Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.0s

Alternatives: 17

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 99.8%

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

   1. 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: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 350:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+197)
     (fma (log y) (- y) y)
     (if (<= t_0 -20000000.0)
       (- x z)
       (if (<= t_0 350.0) (- y (fma (log y) 0.5 z)) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+197) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= -20000000.0) {
		tmp = x - z;
	} else if (t_0 <= 350.0) {
		tmp = y - fma(log(y), 0.5, z);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2e+197)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= -20000000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 350.0)
		tmp = Float64(y - fma(log(y), 0.5, z));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+197], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, -20000000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 350.0], N[(y - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.9999999999999999e197

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 63.0

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

   5. Simplified 63.0%

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

   if -1.9999999999999999e197 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e7 or 350 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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 75.1%

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

   if -2e7 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 350

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --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 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 96.0

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

   8. Simplified 96.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 350:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 350:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+197)
     (fma (log y) (- y) y)
     (if (<= t_0 -20000000.0)
       (- x z)
       (if (<= t_0 350.0) (- (* (log y) -0.5) z) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+197) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= -20000000.0) {
		tmp = x - z;
	} else if (t_0 <= 350.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2e+197)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= -20000000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 350.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+197], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, -20000000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 350.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.9999999999999999e197

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 63.0

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

   5. Simplified 63.0%

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

   if -1.9999999999999999e197 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e7 or 350 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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 75.1%

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

   if -2e7 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 350

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --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 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. log-lowering-log.f64 96.0

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

   8. Simplified 96.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 350:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+197}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 350:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+197)
     (- y (* y (log y)))
     (if (<= t_0 -20000000.0)
       (- x z)
       (if (<= t_0 350.0) (- (* (log y) -0.5) z) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+197) {
		tmp = y - (y * log(y));
	} else if (t_0 <= -20000000.0) {
		tmp = x - z;
	} else if (t_0 <= 350.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - 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) :: t_0
    real(8) :: tmp
    t_0 = y + (x - (log(y) * (y + 0.5d0)))
    if (t_0 <= (-2d+197)) then
        tmp = y - (y * log(y))
    else if (t_0 <= (-20000000.0d0)) then
        tmp = x - z
    else if (t_0 <= 350.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x - (Math.log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2e+197) {
		tmp = y - (y * Math.log(y));
	} else if (t_0 <= -20000000.0) {
		tmp = x - z;
	} else if (t_0 <= 350.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x - (math.log(y) * (y + 0.5)))
	tmp = 0
	if t_0 <= -2e+197:
		tmp = y - (y * math.log(y))
	elif t_0 <= -20000000.0:
		tmp = x - z
	elif t_0 <= 350.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2e+197)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= -20000000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 350.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x - (log(y) * (y + 0.5)));
	tmp = 0.0;
	if (t_0 <= -2e+197)
		tmp = y - (y * log(y));
	elseif (t_0 <= -20000000.0)
		tmp = x - z;
	elseif (t_0 <= 350.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+197], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -20000000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 350.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.9999999999999999e197

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 63.0

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

   5. Simplified 63.0%

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

      1. +-commutative N/A

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

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

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

      3. unsub-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. log-lowering-log.f64 62.9

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

   7. Applied egg-rr 62.9%

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

   if -1.9999999999999999e197 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e7 or 350 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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 75.1%

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

   if -2e7 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 350

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. --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 97.2

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

   5. Simplified 97.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. log-lowering-log.f64 96.0

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

   8. Simplified 96.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+197}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -20000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 350:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y, z\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y (- x (* (log y) (+ y 0.5)))) z)))
   (if (<= t_0 -50000000000.0)
     (- y (fma (log y) y z))
     (if (<= t_0 5e+16) (fma (log y) -0.5 x) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y + (x - (log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = y - fma(log(y), y, z);
	} else if (t_0 <= 5e+16) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(y - fma(log(y), y, z));
	elseif (t_0 <= 5e+16)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[(y - N[(N[Log[y], $MachinePrecision] * y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+16], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -5e10

   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 74.1

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

   5. Simplified 74.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 73.9%

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

   if -5e10 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 5e16

   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

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot \log y} \]
   9. 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 97.8

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

   10. Simplified 97.8%

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

   if 5e16 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq -50000000000:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y, z\right)\\ \mathbf{elif}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z\\ \mathbf{if}\;t\_0 \leq -50000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y (- x (* (log y) (+ y 0.5)))) z)))
   (if (<= t_0 -50000000000.0)
     (- x z)
     (if (<= t_0 500.0) (* (log y) -0.5) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y + (x - (log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = log(y) * -0.5;
	} else {
		tmp = x - 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) :: t_0
    real(8) :: tmp
    t_0 = (y + (x - (log(y) * (y + 0.5d0)))) - z
    if (t_0 <= (-50000000000.0d0)) then
        tmp = x - z
    else if (t_0 <= 500.0d0) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + (x - (Math.log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_0 <= -50000000000.0) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + (x - (math.log(y) * (y + 0.5)))) - z
	tmp = 0
	if t_0 <= -50000000000.0:
		tmp = x - z
	elif t_0 <= 500.0:
		tmp = math.log(y) * -0.5
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
	tmp = 0.0
	if (t_0 <= -50000000000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + (x - (log(y) * (y + 0.5)))) - z;
	tmp = 0.0;
	if (t_0 <= -50000000000.0)
		tmp = x - z;
	elseif (t_0 <= 500.0)
		tmp = log(y) * -0.5;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -50000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -5e10 or 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) 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 x around inf

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

   5. Simplified 66.7%

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

   if -5e10 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. --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 94.5

