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

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 14

Speedup: 0.1×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * N[(y + -1.0), $MachinePrecision] + N[(y * -0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $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. fma-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 50.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+149}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1250000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-191}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-230}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-286}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-282}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e+149)
   (* y -0.5)
   (if (<= y -2.9e+41)
     (* x y)
     (if (<= y -1250000.0)
       (* y -0.5)
       (if (<= y -6.2e-191)
         (- x)
         (if (<= y -2.3e-230)
           0.918938533204673
           (if (<= y -1e-286)
             (- x)
             (if (<= y 5.9e-282)
               0.918938533204673
               (if (<= y 1.0) (- x) (* x y))))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+149) {
		tmp = y * -0.5;
	} else if (y <= -2.9e+41) {
		tmp = x * y;
	} else if (y <= -1250000.0) {
		tmp = y * -0.5;
	} else if (y <= -6.2e-191) {
		tmp = -x;
	} else if (y <= -2.3e-230) {
		tmp = 0.918938533204673;
	} else if (y <= -1e-286) {
		tmp = -x;
	} else if (y <= 5.9e-282) {
		tmp = 0.918938533204673;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d+149)) then
        tmp = y * (-0.5d0)
    else if (y <= (-2.9d+41)) then
        tmp = x * y
    else if (y <= (-1250000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= (-6.2d-191)) then
        tmp = -x
    else if (y <= (-2.3d-230)) then
        tmp = 0.918938533204673d0
    else if (y <= (-1d-286)) then
        tmp = -x
    else if (y <= 5.9d-282) then
        tmp = 0.918938533204673d0
    else if (y <= 1.0d0) then
        tmp = -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+149) {
		tmp = y * -0.5;
	} else if (y <= -2.9e+41) {
		tmp = x * y;
	} else if (y <= -1250000.0) {
		tmp = y * -0.5;
	} else if (y <= -6.2e-191) {
		tmp = -x;
	} else if (y <= -2.3e-230) {
		tmp = 0.918938533204673;
	} else if (y <= -1e-286) {
		tmp = -x;
	} else if (y <= 5.9e-282) {
		tmp = 0.918938533204673;
	} else if (y <= 1.0) {
		tmp = -x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e+149:
		tmp = y * -0.5
	elif y <= -2.9e+41:
		tmp = x * y
	elif y <= -1250000.0:
		tmp = y * -0.5
	elif y <= -6.2e-191:
		tmp = -x
	elif y <= -2.3e-230:
		tmp = 0.918938533204673
	elif y <= -1e-286:
		tmp = -x
	elif y <= 5.9e-282:
		tmp = 0.918938533204673
	elif y <= 1.0:
		tmp = -x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e+149)
		tmp = Float64(y * -0.5);
	elseif (y <= -2.9e+41)
		tmp = Float64(x * y);
	elseif (y <= -1250000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= -6.2e-191)
		tmp = Float64(-x);
	elseif (y <= -2.3e-230)
		tmp = 0.918938533204673;
	elseif (y <= -1e-286)
		tmp = Float64(-x);
	elseif (y <= 5.9e-282)
		tmp = 0.918938533204673;
	elseif (y <= 1.0)
		tmp = Float64(-x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e+149)
		tmp = y * -0.5;
	elseif (y <= -2.9e+41)
		tmp = x * y;
	elseif (y <= -1250000.0)
		tmp = y * -0.5;
	elseif (y <= -6.2e-191)
		tmp = -x;
	elseif (y <= -2.3e-230)
		tmp = 0.918938533204673;
	elseif (y <= -1e-286)
		tmp = -x;
	elseif (y <= 5.9e-282)
		tmp = 0.918938533204673;
	elseif (y <= 1.0)
		tmp = -x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e+149], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -2.9e+41], N[(x * y), $MachinePrecision], If[LessEqual[y, -1250000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -6.2e-191], (-x), If[LessEqual[y, -2.3e-230], 0.918938533204673, If[LessEqual[y, -1e-286], (-x), If[LessEqual[y, 5.9e-282], 0.918938533204673, If[LessEqual[y, 1.0], (-x), N[(x * y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.5999999999999997e149 or -2.89999999999999988e41 < y < -1.25e6

