Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (((x + y) - (log(y) * y)) - (0.5 * log(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) - (log(y) * y)) - (0.5d0 * log(y))) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. associate--r+ N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 54.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ 0.5 y) (log y))) y) z)))
   (if (<= t_0 -5.0)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 1e+50) (fma -0.5 (log y) y) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((0.5 + y) * log(y))) + y) - z;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 1e+50) {
		tmp = fma(-0.5, log(y), y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(0.5 + y) * log(y))) + y) - z)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 1e+50)
		tmp = fma(-0.5, log(y), y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e+50], N[(-0.5 * N[Log[y], $MachinePrecision] + y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -5

   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 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -5 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 1.0000000000000001e50

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log y, y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 87.3%

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

   if 1.0000000000000001e50 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   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 48.8

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

   5. Applied rewrites 48.8%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z \leq -5:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z \leq 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5e+49)
   (- (* (- 1.0 (log y)) y) z)
   (if (<= z 7.6e+21)
     (fma (- -0.5 y) (log y) (+ x y))
     (if (<= z 2.9e+109)
       (- (fma -0.5 (log y) x) z)
       (- y (fma (+ 0.5 y) (log y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5e+49) {
		tmp = ((1.0 - log(y)) * y) - z;
	} else if (z <= 7.6e+21) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else if (z <= 2.9e+109) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5e+49)
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	elseif (z <= 7.6e+21)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	elseif (z <= 2.9e+109)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5e+49], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 7.6e+21], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.9e+109], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.5000000000000005e49

   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 y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -6.5000000000000005e49 < z < 7.6e21

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 7.6e21 < z < 2.9e109

   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 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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if 2.9e109 < z

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 89.2

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

   5. Applied rewrites 89.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+49}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+45}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3400000000000:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e+45) (- z) (if (<= z 3400000000000.0) (* -0.5 (log y)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+45) {
		tmp = -z;
	} else if (z <= 3400000000000.0) {
		tmp = -0.5 * log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.2d+45)) then
        tmp = -z
    else if (z <= 3400000000000.0d0) then
        tmp = (-0.5d0) * log(y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e+45) {
		tmp = -z;
	} else if (z <= 3400000000000.0) {
		tmp = -0.5 * Math.log(y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.2e+45:
		tmp = -z
	elif z <= 3400000000000.0:
		tmp = -0.5 * math.log(y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e+45)
		tmp = Float64(-z);
	elseif (z <= 3400000000000.0)
		tmp = Float64(-0.5 * log(y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.2e+45)
		tmp = -z;
	elseif (z <= 3400000000000.0)
		tmp = -0.5 * log(y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e+45], (-z), If[LessEqual[z, 3400000000000.0], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000003e45 or 3.4e12 < 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 63.3

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

   5. Applied rewrites 63.3%

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

   if -3.2000000000000003e45 < z < 3.4e12

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{-1}{2} \cdot \log y \]
   10. Step-by-step derivation

   11. Applied rewrites 26.2%

       \[\leadsto -0.5 \cdot \log y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+55) (- (fma -0.5 (log y) x) z) (- (* (- 1.0 (log y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.2e+55) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.2e+55)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.2e+55], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.19999999999999962e55

   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 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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 8.19999999999999962e55 < 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

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 83.3

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

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\left(1 - \log y\right) \cdot y} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((x - ((0.5 + y) * 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 - ((0.5d0 + y) * log(y))) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.02 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.02e+93) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.02e+93) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.02e+93)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.02e+93], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.01999999999999998e93

   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 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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 2.01999999999999998e93 < 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 69.4

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

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\left(1 - \log y\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+20}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6e+20) (- (* -0.5 (log y)) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+20) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.6d+20) then
        tmp = ((-0.5d0) * log(y)) - z
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+20) {
		tmp = (-0.5 * Math.log(y)) - z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.6e+20:
		tmp = (-0.5 * math.log(y)) - z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.6e+20)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.6e+20)
		tmp = (-0.5 * log(y)) - z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.6e+20], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6e20

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.6%

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

   if 2.6e20 < 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 61.9

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

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\left(1 - \log y\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 30.1% 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 99.8%

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

3. Taylor expanded in z around 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 29.0

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

5. Applied rewrites 29.0%

   \[\leadsto \color{blue}{-z} \]
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 2024240 
(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))