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

   5. Simplified 94.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. +-commutative N/A

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

      4. sub-neg N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. log-lowering-log.f64 92.3

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

   8. Simplified 92.3%

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

   9. Taylor expanded in z around 0

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

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

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

       2. log-lowering-log.f64 92.3

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

   11. Simplified 92.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq -50000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-131}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+154}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e-131)
   (- (+ x y) z)
   (if (<= y 5.4e-90)
     (fma (log y) -0.5 x)
     (if (<= y 4.2e+154) (- x z) (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e-131) {
		tmp = (x + y) - z;
	} else if (y <= 5.4e-90) {
		tmp = fma(log(y), -0.5, x);
	} else if (y <= 4.2e+154) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e-131)
		tmp = Float64(Float64(x + y) - z);
	elseif (y <= 5.4e-90)
		tmp = fma(log(y), -0.5, x);
	elseif (y <= 4.2e+154)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e-131], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.4e-90], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], If[LessEqual[y, 4.2e+154], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.5999999999999999e-131

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.9%

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

   if 3.5999999999999999e-131 < y < 5.39999999999999993e-90

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot \log y} \]
   9. 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 100.0

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

   10. Simplified 100.0%

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

   if 5.39999999999999993e-90 < y < 4.19999999999999989e154

   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 4.19999999999999989e154 < 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. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 75.4

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

   5. Simplified 75.4%

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

      1. +-commutative N/A

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

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

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

      3. unsub-neg N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. log-lowering-log.f64 75.4

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

   7. Applied egg-rr 75.4%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $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. *-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 99.4

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

   5. Simplified 99.4%

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

   if 0.28000000000000003 < 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

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

      1. log-rec N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. log-lowering-log.f64 99.7

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

   7. Simplified 99.7%

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

Alternative 9: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+19}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 165:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+19) (- x z) (if (<= z 165.0) (fma (log y) -0.5 x) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+19) {
		tmp = x - z;
	} else if (z <= 165.0) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+19)
		tmp = Float64(x - z);
	elseif (z <= 165.0)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+19], N[(x - z), $MachinePrecision], If[LessEqual[z, 165.0], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e19 or 165 < 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 80.1%

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

   if -2.3e19 < z < 165

   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. sub-neg N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. difference-of-squares N/A

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

      5. associate-*l* N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in z around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 99.8

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot \log y} \]
   9. 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 62.3

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

   10. Simplified 62.3%

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

Alternative 10: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $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. *-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 99.4

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

   5. Simplified 99.4%

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

   if 0.28000000000000003 < 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

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

      1. log-rec N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. log-lowering-log.f64 99.7

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

   7. Simplified 99.7%

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

      1. associate-+r- N/A

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

      2. associate--l- N/A

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

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

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

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

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

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

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

      6. log-lowering-log.f64 99.6

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

   9. Applied egg-rr 99.6%

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

Alternative 11: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y + 0.5, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e-6) (- (fma (log y) -0.5 x) z) (- y (fma (log y) (+ y 0.5) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e-6) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(log(y), (y + 0.5), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e-6)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(log(y), Float64(y + 0.5), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e-6], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.1999999999999999e-6

   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 100.0

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

   5. Simplified 100.0%

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

   if 6.1999999999999999e-6 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 83.3

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

   5. Simplified 83.3%

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

Alternative 12: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 42000:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 42000.0) (- (fma (log y) -0.5 x) z) (- (fma (log y) (- y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 42000.0) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), -y, y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 42000.0)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 42000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 42000

   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 99.4

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

   5. Simplified 99.4%

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

   if 42000 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 83.0

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

   5. Simplified 83.0%

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

Alternative 13: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 52000.0) (- (fma (log y) -0.5 x) z) (- (- y (* y (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 52000.0) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = (y - (y * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 52000.0)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 52000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52000

   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 99.4

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

   5. Simplified 99.4%

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

   if 52000 < 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. log-rec N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-out-- N/A

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

      4. *-rgt-identity N/A

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

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

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

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

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

      7. log-lowering-log.f64 82.9

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

   7. Simplified 82.9%

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

Alternative 14: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 52000:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 52000.0) (- (fma (log y) -0.5 x) z) (- y (fma (log y) y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 52000.0) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(log(y), y, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 52000.0)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(log(y), y, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 52000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 52000

   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 99.4

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

   5. Simplified 99.4%

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

   if 52000 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 83.0

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

   5. Simplified 83.0%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 82.9%

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

Alternative 15: 47.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+106}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e+106) x (if (<= x 3.4e-5) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+106) {
		tmp = x;
	} else if (x <= 3.4e-5) {
		tmp = -z;
	} else {
		tmp = x;
	}
	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 (x <= (-1.7d+106)) then
        tmp = x
    else if (x <= 3.4d-5) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+106) {
		tmp = x;
	} else if (x <= 3.4e-5) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.7e+106:
		tmp = x
	elif x <= 3.4e-5:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e+106)
		tmp = x;
	elseif (x <= 3.4e-5)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.7e+106)
		tmp = x;
	elseif (x <= 3.4e-5)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e+106], x, If[LessEqual[x, 3.4e-5], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999997e106 or 3.4e-5 < 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 x around inf

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

   5. Simplified 64.2%

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

   if -1.69999999999999997e106 < x < 3.4e-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 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-lowering-neg.f64 40.3

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

   5. Simplified 40.3%

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

Alternative 16: 58.2% 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 99.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 58.7%

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

Alternative 17: 30.2% 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 99.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 29.9%

   \[\leadsto \color{blue}{x} \]
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 2024198 
(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))