   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. sub-neg 100.0%

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

      3. 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 100.0%

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

   5. Taylor expanded in y around inf 99.9%

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

   6. Taylor expanded in x around 0 60.0%

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

      1. *-commutative 60.0%

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

   8. Simplified 60.0%

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

   if -4.5999999999999997e149 < y < -2.89999999999999988e41 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 62.9%

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

   7. Taylor expanded in y around inf 62.2%

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

   if -1.25e6 < y < -6.2000000000000004e-191 or -2.2999999999999998e-230 < y < -1.00000000000000005e-286 or 5.8999999999999997e-282 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 61.9%

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

   7. Taylor expanded in y around 0 59.6%

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

      1. neg-mul-1 59.6%

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

   9. Simplified 59.6%

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

   if -6.2000000000000004e-191 < y < -2.2999999999999998e-230 or -1.00000000000000005e-286 < y < 5.8999999999999997e-282

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-commutative 100.0%

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

      6. prod-diff 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-udef 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-commutative 100.0%

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

      16. add-sqr-sqrt 36.4%

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

      17. sqrt-unprod 100.0%

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

      18. swap-sqr 100.0%

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

      19. metadata-eval 100.0%

          \[\leadsto \left(\mathsf{fma}\left(x, y + -1, y \cdot -0.5\right) + \mathsf{fma}\left(-0.5, y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.25}}\right)\right) + 0.918938533204673 \]

      20. metadata-eval 100.0%

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

      21. swap-sqr 100.0%

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

      22. sqrt-unprod 63.6%

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

      23. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-rgt-out 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-+r+ 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. sub-neg 100.0%

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

      14. metadata-eval 100.0%

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

      15. +-commutative 100.0%

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

      16. fma-udef 100.0%

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

      17. *-commutative 100.0%

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

      18. distribute-lft-out 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-commutative 100.0%

