Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 12.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. unsub-neg N/A

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

   11. --lowering--.f64 N/A

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

   12. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+29}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y + 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+29)
   (- x z)
   (if (<= x 1.8e+16)
     (- y (fma (log y) (+ y 0.5) z))
     (+ y (fma (log y) (- -0.5 y) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+29) {
		tmp = x - z;
	} else if (x <= 1.8e+16) {
		tmp = y - fma(log(y), (y + 0.5), z);
	} else {
		tmp = y + fma(log(y), (-0.5 - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+29)
		tmp = Float64(x - z);
	elseif (x <= 1.8e+16)
		tmp = Float64(y - fma(log(y), Float64(y + 0.5), z));
	else
		tmp = Float64(y + fma(log(y), Float64(-0.5 - y), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+29], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.8e+16], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.0000000000000005e29

   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 inf

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

   5. Simplified 87.1%

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

   if -9.0000000000000005e29 < x < 1.8e16

   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. --lowering--.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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 98.6

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

   5. Simplified 98.6%

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

   if 1.8e16 < x

   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 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 89.9

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

   5. Simplified 89.9%

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

Alternative 3: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+89}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+97}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e+89)
   (- x z)
   (if (<= z 1.9e+97) (+ y (fma (log y) (- -0.5 y) x)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+89) {
		tmp = x - z;
	} else if (z <= 1.9e+97) {
		tmp = y + fma(log(y), (-0.5 - y), x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e+89)
		tmp = Float64(x - z);
	elseif (z <= 1.9e+97)
		tmp = Float64(y + fma(log(y), Float64(-0.5 - y), x));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.25e+89], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.9e+97], N[(y + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.24999999999999996e89 or 1.90000000000000018e97 < 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 x around inf

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

   5. Simplified 89.8%

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

   if -1.24999999999999996e89 < z < 1.90000000000000018e97

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 94.4

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

   5. Simplified 94.4%

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

Alternative 4: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1620:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1620.0) (- x z) (if (<= z 1.45e-5) (fma (log y) -0.5 x) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1620.0) {
		tmp = x - z;
	} else if (z <= 1.45e-5) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1620.0)
		tmp = Float64(x - z);
	elseif (z <= 1.45e-5)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1620.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.45e-5], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1620 or 1.45e-5 < 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 x around inf

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

   5. Simplified 80.3%

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

   if -1620 < z < 1.45e-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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 67.0

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

   8. Simplified 67.0%

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

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e+50) (- (fma (log y) -0.5 x) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+50) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e+50)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e+50], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000006e50

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 96.6

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

   5. Simplified 96.6%

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

   if 6.20000000000000006e50 < y

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-rec N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 82.1

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

   7. Simplified 82.1%

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

Alternative 6: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.02e+148) (- (fma (log y) -0.5 x) z) (fma (log y) (- -0.5 y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e+148) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), (-0.5 - y), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.02e+148)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = fma(log(y), Float64(-0.5 - y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.02e+148], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e148

   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. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 89.0

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

   5. Simplified 89.0%

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

   if 1.02e148 < 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 85.7

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

   5. Simplified 85.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 74.8

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

   8. Simplified 74.8%

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

Alternative 7: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e+148) (- (fma (log y) -0.5 x) z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e+148) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e+148)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e+148], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000007e148

   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. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 89.0

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

   5. Simplified 89.0%

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

   if 1.50000000000000007e148 < 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. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-rec N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 74.8

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

   7. Simplified 74.8%

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

Alternative 8: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+148}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.22e+148) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.22e+148) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(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 <= 1.22d+148) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.22e+148) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.22e+148:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.22e+148)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.22e+148)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.22e+148], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.22000000000000007e148

   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 inf

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

   5. Simplified 71.6%

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

   if 1.22000000000000007e148 < 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. Step-by-step derivation

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. metadata-eval 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-rec N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 74.8

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

   7. Simplified 74.8%

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

Alternative 9: 49.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+71}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.6e+71) (- 0.0 z) (if (<= z 1.05e+101) x (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+71) {
		tmp = 0.0 - z;
	} else if (z <= 1.05e+101) {
		tmp = x;
	} else {
		tmp = 0.0 - 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 <= (-5.6d+71)) then
        tmp = 0.0d0 - z
    else if (z <= 1.05d+101) then
        tmp = x
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+71) {
		tmp = 0.0 - z;
	} else if (z <= 1.05e+101) {
		tmp = x;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.6e+71:
		tmp = 0.0 - z
	elif z <= 1.05e+101:
		tmp = x
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.6e+71)
		tmp = Float64(0.0 - z);
	elseif (z <= 1.05e+101)
		tmp = x;
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.6e+71)
		tmp = 0.0 - z;
	elseif (z <= 1.05e+101)
		tmp = x;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.6e+71], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 1.05e+101], x, N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.60000000000000004e71 or 1.05e101 < 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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - z} \]

      3. --lowering--.f64 73.1

         \[\leadsto \color{blue}{0 - z} \]

   5. Simplified 73.1%

      \[\leadsto \color{blue}{0 - z} \]

   if -5.60000000000000004e71 < z < 1.05e101

   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 inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 44.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.4% accurate, 29.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - 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 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 60.5%

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

Alternative 11: 31.8% accurate, 118.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 35.0%

   \[\leadsto \color{blue}{x} \]
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 2024195 
(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))