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

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 11

Speedup: 1.2×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. associate-+r+ 100.0%

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

   6. associate-+l+ 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. distribute-lft-out 100.0%

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

   9. fma-def 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

   12. +-commutative 100.0%

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

   13. cancel-sign-sub-inv 100.0%

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

   14. *-lft-identity 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -440 \lor \neg \left(x \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -440.0) (not (<= x 1.5e+15)))
   (* x (+ y -1.0))
   (- (+ 0.918938533204673 (* y -0.5)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -440.0) || !(x <= 1.5e+15)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = (0.918938533204673 + (y * -0.5)) - 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 <= (-440.0d0)) .or. (.not. (x <= 1.5d+15))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = (0.918938533204673d0 + (y * (-0.5d0))) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -440.0) || !(x <= 1.5e+15)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -440.0) or not (x <= 1.5e+15):
		tmp = x * (y + -1.0)
	else:
		tmp = (0.918938533204673 + (y * -0.5)) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -440.0) || !(x <= 1.5e+15))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(Float64(0.918938533204673 + Float64(y * -0.5)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -440.0) || ~((x <= 1.5e+15)))
		tmp = x * (y + -1.0);
	else
		tmp = (0.918938533204673 + (y * -0.5)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -440.0], N[Not[LessEqual[x, 1.5e+15]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.918938533204673 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -440 or 1.5e15 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
   8. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

         \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

      4. unsub-neg 100.0%

         \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

   10. Taylor expanded in x around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. neg-mul-1 98.7%

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

       3. sub-neg 98.7%

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

       4. neg-sub0 98.7%

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

       5. sub-neg 98.7%

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

       6. +-commutative 98.7%

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

       7. distribute-rgt1-in 98.7%

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

       8. distribute-lft-neg-in 98.7%

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

       9. unsub-neg 98.7%

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

       10. associate-+l- 98.7%

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

       11. *-rgt-identity 98.7%

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

       12. neg-sub0 98.7%

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

       13. *-rgt-identity 98.7%

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

       14. mul-1-neg 98.7%

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

       15. distribute-rgt-in 98.7%

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

       16. +-commutative 98.7%

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

   12. Simplified 98.7%

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

   if -440 < x < 1.5e15

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.5%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.5%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around 0 99.5%

      \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right) - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -440 \lor \neg \left(x \leq 1.5 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.918938533204673 + y \cdot -0.5\right) - x\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 4: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.25) (not (<= y 1.0)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.25) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - 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 <= (-1.25d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.25) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.25) or not (y <= 1.0):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.25) || !(y <= 1.0))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.25) || ~((y <= 1.0)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.25], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25 or 1 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 96.4%

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

   if -1.25 < y < 1

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.3%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.3%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 5: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.62)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.62)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (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 ((x <= (-0.72d0)) .or. (.not. (x <= 0.62d0))) then
        tmp = x * (y + (-1.0d0))
    else
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.62)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.72) or not (x <= 0.62):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.62))
		tmp = Float64(x * Float64(y + -1.0));
	else
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.72) || ~((x <= 0.62)))
		tmp = x * (y + -1.0);
	else
		tmp = 0.918938533204673 - (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.62]], $MachinePrecision]], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.619999999999999996 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
   8. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. mul-1-neg 100.0%

         \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

      4. unsub-neg 100.0%

         \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

   10. Taylor expanded in x around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. neg-mul-1 96.4%

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

       3. sub-neg 96.4%

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

       4. neg-sub0 96.4%

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

       5. sub-neg 96.4%

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

       6. +-commutative 96.4%

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

       7. distribute-rgt1-in 96.4%

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

       8. distribute-lft-neg-in 96.4%

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

       9. unsub-neg 96.4%

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

       10. associate-+l- 96.4%

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

       11. *-rgt-identity 96.4%

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

       12. neg-sub0 96.4%

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

       13. *-rgt-identity 96.4%

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

       14. mul-1-neg 96.4%

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

       15. distribute-rgt-in 96.4%

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

       16. +-commutative 96.4%

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

   12. Simplified 96.4%

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

   if -0.71999999999999997 < x < 0.619999999999999996

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.62\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. fma-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. remove-double-neg 100.0%

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

   6. sub-neg 100.0%

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

   7. fma-neg 100.0%

      \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   8. sub-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
5. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. metadata-eval 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-rgt-in 100.0%

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

   5. fma-def 100.0%

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

   6. neg-mul-1 100.0%

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

   7. fma-neg 100.0%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

7. Taylor expanded in y around -inf 100.0%

   \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
8. Step-by-step derivation