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

      21. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around -inf 78.0%

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

      1. mul-1-neg 78.0%

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

      2. *-commutative 78.0%

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

      3. distribute-rgt-neg-in 78.0%

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

      4. sub-neg 78.0%

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

      5. metadata-eval 78.0%

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

      6. mul-1-neg 78.0%

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

      7. unsub-neg 78.0%

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

   10. Simplified 78.0%

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

   11. Taylor expanded in y around 0 78.0%

       \[\leadsto \color{blue}{0.918938533204673} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+149}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1250000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-191}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-230}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-286}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-282}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-292}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{-273}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-129}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -640000000.0)
   (- x)
   (if (<= x -3.7e-65)
     (* y -0.5)
     (if (<= x -4.7e-292)
       0.918938533204673
       (if (<= x 3.15e-273)
         (* y -0.5)
         (if (<= x 1.8e-129)
           0.918938533204673
           (if (<= x 0.9) (* y -0.5) (- x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -640000000.0) {
		tmp = -x;
	} else if (x <= -3.7e-65) {
		tmp = y * -0.5;
	} else if (x <= -4.7e-292) {
		tmp = 0.918938533204673;
	} else if (x <= 3.15e-273) {
		tmp = y * -0.5;
	} else if (x <= 1.8e-129) {
		tmp = 0.918938533204673;
	} else if (x <= 0.9) {
		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 <= (-640000000.0d0)) then
        tmp = -x
    else if (x <= (-3.7d-65)) then
        tmp = y * (-0.5d0)
    else if (x <= (-4.7d-292)) then
        tmp = 0.918938533204673d0
    else if (x <= 3.15d-273) then
        tmp = y * (-0.5d0)
    else if (x <= 1.8d-129) then
        tmp = 0.918938533204673d0
    else if (x <= 0.9d0) 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 <= -640000000.0) {
		tmp = -x;
	} else if (x <= -3.7e-65) {
		tmp = y * -0.5;
	} else if (x <= -4.7e-292) {
		tmp = 0.918938533204673;
	} else if (x <= 3.15e-273) {
		tmp = y * -0.5;
	} else if (x <= 1.8e-129) {
		tmp = 0.918938533204673;
	} else if (x <= 0.9) {
		tmp = y * -0.5;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -640000000.0:
		tmp = -x
	elif x <= -3.7e-65:
		tmp = y * -0.5
	elif x <= -4.7e-292:
		tmp = 0.918938533204673
	elif x <= 3.15e-273:
		tmp = y * -0.5
	elif x <= 1.8e-129:
		tmp = 0.918938533204673
	elif x <= 0.9:
		tmp = y * -0.5
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -640000000.0)
		tmp = Float64(-x);
	elseif (x <= -3.7e-65)
		tmp = Float64(y * -0.5);
	elseif (x <= -4.7e-292)
		tmp = 0.918938533204673;
	elseif (x <= 3.15e-273)
		tmp = Float64(y * -0.5);
	elseif (x <= 1.8e-129)
		tmp = 0.918938533204673;
	elseif (x <= 0.9)
		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 <= -640000000.0)
		tmp = -x;
	elseif (x <= -3.7e-65)
		tmp = y * -0.5;
	elseif (x <= -4.7e-292)
		tmp = 0.918938533204673;
	elseif (x <= 3.15e-273)
		tmp = y * -0.5;
	elseif (x <= 1.8e-129)
		tmp = 0.918938533204673;
	elseif (x <= 0.9)
		tmp = y * -0.5;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -640000000.0], (-x), If[LessEqual[x, -3.7e-65], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, -4.7e-292], 0.918938533204673, If[LessEqual[x, 3.15e-273], N[(y * -0.5), $MachinePrecision], If[LessEqual[x, 1.8e-129], 0.918938533204673, If[LessEqual[x, 0.9], N[(y * -0.5), $MachinePrecision], (-x)]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4e8 or 0.900000000000000022 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.5%

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

   7. Taylor expanded in y around 0 51.5%

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

      1. neg-mul-1 51.5%

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

   9. Simplified 51.5%

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

   if -6.4e8 < x < -3.7e-65 or -4.70000000000000002e-292 < x < 3.14999999999999989e-273 or 1.8e-129 < x < 0.900000000000000022

   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. sub-neg 100.0%

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

      3. 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 98.6%

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

   5. Taylor expanded in y around inf 72.0%

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

   6. Taylor expanded in x around 0 63.9%

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

      1. *-commutative 63.9%

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

   8. Simplified 63.9%

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

   if -3.7e-65 < x < -4.70000000000000002e-292 or 3.14999999999999989e-273 < x < 1.8e-129

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-commutative 100.0%

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

      6. prod-diff 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-udef 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-commutative 100.0%

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

      16. add-sqr-sqrt 47.8%

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

      17. sqrt-unprod 76.9%

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

      18. swap-sqr 76.9%

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

      19. metadata-eval 76.9%

          \[\leadsto \left(\mathsf{fma}\left(x, y + -1, y \cdot -0.5\right) + \mathsf{fma}\left(-0.5, y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.25}}\right)\right) + 0.918938533204673 \]

      20. metadata-eval 76.9%

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

      21. swap-sqr 76.9%

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

      22. sqrt-unprod 34.7%

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

      23. add-sqr-sqrt 69.6%

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

   5. Applied egg-rr 69.6%

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

      1. fma-udef 69.6%

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

      2. distribute-rgt-out 69.6%

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

      3. neg-mul-1 69.6%

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

      4. +-commutative 69.6%

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

      5. *-commutative 69.6%

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

      6. associate-+r+ 69.6%

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

      7. *-commutative 69.6%

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

      8. distribute-lft-in 69.6%

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

      9. +-commutative 69.6%

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

      10. sub-neg 69.6%

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

      11. metadata-eval 69.6%

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

      12. sub-neg 69.6%

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

      13. sub-neg 69.6%

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

      14. metadata-eval 69.6%

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

      15. +-commutative 69.6%

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

      16. fma-udef 69.6%

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

      17. *-commutative 69.6%

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

      18. distribute-lft-out 69.6%

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

      19. metadata-eval 69.6%

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

      20. *-commutative 69.6%

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

      21. neg-mul-1 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in y around -inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. *-commutative 69.6%

