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

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 10

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-+l- N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. sub-neg N/A

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

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

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

   7. metadata-eval N/A

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

   8. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

   9. associate-+l- N/A

      \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

   10. neg-sub0 N/A

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

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

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

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

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

   13. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, \color{blue}{-0.5}, 0.918938533204673\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\right)} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+204}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.06e+204)
   (* y x)
   (if (<= x -3.2e-6)
     (- 0.918938533204673 x)
     (if (<= x 0.52) (fma -0.5 y 0.918938533204673) (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.06e+204) {
		tmp = y * x;
	} else if (x <= -3.2e-6) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 0.52) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.06e+204)
		tmp = Float64(y * x);
	elseif (x <= -3.2e-6)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 0.52)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.06e+204], N[(y * x), $MachinePrecision], If[LessEqual[x, -3.2e-6], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 0.52], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05999999999999997e204 or 0.52000000000000002 < x

   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 inf

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 55.1

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

   8. Simplified 55.1%

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

   if -1.05999999999999997e204 < x < -3.1999999999999999e-6

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   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 69.1

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

   5. Simplified 69.1%

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

   if -3.1999999999999999e-6 < x < 0.52000000000000002

   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 97.6

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

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+204}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 0.52:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+67}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -480000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+67)
   (* y x)
   (if (<= y -480000.0)
     (* y -0.5)
     (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+67) {
		tmp = y * x;
	} else if (y <= -480000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		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 <= (-7d+67)) then
        tmp = y * x
    else if (y <= (-480000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.85d0) 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 <= -7e+67) {
		tmp = y * x;
	} else if (y <= -480000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.85) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+67:
		tmp = y * x
	elif y <= -480000.0:
		tmp = y * -0.5
	elif y <= 1.85:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+67)
		tmp = Float64(y * x);
	elseif (y <= -480000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.85)
		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 <= -7e+67)
		tmp = y * x;
	elseif (y <= -480000.0)
		tmp = y * -0.5;
	elseif (y <= 1.85)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+67], N[(y * x), $MachinePrecision], If[LessEqual[y, -480000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7e67

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 53.4

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

   8. Simplified 53.4%

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

   if -7e67 < y < -4.8e5 or 1.8500000000000001 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

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

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

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

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

      13. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, \color{blue}{-0.5}, 0.918938533204673\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   6. 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. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 98.2

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

   7. Simplified 98.2%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 53.9%

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

   if -4.8e5 < y < 1.8500000000000001

   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 96.6

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

   5. Simplified 96.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+67}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -480000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, y \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35)
   (* y (+ x -0.5))
   (if (<= y 1.0) (- 0.918938533204673 x) (fma y x (* y -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35) {
		tmp = y * (x + -0.5);
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = fma(y, x, (y * -0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35)
		tmp = Float64(y * Float64(x + -0.5));
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = fma(y, x, Float64(y * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35], N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3500000000000001

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

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

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

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

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

      13. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, \color{blue}{-0.5}, 0.918938533204673\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   6. 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. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 94.3

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

   7. Simplified 94.3%

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

   if -1.3500000000000001 < y < 1

   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 98.4

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

   5. Simplified 98.4%

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

   if 1 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

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

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

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

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

      13. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, \color{blue}{-0.5}, 0.918938533204673\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   6. 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. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 100.0

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

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

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

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

      3. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, y \cdot -0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.35) t_0 (if (<= y 1.85) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.35) {
		tmp = t_0;
	} else if (y <= 1.85) {
		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 <= (-1.35d0)) then
        tmp = t_0
    else if (y <= 1.85d0) 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 <= -1.35) {
		tmp = t_0;
	} else if (y <= 1.85) {
		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 <= -1.35:
		tmp = t_0
	elif y <= 1.85:
		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 <= -1.35)
		tmp = t_0;
	elseif (y <= 1.85)
		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 <= -1.35)
		tmp = t_0;
	elseif (y <= 1.85)
		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, -1.35], t$95$0, If[LessEqual[y, 1.85], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001 or 1.8500000000000001 < y

