Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq 10^{+134}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -1e+44)
     (- y (* y (log y)))
     (if (<= t_0 1e+134) (- (* (log y) -0.5) z) (fma (log y) -0.5 x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -1e+44) {
		tmp = y - (y * log(y));
	} else if (t_0 <= 1e+134) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(log(y), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -1e+44)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= 1e+134)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(log(y), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+44], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+134], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.0000000000000001e44

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.5

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   5. Taylor expanded in y around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub N/A

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

      7. log-rec N/A

         \[\leadsto y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right) \cdot y \]

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if -1.0000000000000001e44 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 9.99999999999999921e133

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.9

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   4. Applied rewrites 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   5. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 95.2

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.8%

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

   if 9.99999999999999921e133 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+44}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 10^{+134}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.75e-10)
   (fma (log y) -0.5 x)
   (if (<= y 11000000000.0) (- z) (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e-10) {
		tmp = fma(log(y), -0.5, x);
	} else if (y <= 11000000000.0) {
		tmp = -z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.75e-10)
		tmp = fma(log(y), -0.5, x);
	elseif (y <= 11000000000.0)
		tmp = Float64(-z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.75e-10], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], If[LessEqual[y, 11000000000.0], (-z), N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7499999999999999e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

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

   if 1.7499999999999999e-10 < y < 1.1e10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 1.1e10 < y

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.5

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   4. Applied rewrites 99.5%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   5. Taylor expanded in y around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub N/A

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

      7. log-rec N/A

         \[\leadsto y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right) \cdot y \]

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 73.4

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

   7. Applied rewrites 73.4%

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

Alternative 4: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 11000000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.75e-10)
   (fma (log y) -0.5 x)
   (if (<= y 11000000000.0) (- z) (fma (log y) (- y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.75e-10) {
		tmp = fma(log(y), -0.5, x);
	} else if (y <= 11000000000.0) {
		tmp = -z;
	} else {
		tmp = fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.75e-10)
		tmp = fma(log(y), -0.5, x);
	elseif (y <= 11000000000.0)
		tmp = Float64(-z);
	else
		tmp = fma(log(y), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.75e-10], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], If[LessEqual[y, 11000000000.0], (-z), N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7499999999999999e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

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

   if 1.7499999999999999e-10 < y < 1.1e10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 1.1e10 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 73.4

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

   5. Applied rewrites 73.4%

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

Alternative 5: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+42}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.2e+42) (- z) (if (<= z 1.52e+57) (fma (log y) -0.5 x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.2e+42) {
		tmp = -z;
	} else if (z <= 1.52e+57) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.2e+42)
		tmp = Float64(-z);
	elseif (z <= 1.52e+57)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.2e+42], (-z), If[LessEqual[z, 1.52e+57], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000003e42 or 1.51999999999999998e57 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if -6.2000000000000003e42 < z < 1.51999999999999998e57

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2} \cdot \log y} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 10500000000.0)
   (- (fma (log y) -0.5 x) z)
   (+ y (fma (log y) (- -0.5 y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 10500000000.0) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y + fma(log(y), (-0.5 - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 10500000000.0)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y + fma(log(y), Float64(-0.5 - y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 10500000000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.05e10 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 11000000000:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 11000000000.0)
   (- (fma (log y) -0.5 x) z)
   (+ y (fma (log y) (- y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 11000000000.0) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y + fma(log(y), -y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 11000000000.0)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y + fma(log(y), Float64(-y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 11000000000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y + N[(N[Log[y], $MachinePrecision] * (-y) + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1.1e10 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 87.6%

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

Alternative 8: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e+40) (- (fma (log y) -0.5 x) z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e+40) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e+40)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e+40], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.59999999999999996e40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if 3.59999999999999996e40 < y

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.4

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   4. Applied rewrites 99.4%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   5. Taylor expanded in y around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. cancel-sign-sub N/A

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

      7. log-rec N/A

         \[\leadsto y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right) \cdot y \]

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 75.9

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

   7. Applied rewrites 75.9%

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

Alternative 9: 29.3% accurate, 39.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 24.4

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

5. Applied rewrites 24.4%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * Math.log(y));
}

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- (+ y x) z) (* (+ y 1/2) (log y))))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))