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

      3. distribute-rgt-neg-in 69.6%

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

      4. sub-neg 69.6%

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

      5. metadata-eval 69.6%

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

      6. mul-1-neg 69.6%

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

      7. unsub-neg 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in y around 0 64.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-292}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{-273}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-129}:\\ \;\;\;\;0.918938533204673\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;0.918938533204673 + \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -640000000.0)
   (+ 0.918938533204673 (- (* x y) x))
   (if (<= x 0.55)
     (+ (* x y) (- 0.918938533204673 (* y 0.5)))
     (+ 0.918938533204673 (* x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -640000000.0) {
		tmp = 0.918938533204673 + ((x * y) - x);
	} else if (x <= 0.55) {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-640000000.0d0)) then
        tmp = 0.918938533204673d0 + ((x * y) - x)
    else if (x <= 0.55d0) then
        tmp = (x * y) + (0.918938533204673d0 - (y * 0.5d0))
    else
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -640000000.0) {
		tmp = 0.918938533204673 + ((x * y) - x);
	} else if (x <= 0.55) {
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -640000000.0:
		tmp = 0.918938533204673 + ((x * y) - x)
	elif x <= 0.55:
		tmp = (x * y) + (0.918938533204673 - (y * 0.5))
	else:
		tmp = 0.918938533204673 + (x * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -640000000.0)
		tmp = Float64(0.918938533204673 + Float64(Float64(x * y) - x));
	elseif (x <= 0.55)
		tmp = Float64(Float64(x * y) + Float64(0.918938533204673 - Float64(y * 0.5)));
	else
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -640000000.0)
		tmp = 0.918938533204673 + ((x * y) - x);
	elseif (x <= 0.55)
		tmp = (x * y) + (0.918938533204673 - (y * 0.5));
	else
		tmp = 0.918938533204673 + (x * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -640000000.0], N[(0.918938533204673 + N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.55], N[(N[(x * y), $MachinePrecision] + N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4e8

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 100.0%

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

   6. Simplified 100.0%

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

   if -6.4e8 < x < 0.55000000000000004

   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. sub-neg 100.0%

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

      3. 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 99.4%

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

   if 0.55000000000000004 < 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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} + 0.918938533204673 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;0.918938533204673 + \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \leq 0.55:\\ \;\;\;\;x \cdot y + \left(0.918938533204673 - y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 5: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+144}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+46}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1150000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.45e+144)
   (* y -0.5)
   (if (<= y -5e+46)
     (* x y)
     (if (<= y -1150000.0)
       (* y -0.5)
       (if (<= y 1.0) (- 0.918938533204673 x) (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+144) {
		tmp = y * -0.5;
	} else if (y <= -5e+46) {
		tmp = x * y;
	} else if (y <= -1150000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.45d+144)) then
        tmp = y * (-0.5d0)
    else if (y <= (-5d+46)) then
        tmp = x * y
    else if (y <= (-1150000.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.0d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+144) {
		tmp = y * -0.5;
	} else if (y <= -5e+46) {
		tmp = x * y;
	} else if (y <= -1150000.0) {
		tmp = y * -0.5;
	} else if (y <= 1.0) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.45e+144:
		tmp = y * -0.5
	elif y <= -5e+46:
		tmp = x * y
	elif y <= -1150000.0:
		tmp = y * -0.5
	elif y <= 1.0:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.45e+144)
		tmp = Float64(y * -0.5);
	elseif (y <= -5e+46)
		tmp = Float64(x * y);
	elseif (y <= -1150000.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.0)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.45e+144)
		tmp = y * -0.5;
	elseif (y <= -5e+46)
		tmp = x * y;
	elseif (y <= -1150000.0)
		tmp = y * -0.5;
	elseif (y <= 1.0)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.45e+144], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, -5e+46], N[(x * y), $MachinePrecision], If[LessEqual[y, -1150000.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.0], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.45e144 or -5.0000000000000002e46 < y < -1.15e6