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l- N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{0 - \left(y \cdot \frac{1}{2} - \frac{918938533204673}{1000000000000000}\right)}\right) \]

      9. associate-+l- N/A

         \[\leadsto \mathsf{fma}\left(y + -1, x, \color{blue}{\left(0 - y \cdot \frac{1}{2}\right) + \frac{918938533204673}{1000000000000000}}\right) \]

      10. neg-sub0 N/A

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

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

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

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

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

      13. metadata-eval 100.0

          \[\leadsto \mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, \color{blue}{-0.5}, 0.918938533204673\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + -1, x, \mathsf{fma}\left(y, -0.5, 0.918938533204673\right)\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(x - \frac{1}{2}\right)} \]
   6. 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. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 97.2

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

   7. Simplified 97.2%

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

   if -1.3500000000000001 < y < 1.8500000000000001

   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 98.4

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

   5. Simplified 98.4%

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

Alternative 6: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15.2:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -15.2) (* y x) (if (<= y 1.0) (- 0.918938533204673 x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -15.2) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-15.2d0)) then
        tmp = y * x
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -15.2) {
		tmp = y * x;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -15.2:
		tmp = y * x
	elif y <= 1.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -15.2)
		tmp = Float64(y * x);
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -15.2)
		tmp = y * x;
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -15.2], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -15.199999999999999 or 1 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 47.4

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

   8. Simplified 47.4%

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

   if -15.199999999999999 < y < 1

   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 97.2

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

   5. Simplified 97.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15.2:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- 0.0 x) (if (<= x 0.92) 0.918938533204673 (- 0.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.92) {
		tmp = 0.0 - x;
	} else if (x <= 0.92) {
		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) :: tmp
    if (x <= (-0.92d0)) then
        tmp = 0.0d0 - x
    else if (x <= 0.92d0) 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 tmp;
	if (x <= -0.92) {
		tmp = 0.0 - x;
	} else if (x <= 0.92) {
		tmp = 0.918938533204673;
	} else {
		tmp = 0.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = 0.0 - x
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = 0.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.92)
		tmp = Float64(0.0 - x);
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = Float64(0.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.92)
		tmp = 0.0 - x;
	elseif (x <= 0.92)
		tmp = 0.918938533204673;
	else
		tmp = 0.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], N[(0.0 - x), $MachinePrecision], If[LessEqual[x, 0.92], 0.918938533204673, N[(0.0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 or 0.92000000000000004 < x

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   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 50.9

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

   5. Simplified 50.9%

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

   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 50.4

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

   8. Simplified 50.4%

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

      1. sub-neg N/A

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

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} + {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      4. cube-neg N/A

         \[\leadsto \frac{0 + \color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{0 - {x}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      7. sqr-pow N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      9. sqr-neg N/A

         \[\leadsto \frac{{0}^{3} - {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      11. sqr-pow N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      12. sqr-neg N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{x \cdot x} - 0 \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      13. mul0-lft N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(x \cdot x - \color{blue}{0}\right)} \]

      14. --rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \color{blue}{x \cdot x}} \]

      15. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \color{blue}{\left(x \cdot x + 0\right)}} \]

      16. sqr-neg N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} + 0\right)} \]

      17. mul0-lft N/A

          \[\leadsto \frac{{0}^{3} - {\left(\mathsf{neg}\left(x\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) + \color{blue}{0 \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)} \]

      18. flip3-- N/A

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

      19. neg-sub0 N/A

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

      20. neg-lowering-neg.f64 N/A

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

      21. +-lft-identity N/A

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

   10. Applied egg-rr 50.4%

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

   if -0.92000000000000004 < x < 0.92000000000000004

   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 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 51.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;0 - x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;0 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 9: 50.4% 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 52.1

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

5. Simplified 52.1%

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

Alternative 10: 25.8% 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 52.1

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

5. Simplified 52.1%

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

6. Taylor expanded in x around 0

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

8. Simplified 28.1%

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

Reproduce

herbie shell --seed 2024198 
(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))