   1. mul-1-neg 100.0%

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

   2. unsub-neg 100.0%

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

   3. mul-1-neg 100.0%

      \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

   4. unsub-neg 100.0%

      \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

9. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

10. Final simplification 100.0%

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

Alternative 7: 49.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.7) (- x) (if (<= x 0.5) (* y -0.5) (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.7) {
		tmp = -x;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} else {
		tmp = -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 <= (-0.7d0)) then
        tmp = -x
    else if (x <= 0.5d0) then
        tmp = y * (-0.5d0)
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.7) {
		tmp = -x;
	} else if (x <= 0.5) {
		tmp = y * -0.5;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.7:
		tmp = -x
	elif x <= 0.5:
		tmp = y * -0.5
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.7)
		tmp = Float64(-x);
	elseif (x <= 0.5)
		tmp = Float64(y * -0.5);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.7)
		tmp = -x;
	elseif (x <= 0.5)
		tmp = y * -0.5;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.7], (-x), If[LessEqual[x, 0.5], N[(y * -0.5), $MachinePrecision], (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.69999999999999996 or 0.5 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 50.6%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 50.6%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 50.6%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around 0 50.6%

      \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right) - x} \]

   8. Taylor expanded in x around inf 48.0%

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

      1. mul-1-neg 48.0%

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

   10. Simplified 48.0%

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

   if -0.69999999999999996 < x < 0.5

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 48.5%

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

   5. Taylor expanded in x around 0 47.9%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 47.9%

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

   7. Simplified 47.9%

      \[\leadsto \color{blue}{y \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 8: 49.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12600:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -12600.0) (* y x) (if (<= y 1.25) (- x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -12600.0) {
		tmp = y * x;
	} else if (y <= 1.25) {
		tmp = -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 <= (-12600.0d0)) then
        tmp = y * x
    else if (y <= 1.25d0) then
        tmp = -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 <= -12600.0) {
		tmp = y * x;
	} else if (y <= 1.25) {
		tmp = -x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -12600.0:
		tmp = y * x
	elif y <= 1.25:
		tmp = -x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -12600.0)
		tmp = Float64(y * x);
	elseif (y <= 1.25)
		tmp = Float64(-x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -12600.0)
		tmp = y * x;
	elseif (y <= 1.25)
		tmp = -x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -12600.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.25], (-x), N[(y * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -12600

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-in 99.9%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 99.9%

         \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. mul-1-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

      4. unsub-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

   10. Taylor expanded in x around inf 57.6%

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

       1. mul-1-neg 57.6%

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

       2. neg-mul-1 57.6%

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

       3. sub-neg 57.6%

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

       4. neg-sub0 57.6%

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

       5. sub-neg 57.6%

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

       6. +-commutative 57.6%

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

       7. distribute-rgt1-in 57.6%

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

       8. distribute-lft-neg-in 57.6%

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

       9. unsub-neg 57.6%

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

       10. associate-+l- 57.6%

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

       11. *-rgt-identity 57.6%

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

       12. neg-sub0 57.6%

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

       13. *-rgt-identity 57.6%

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

       14. mul-1-neg 57.6%

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

       15. distribute-rgt-in 57.6%

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

       16. +-commutative 57.6%

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

   12. Simplified 57.6%

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

   13. Taylor expanded in y around inf 56.1%

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

   if -12600 < y < 1.25

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.4%

         \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.4%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around 0 99.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right) - x} \]

   8. Taylor expanded in x around inf 48.6%

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

      1. mul-1-neg 48.6%

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

   10. Simplified 48.6%

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

   if 1.25 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 97.1%

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

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12600:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 9: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -58000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -58000.0)
   (* y x)
   (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -58000.0) {
		tmp = y * x;
	} 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 <= (-58000.0d0)) then
        tmp = y * x
    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 <= -58000.0) {
		tmp = y * x;
	} 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 <= -58000.0:
		tmp = y * x
	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 <= -58000.0)
		tmp = Float64(y * x);
	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 <= -58000.0)
		tmp = y * x;
	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, -58000.0], N[(y * x), $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 < -58000

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-in 99.9%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 99.9%

         \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. mul-1-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