   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. sub-neg 100.0%

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

      3. 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 100.0%

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

   5. Taylor expanded in y around inf 99.9%

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

   6. Taylor expanded in x around 0 60.0%

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

      1. *-commutative 60.0%

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

   8. Simplified 60.0%

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

   if -2.45e144 < y < -5.0000000000000002e46 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 62.9%

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

   7. Taylor expanded in y around inf 62.2%

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

   if -1.15e6 < 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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. neg-mul-1 96.6%

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

      2. sub-neg 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+144}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+46}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq -1150000:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1450000000 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1450000000.0) (not (<= x 0.85)))
   (+ 0.918938533204673 (* x (+ y -1.0)))
   (+ 0.918938533204673 (* y (- x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1450000000.0) || !(x <= 0.85)) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else {
		tmp = 0.918938533204673 + (y * (x - 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 <= (-1450000000.0d0)) .or. (.not. (x <= 0.85d0))) then
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    else
        tmp = 0.918938533204673d0 + (y * (x - 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1450000000.0) || !(x <= 0.85)) {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	} else {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1450000000.0) or not (x <= 0.85):
		tmp = 0.918938533204673 + (x * (y + -1.0))
	else:
		tmp = 0.918938533204673 + (y * (x - 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1450000000.0) || !(x <= 0.85))
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	else
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x - 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1450000000.0) || ~((x <= 0.85)))
		tmp = 0.918938533204673 + (x * (y + -1.0));
	else
		tmp = 0.918938533204673 + (y * (x - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1450000000.0], N[Not[LessEqual[x, 0.85]], $MachinePrecision]], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e9 or 0.849999999999999978 < 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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.8%

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

   if -1.45e9 < x < 0.849999999999999978

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1450000000 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -16500000000:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 180000:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -16500000000.0)
   (- (* x y) x)
   (if (<= x 180000.0)
     (+ 0.918938533204673 (* y (- x 0.5)))
     (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -16500000000.0) {
		tmp = (x * y) - x;
	} else if (x <= 180000.0) {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-16500000000.0d0)) then
        tmp = (x * y) - x
    else if (x <= 180000.0d0) then
        tmp = 0.918938533204673d0 + (y * (x - 0.5d0))
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -16500000000.0) {
		tmp = (x * y) - x;
	} else if (x <= 180000.0) {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -16500000000.0:
		tmp = (x * y) - x
	elif x <= 180000.0:
		tmp = 0.918938533204673 + (y * (x - 0.5))
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -16500000000.0)
		tmp = Float64(Float64(x * y) - x);
	elseif (x <= 180000.0)
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x - 0.5)));
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -16500000000.0)
		tmp = (x * y) - x;
	elseif (x <= 180000.0)
		tmp = 0.918938533204673 + (y * (x - 0.5));
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -16500000000.0], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 180000.0], N[(0.918938533204673 + N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e10

   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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.9%

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

   7. Taylor expanded in y around 0 99.9%

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

      1. neg-mul-1 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   if -1.65e10 < x < 1.8e5

