Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

Alternatives: 12

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] + x), $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. 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. distribute-lft-neg-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e+28)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 5.1e+132)
     (+ (fma (log y) (- -0.5 y) y) x)
     (- (* (- 1.0 (log y)) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e+28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 5.1e+132) {
		tmp = fma(log(y), (-0.5 - y), y) + x;
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e+28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 5.1e+132)
		tmp = Float64(fma(log(y), Float64(-0.5 - y), y) + x);
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e+28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.1e+132], N[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7999999999999999e28

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.7999999999999999e28 < y < 5.1000000000000001e132

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   7. Applied rewrites 89.2%

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

   if 5.1000000000000001e132 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e+28)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 5.1e+132)
     (fma (- y) (log y) (+ y x))
     (- (* (- 1.0 (log y)) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e+28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 5.1e+132) {
		tmp = fma(-y, log(y), (y + x));
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e+28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 5.1e+132)
		tmp = fma(Float64(-y), log(y), Float64(y + x));
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e+28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.1e+132], N[((-y) * N[Log[y], $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7999999999999999e28

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.7999999999999999e28 < y < 5.1000000000000001e132

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 89.2%

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

   if 5.1000000000000001e132 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 4: 65.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\left(1 \cdot x - z\right) + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e-103)
   (fma -0.5 (log y) x)
   (if (<= y 1.7e+30) (+ (- (* 1.0 x) z) y) (* (- 1.0 (log y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e-103) {
		tmp = fma(-0.5, log(y), x);
	} else if (y <= 1.7e+30) {
		tmp = ((1.0 * x) - z) + y;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e-103)
		tmp = fma(-0.5, log(y), x);
	elseif (y <= 1.7e+30)
		tmp = Float64(Float64(Float64(1.0 * x) - z) + y);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e-103], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 1.7e+30], N[(N[(N[(1.0 * x), $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.29999999999999998e-103

   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. 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 72.1

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

   5. Applied rewrites 72.1%

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

   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 72.1%

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

   if 1.29999999999999998e-103 < y < 1.7000000000000001e30

   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

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   10. Step-by-step derivation

   11. Applied rewrites 80.8%

       \[\leadsto \left(1 \cdot x + y\right) - z \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate--l+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower--.f64 80.8

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

   13. Applied rewrites 80.8%

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

   if 1.7000000000000001e30 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\left(1 \cdot x - z\right) + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - \left(\log y \cdot y - y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (fma -0.5 (log y) x) z) (- (- x (- (* (log y) y) y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (x - ((log(y) * y) - y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(x - Float64(Float64(log(y) * y) - y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x - N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.28000000000000003 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (fma -0.5 (log y) x) z) (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.28000000000000003 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 7: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 \cdot x - z\right) + y\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 540000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- (* 1.0 x) z) y)))
   (if (<= z -4.3e+23)
     t_0
     (if (<= z 540000000000.0) (fma -0.5 (log y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 * x) - z) + y;
	double tmp;
	if (z <= -4.3e+23) {
		tmp = t_0;
	} else if (z <= 540000000000.0) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 * x) - z) + y)
	tmp = 0.0
	if (z <= -4.3e+23)
		tmp = t_0;
	elseif (z <= 540000000000.0)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 * x), $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -4.3e+23], t$95$0, If[LessEqual[z, 540000000000.0], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2999999999999999e23 or 5.4e11 < z

   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

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 82.5

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

   8. Applied rewrites 82.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   10. Step-by-step derivation

   11. Applied rewrites 72.0%

       \[\leadsto \left(1 \cdot x + y\right) - z \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate--l+ N/A

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

       5. lower-+.f64 N/A

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

       6. lower--.f64 72.0

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

   13. Applied rewrites 72.0%

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

   if -4.2999999999999999e23 < z < 5.4e11

   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. 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 98.4

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

   5. Applied rewrites 98.4%

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

   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 53.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+23}:\\ \;\;\;\;\left(1 \cdot x - z\right) + y\\ \mathbf{elif}\;z \leq 540000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x - z\right) + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e+28) (- (fma -0.5 (log y) x) z) (fma (- y) (log y) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e+28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(-y, log(y), (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e+28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-y), log(y), Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e+28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[((-y) * N[Log[y], $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999999e28

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.7999999999999999e28 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 87.9%

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

Alternative 9: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, x\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e+28) (- (fma -0.5 (log y) x) z) (+ (fma (log y) (- y) x) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e+28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(log(y), -y, x) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e+28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-y), x) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e+28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + x), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999999e28

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.7999999999999999e28 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 87.8%

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

Alternative 10: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e+30) (- (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 <= 1.7e+30) {
		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 <= 1.7e+30)
		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, 1.7e+30], 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 < 1.7000000000000001e30

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.7000000000000001e30 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 11: 56.8% accurate, 9.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((1.0 * x) - z) + 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 = ((1.0d0 * x) - z) + y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(1.0 * x), $MachinePrecision] - z), $MachinePrecision] + y), $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. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

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

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 85.5

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

5. Applied rewrites 85.5%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. lower-/.f64 N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 84.9

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

8. Applied rewrites 84.9%

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

9. Taylor expanded in x around inf

   \[\leadsto \left(1 \cdot x + y\right) - z \]
10. Step-by-step derivation

11. Applied rewrites 47.1%

    \[\leadsto \left(1 \cdot x + y\right) - z \]
12. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. lower-+.f64 N/A

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

    6. lower--.f64 47.1

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

13. Applied rewrites 47.1%

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

14. Final simplification 47.1%

    \[\leadsto \left(1 \cdot x - z\right) + y \]
15. Add Preprocessing

Alternative 12: 29.8% 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 24.0

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

5. Applied rewrites 24.0%

   \[\leadsto \color{blue}{-z} \]
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 2024270 
(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))