      4. unsub-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

   10. Taylor expanded in x around inf 58.3%

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

       1. mul-1-neg 58.3%

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

       2. neg-mul-1 58.3%

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

       3. sub-neg 58.3%

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

       4. neg-sub0 58.3%

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

       5. sub-neg 58.3%

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

       6. +-commutative 58.3%

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

       7. distribute-rgt1-in 58.3%

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

       8. distribute-lft-neg-in 58.3%

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

       9. unsub-neg 58.3%

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

       10. associate-+l- 58.3%

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

       11. *-rgt-identity 58.3%

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

       12. neg-sub0 58.3%

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

       13. *-rgt-identity 58.3%

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

       14. mul-1-neg 58.3%

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

       15. distribute-rgt-in 58.3%

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

       16. +-commutative 58.3%

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

   12. Simplified 58.3%

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

   13. Taylor expanded in y around inf 56.8%

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

   if -58000 < y < 1.8500000000000001

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.0%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.0%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 97.0%

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

   6. Simplified 97.0%

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

   if 1.8500000000000001 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 97.1%

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

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -58000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 10: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.9e-9)
   (* x (+ y -1.0))
   (if (<= y 1.85) (- 0.918938533204673 x) (* y -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.9e-9) {
		tmp = x * (y + -1.0);
	} 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 <= (-6.9d-9)) then
        tmp = x * (y + (-1.0d0))
    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 <= -6.9e-9) {
		tmp = x * (y + -1.0);
	} 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 <= -6.9e-9:
		tmp = x * (y + -1.0)
	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 <= -6.9e-9)
		tmp = Float64(x * Float64(y + -1.0));
	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 <= -6.9e-9)
		tmp = x * (y + -1.0);
	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, -6.9e-9], N[(x * N[(y + -1.0), $MachinePrecision]), $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 < -6.89999999999999975e-9

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
   5. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-in 99.9%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 99.9%

         \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(y \cdot x - x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   7. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 + -1 \cdot \left(y \cdot \left(0.5 + -1 \cdot x\right)\right)\right) - x} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. mul-1-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \left(0.5 + \color{blue}{\left(-x\right)}\right)\right) - x \]

      4. unsub-neg 99.9%

         \[\leadsto \left(0.918938533204673 - y \cdot \color{blue}{\left(0.5 - x\right)}\right) - x \]

   9. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.918938533204673 - y \cdot \left(0.5 - x\right)\right) - x} \]

   10. Taylor expanded in x around inf 56.8%

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

       1. mul-1-neg 56.8%

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

       2. neg-mul-1 56.8%

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

       3. sub-neg 56.8%

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

       4. neg-sub0 56.8%

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

       5. sub-neg 56.8%

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

       6. +-commutative 56.8%

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

       7. distribute-rgt1-in 56.8%

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

       8. distribute-lft-neg-in 56.8%

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

       9. unsub-neg 56.8%

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

       10. associate-+l- 56.8%

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

       11. *-rgt-identity 56.8%

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

       12. neg-sub0 56.8%

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

       13. *-rgt-identity 56.8%

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

       14. mul-1-neg 56.8%

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

       15. distribute-rgt-in 56.8%

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

       16. +-commutative 56.8%

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

   12. Simplified 56.8%

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

   if -6.89999999999999975e-9 < y < 1.8500000000000001

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.2%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 99.2%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 99.2%

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

   6. Simplified 99.2%

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

   if 1.8500000000000001 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. fma-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. fma-neg 100.0%

         \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      8. sub-neg 100.0%

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

      9. remove-double-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 97.1%

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

   5. Taylor expanded in x around 0 50.1%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
   6. Step-by-step derivation

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.85:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]

Alternative 11: 25.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 100.0%

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

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. fma-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. remove-double-neg 100.0%

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

   6. sub-neg 100.0%

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

   7. fma-neg 100.0%

      \[\leadsto \color{blue}{x \cdot \left(\left(-1\right) - \left(-y\right)\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   8. sub-neg 100.0%

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

   9. remove-double-neg 100.0%

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

   10. +-commutative 100.0%

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

   11. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around 0 74.3%

   \[\leadsto \color{blue}{-1 \cdot x} - \left(y \cdot 0.5 - 0.918938533204673\right) \]
5. Step-by-step derivation

   1. neg-mul-1 74.3%

      \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

6. Simplified 74.3%

   \[\leadsto \color{blue}{\left(-x\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

7. Taylor expanded in y around 0 74.3%

   \[\leadsto \color{blue}{\left(-0.5 \cdot y + 0.918938533204673\right) - x} \]

8. Taylor expanded in x around inf 26.4%

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

   1. mul-1-neg 26.4%

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

10. Simplified 26.4%

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

11. Final simplification 26.4%

    \[\leadsto -x \]

Reproduce

herbie shell --seed 2023171 
(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))