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.4%

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

   if 1.8e5 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -16500000000:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 180000:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 8: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;0.918938533204673 + \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \leq 0.00084:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -640000000.0)
   (+ 0.918938533204673 (- (* x y) x))
   (if (<= x 0.00084)
     (+ 0.918938533204673 (* y (- x 0.5)))
     (+ 0.918938533204673 (* x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -640000000.0) {
		tmp = 0.918938533204673 + ((x * y) - x);
	} else if (x <= 0.00084) {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-640000000.0d0)) then
        tmp = 0.918938533204673d0 + ((x * y) - x)
    else if (x <= 0.00084d0) then
        tmp = 0.918938533204673d0 + (y * (x - 0.5d0))
    else
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -640000000.0) {
		tmp = 0.918938533204673 + ((x * y) - x);
	} else if (x <= 0.00084) {
		tmp = 0.918938533204673 + (y * (x - 0.5));
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -640000000.0:
		tmp = 0.918938533204673 + ((x * y) - x)
	elif x <= 0.00084:
		tmp = 0.918938533204673 + (y * (x - 0.5))
	else:
		tmp = 0.918938533204673 + (x * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -640000000.0)
		tmp = Float64(0.918938533204673 + Float64(Float64(x * y) - x));
	elseif (x <= 0.00084)
		tmp = Float64(0.918938533204673 + Float64(y * Float64(x - 0.5)));
	else
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -640000000.0)
		tmp = 0.918938533204673 + ((x * y) - x);
	elseif (x <= 0.00084)
		tmp = 0.918938533204673 + (y * (x - 0.5));
	else
		tmp = 0.918938533204673 + (x * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -640000000.0], N[(0.918938533204673 + N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00084], N[(0.918938533204673 + N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4e8

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. neg-mul-1 100.0%

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

      7. fma-neg 100.0%

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

   6. Simplified 100.0%

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

   if -6.4e8 < x < 8.4000000000000003e-4

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 99.4%

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

   if 8.4000000000000003e-4 < 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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} + 0.918938533204673 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -640000000:\\ \;\;\;\;0.918938533204673 + \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \leq 0.00084:\\ \;\;\;\;0.918938533204673 + y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 9: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.22) (not (<= y 1.0)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.22) || !(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 <= (-1.22d0)) .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 <= -1.22) || !(y <= 1.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.22) 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 <= -1.22) || !(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 <= -1.22) || ~((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, -1.22], 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 < -1.21999999999999997 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. sub-neg 100.0%

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

      3. 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 99.1%

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

   5. Taylor expanded in y around inf 98.5%

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

   if -1.21999999999999997 < 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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.3%

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

      1. neg-mul-1 97.3%

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

      2. sub-neg 97.3%

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

   6. Simplified 97.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]

Alternative 10: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.9\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.66) (not (<= x 0.9)))
   (* x (+ y -1.0))
   (- 0.918938533204673 (* y 0.5))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.66) || !(x <= 0.9)) {
		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 <= (-0.66d0)) .or. (.not. (x <= 0.9d0))) 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 <= -0.66) || !(x <= 0.9)) {
		tmp = x * (y + -1.0);
	} else {
		tmp = 0.918938533204673 - (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.66) or not (x <= 0.9):
		tmp = x * (y + -1.0)
	else:
		tmp = 0.918938533204673 - (y * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.66) || !(x <= 0.9))
		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 <= -0.66) || ~((x <= 0.9)))
		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, -0.66], N[Not[LessEqual[x, 0.9]], $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 < -0.660000000000000031 or 0.900000000000000022 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.6%

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

   if -0.660000000000000031 < x < 0.900000000000000022

   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. sub-neg 100.0%

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

      3. 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 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.9\right):\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \end{array} \]

Alternative 11: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.72)
   (- (* x y) x)
   (if (<= x 0.9) (- 0.918938533204673 (* y 0.5)) (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.72) {
		tmp = (x * y) - x;
	} else if (x <= 0.9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.72d0)) then
        tmp = (x * y) - x
    else if (x <= 0.9d0) then
        tmp = 0.918938533204673d0 - (y * 0.5d0)
    else
        tmp = x * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.72) {
		tmp = (x * y) - x;
	} else if (x <= 0.9) {
		tmp = 0.918938533204673 - (y * 0.5);
	} else {
		tmp = x * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.72:
		tmp = (x * y) - x
	elif x <= 0.9:
		tmp = 0.918938533204673 - (y * 0.5)
	else:
		tmp = x * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.72)
		tmp = Float64(Float64(x * y) - x);
	elseif (x <= 0.9)
		tmp = Float64(0.918938533204673 - Float64(y * 0.5));
	else
		tmp = Float64(x * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.72)
		tmp = (x * y) - x;
	elseif (x <= 0.9)
		tmp = 0.918938533204673 - (y * 0.5);
	else
		tmp = x * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.72], N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 0.9], N[(0.918938533204673 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.71999999999999997

