1.05) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.36) {
		tmp = t_0;
	} else if (y <= 1.05) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x + (-0.5d0))
    if (y <= (-1.36d0)) then
        tmp = t_0
    else if (y <= 1.05d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.36) {
		tmp = t_0;
	} else if (y <= 1.05) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.36:
		tmp = t_0
	elif y <= 1.05:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.36)
		tmp = t_0;
	elseif (y <= 1.05)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.36)
		tmp = t_0;
	elseif (y <= 1.05)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.36], t$95$0, If[LessEqual[y, 1.05], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3600000000000001 or 1.05000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 98.2

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.2

         \[\leadsto \color{blue}{\left(x + -0.5\right)} \cdot y \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot y} \]

   if -1.3600000000000001 < y < 1.05000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 98.3

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.82:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+18)
   (* y -0.5)
   (if (<= y 1.82) (- 0.918938533204673 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+18) {
		tmp = y * -0.5;
	} else if (y <= 1.82) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.9d+18)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.82d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+18) {
		tmp = y * -0.5;
	} else if (y <= 1.82) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+18:
		tmp = y * -0.5
	elif y <= 1.82:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+18)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.82)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+18)
		tmp = y * -0.5;
	elseif (y <= 1.82)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+18], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.82], N[(0.918938533204673 - x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e18 or 1.82000000000000006 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{2}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + -1 \cdot x\right)}\right)\right) \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)} \]

      7. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)} \]

      8. --rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right) - 0\right)} \]

      9. associate-+l- N/A

         \[\leadsto \color{blue}{\left(0 - y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right) + 0} \]

      10. neg-sub0 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right)\right)} + 0 \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\left(\frac{1}{2} + -1 \cdot x\right)\right), 0\right)} \]

      13. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)}, 0\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(-1 \cdot x\right)\right), 0\right) \]

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right), 0\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-1}{2} + \color{blue}{x}, 0\right) \]

      17. +-lowering-+.f64 98.8

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.5 + x}, 0\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.8

         \[\leadsto \color{blue}{\left(x + -0.5\right)} \cdot y \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(x + -0.5\right) \cdot y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y \]
   9. Step-by-step derivation

   10. Simplified 43.4%

       \[\leadsto \color{blue}{-0.5} \cdot y \]

   if -2.9e18 < y < 1.82000000000000006

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. --lowering--.f64 96.4

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Simplified 96.4%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.82:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 x))

C

double code(double x, double y) {
	return 0.918938533204673 - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 - x
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 - x;
}

Python

def code(x, y):
	return 0.918938533204673 - x

Julia

function code(x, y)
	return Float64(0.918938533204673 - x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 - x;
end

Wolfram

code[x_, y_] := N[(0.918938533204673 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. --lowering--.f64 52.0

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Simplified 52.0%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]
6. Add Preprocessing

Alternative 9: 26.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. --lowering--.f64 52.0

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Simplified 52.0%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000}} \]
7. Step-by-step derivation

8. Simplified 26.3%

   \[\leadsto \color{blue}{0.918938533204673} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))