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 98.1%

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

   7. Taylor expanded in y around 0 98.1%

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

      1. neg-mul-1 98.1%

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

      2. unsub-neg 98.1%

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

      3. *-commutative 98.1%

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

   9. Simplified 98.1%

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

   if -0.71999999999999997 < x < 0.900000000000000022

   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. sub-neg 100.0%

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

      3. 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 98.3%

      \[\leadsto \color{blue}{0.918938533204673 - 0.5 \cdot y} \]

   if 0.900000000000000022 < 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. cancel-sign-sub-inv 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{\left(y - 1\right) \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;x \cdot y - x\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;0.918938533204673 - y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 12: 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. cancel-sign-sub-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-out-- 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. *-rgt-identity 100.0%

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

   6. associate-+r+ 100.0%

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

   7. associate-+l+ 100.0%

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

   8. +-commutative 100.0%

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

   9. unsub-neg 100.0%

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

   10. associate-+r- 100.0%

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

   11. *-commutative 100.0%

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

   12. distribute-lft-neg-out 100.0%

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

   13. distribute-rgt-neg-in 100.0%

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

   14. distribute-lft-out 100.0%

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

   15. fma-def 100.0%

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

   16. +-commutative 100.0%

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

   17. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 13: 49.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.91:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.91) (- x) (if (<= x 1.25e-15) 0.918938533204673 (- x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.91) {
		tmp = -x;
	} else if (x <= 1.25e-15) {
		tmp = 0.918938533204673;
	} 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 <= (-0.91d0)) then
        tmp = -x
    else if (x <= 1.25d-15) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.91) {
		tmp = -x;
	} else if (x <= 1.25e-15) {
		tmp = 0.918938533204673;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.91:
		tmp = -x
	elif x <= 1.25e-15:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.91)
		tmp = Float64(-x);
	elseif (x <= 1.25e-15)
		tmp = 0.918938533204673;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.91)
		tmp = -x;
	elseif (x <= 1.25e-15)
		tmp = 0.918938533204673;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.91], (-x), If[LessEqual[x, 1.25e-15], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.910000000000000031 or 1.25e-15 < x

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-out-- 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-+r+ 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

      10. associate-+r- 100.0%

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

      11. *-commutative 100.0%

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

      12. distribute-lft-neg-out 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. distribute-lft-out 100.0%

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

      15. fma-def 100.0%

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

      16. +-commutative 100.0%

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

      17. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x - 0.5, 0.918938533204673\right) - x} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 97.9%

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

   7. Taylor expanded in y around 0 50.2%

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

      1. neg-mul-1 50.2%

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

   9. Simplified 50.2%

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

   if -0.910000000000000031 < x < 1.25e-15

   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. fma-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. *-commutative 100.0%

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

      6. prod-diff 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. *-commutative 100.0%

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

      13. fma-udef 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-commutative 100.0%

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

      16. add-sqr-sqrt 45.9%

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

      17. sqrt-unprod 66.1%

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

      18. swap-sqr 66.1%

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

      19. metadata-eval 66.1%

          \[\leadsto \left(\mathsf{fma}\left(x, y + -1, y \cdot -0.5\right) + \mathsf{fma}\left(-0.5, y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.25}}\right)\right) + 0.918938533204673 \]

      20. metadata-eval 66.1%

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

      21. swap-sqr 66.1%

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

      22. sqrt-unprod 30.6%

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

      23. add-sqr-sqrt 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. fma-udef 59.4%

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

      2. distribute-rgt-out 59.4%

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

      3. neg-mul-1 59.4%

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

      4. +-commutative 59.4%

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

      5. *-commutative 59.4%

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

      6. associate-+r+ 59.4%

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

      7. *-commutative 59.4%

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

      8. distribute-lft-in 59.4%

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

      9. +-commutative 59.4%

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

      10. sub-neg 59.4%

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

      11. metadata-eval 59.4%

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

      12. sub-neg 59.4%

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

      13. sub-neg 59.4%

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

      14. metadata-eval 59.4%

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

      15. +-commutative 59.4%

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

      16. fma-udef 59.4%

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

      17. *-commutative 59.4%

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

      18. distribute-lft-out 59.4%

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

      19. metadata-eval 59.4%

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

      20. *-commutative 59.4%

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

      21. neg-mul-1 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in y around -inf 58.8%

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

      1. mul-1-neg 58.8%

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

      2. *-commutative 58.8%

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

      3. distribute-rgt-neg-in 58.8%

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

      4. sub-neg 58.8%

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

      5. metadata-eval 58.8%

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

      6. mul-1-neg 58.8%

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

      7. unsub-neg 58.8%

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

   10. Simplified 58.8%

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

   11. Taylor expanded in y around 0 51.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.91:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 14: 25.1% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

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

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

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

   1. fma-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. fma-udef 100.0%

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

   2. *-commutative 100.0%

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

   3. metadata-eval 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. *-commutative 100.0%

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

   6. prod-diff 100.0%

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

   7. *-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. *-commutative 100.0%

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

   10. cancel-sign-sub-inv 100.0%

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

   11. metadata-eval 100.0%

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

   12. *-commutative 100.0%

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

   13. fma-udef 100.0%

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

   14. metadata-eval 100.0%

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

   15. *-commutative 100.0%

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

   16. add-sqr-sqrt 48.3%

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

   17. sqrt-unprod 76.0%

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

   18. swap-sqr 76.0%

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

   19. metadata-eval 76.0%

       \[\leadsto \left(\mathsf{fma}\left(x, y + -1, y \cdot -0.5\right) + \mathsf{fma}\left(-0.5, y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{0.25}}\right)\right) + 0.918938533204673 \]

   20. metadata-eval 76.0%

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

   21. swap-sqr 76.0%

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

   22. sqrt-unprod 40.6%

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

   23. add-sqr-sqrt 80.8%

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

5. Applied egg-rr 80.8%

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

   1. fma-udef 80.8%

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

   2. distribute-rgt-out 80.8%

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

   3. neg-mul-1 80.8%

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

   4. +-commutative 80.8%

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

   5. *-commutative 80.8%

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

   6. associate-+r+ 80.8%

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

   7. *-commutative 80.8%

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

   8. distribute-lft-in 80.8%

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

   9. +-commutative 80.8%

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

   10. sub-neg 80.8%

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

   11. metadata-eval 80.8%

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

   12. sub-neg 80.8%

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

   13. sub-neg 80.8%

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

   14. metadata-eval 80.8%

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

   15. +-commutative 80.8%

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

   16. fma-udef 80.8%

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

   17. *-commutative 80.8%

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

   18. distribute-lft-out 80.8%

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

   19. metadata-eval 80.8%

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

   20. *-commutative 80.8%

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

   21. neg-mul-1 80.8%

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

7. Simplified 80.8%

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

8. Taylor expanded in y around -inf 53.5%

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

   1. mul-1-neg 53.5%

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

   2. *-commutative 53.5%

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

   3. distribute-rgt-neg-in 53.5%

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

   4. sub-neg 53.5%

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

   5. metadata-eval 53.5%

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

   6. mul-1-neg 53.5%

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

   7. unsub-neg 53.5%

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

10. Simplified 53.5%

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

11. Taylor expanded in y around 0 24.9%

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

12. Final simplification 24.9%

    \[\leadsto 0.918938533204673 \]

Reproduce

herbie shell --seed 2023268 
(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))