Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -3.6 \cdot 10^{+127}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \mathbf{elif}\;t\_0 \leq 15.5:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y + 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -3.6e+127)
     (+ y (fma (log y) (- y) x))
     (if (<= t_0 15.5)
       (- y (fma (log y) (+ y 0.5) z))
       (- (fma (log y) -0.5 x) z)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -3.6e+127) {
		tmp = y + fma(log(y), -y, x);
	} else if (t_0 <= 15.5) {
		tmp = y - fma(log(y), (y + 0.5), z);
	} else {
		tmp = fma(log(y), -0.5, x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -3.6e+127)
		tmp = Float64(y + fma(log(y), Float64(-y), x));
	elseif (t_0 <= 15.5)
		tmp = Float64(y - fma(log(y), Float64(y + 0.5), z));
	else
		tmp = Float64(fma(log(y), -0.5, x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.6e+127], N[(y + N[(N[Log[y], $MachinePrecision] * (-y) + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 15.5], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - 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) < -3.59999999999999979e127

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -3.59999999999999979e127 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 15.5

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-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. lower-+.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if 15.5 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -3.6 \cdot 10^{+127}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 15.5:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y + 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -3.6 \cdot 10^{+127}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+61}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -3.6e+127)
     (+ y (fma (log y) (- y) x))
     (if (<= t_0 -1e+61) (- y (fma y (log y) z)) (- (fma (log y) -0.5 x) z)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -3.6e+127) {
		tmp = y + fma(log(y), -y, x);
	} else if (t_0 <= -1e+61) {
		tmp = y - fma(y, log(y), z);
	} else {
		tmp = fma(log(y), -0.5, x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -3.6e+127)
		tmp = Float64(y + fma(log(y), Float64(-y), x));
	elseif (t_0 <= -1e+61)
		tmp = Float64(y - fma(y, log(y), z));
	else
		tmp = Float64(fma(log(y), -0.5, x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -3.6e+127], N[(y + N[(N[Log[y], $MachinePrecision] * (-y) + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+61], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - 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) < -3.59999999999999979e127

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if -3.59999999999999979e127 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999949e60

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   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. lower-fma.f64 N/A

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

      12. lower-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. lower-neg.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. log-rec N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 96.5

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

   8. Applied rewrites 96.5%

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

   if -9.99999999999999949e60 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -3.6 \cdot 10^{+127}:\\ \;\;\;\;y + \mathsf{fma}\left(\log y, -y, x\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1 \cdot 10^{+61}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -1000.0)
     (- y (fma y (log y) z))
     (if (<= t_0 1e+29) (- (* (log y) -0.5) z) (fma (log y) -0.5 x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = y - fma(y, log(y), z);
	} else if (t_0 <= 1e+29) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(log(y), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = Float64(y - fma(y, log(y), z));
	elseif (t_0 <= 1e+29)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(log(y), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+29], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e3

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   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. lower-fma.f64 N/A

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

      12. lower-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. lower-neg.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. log-rec N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if -1e3 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 9.99999999999999914e28

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 96.5

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

   8. Applied rewrites 96.5%

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

   if 9.99999999999999914e28 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 81.2

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

   8. Applied rewrites 81.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -1e+150)
     (fma (log y) (- y) y)
     (if (<= t_0 1e+29) (- (* (log y) -0.5) z) (fma (log y) -0.5 x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -1e+150) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= 1e+29) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(log(y), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -1e+150)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= 1e+29)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(log(y), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+150], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, 1e+29], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999981e149

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if -9.99999999999999981e149 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 9.99999999999999914e28

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 77.6

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

   8. Applied rewrites 77.6%

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

   if 9.99999999999999914e28 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 81.2

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

   8. Applied rewrites 81.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+150}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -1e+150)
     (- y (* y (log y)))
     (if (<= t_0 1e+29) (- (* (log y) -0.5) z) (fma (log y) -0.5 x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -1e+150) {
		tmp = y - (y * log(y));
	} else if (t_0 <= 1e+29) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = fma(log(y), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -1e+150)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= 1e+29)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = fma(log(y), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+150], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+29], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999981e149

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-log.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   if -9.99999999999999981e149 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 9.99999999999999914e28

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 77.6

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

   8. Applied rewrites 77.6%

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

   if 9.99999999999999914e28 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 81.2

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

   8. Applied rewrites 81.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1 \cdot 10^{+150}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 10^{+29}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \left(y + 0.5\right) \cdot \log y\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+150}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq 260:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (+ y 0.5) (log y))))))
   (if (<= t_0 -1e+150)
     (- y (* y (log y)))
     (if (<= t_0 260.0) (- z) (fma (log y) -0.5 x)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - ((y + 0.5) * log(y)));
	double tmp;
	if (t_0 <= -1e+150) {
		tmp = y - (y * log(y));
	} else if (t_0 <= 260.0) {
		tmp = -z;
	} else {
		tmp = fma(log(y), -0.5, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(Float64(y + 0.5) * log(y))))
	tmp = 0.0
	if (t_0 <= -1e+150)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= 260.0)
		tmp = Float64(-z);
	else
		tmp = fma(log(y), -0.5, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+150], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 260.0], (-z), N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999981e149

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-log.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

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

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   if -9.99999999999999981e149 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 260

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if 260 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 76.2

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

   8. Applied rewrites 76.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq -1 \cdot 10^{+150}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \left(y + 0.5\right) \cdot \log y\right) \leq 260:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-175}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-298}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= x -4.5e+123)
     t_0
     (if (<= x -7e-175)
       (- z)
       (if (<= x -6e-298) (* (log y) -0.5) (if (<= x 1.4e+50) (- z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (x <= -4.5e+123) {
		tmp = t_0;
	} else if (x <= -7e-175) {
		tmp = -z;
	} else if (x <= -6e-298) {
		tmp = log(y) * -0.5;
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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 = 1.0d0 / (1.0d0 / x)
    if (x <= (-4.5d+123)) then
        tmp = t_0
    else if (x <= (-7d-175)) then
        tmp = -z
    else if (x <= (-6d-298)) then
        tmp = log(y) * (-0.5d0)
    else if (x <= 1.4d+50) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (x <= -4.5e+123) {
		tmp = t_0;
	} else if (x <= -7e-175) {
		tmp = -z;
	} else if (x <= -6e-298) {
		tmp = Math.log(y) * -0.5;
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 / (1.0 / x)
	tmp = 0
	if x <= -4.5e+123:
		tmp = t_0
	elif x <= -7e-175:
		tmp = -z
	elif x <= -6e-298:
		tmp = math.log(y) * -0.5
	elif x <= 1.4e+50:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (x <= -4.5e+123)
		tmp = t_0;
	elseif (x <= -7e-175)
		tmp = Float64(-z);
	elseif (x <= -6e-298)
		tmp = Float64(log(y) * -0.5);
	elseif (x <= 1.4e+50)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 / (1.0 / x);
	tmp = 0.0;
	if (x <= -4.5e+123)
		tmp = t_0;
	elseif (x <= -7e-175)
		tmp = -z;
	elseif (x <= -6e-298)
		tmp = log(y) * -0.5;
	elseif (x <= 1.4e+50)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+123], t$95$0, If[LessEqual[x, -7e-175], (-z), If[LessEqual[x, -6e-298], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 1.4e+50], (-z), t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999983e123 or 1.3999999999999999e50 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 73.0

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 73.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -4.49999999999999983e123 < x < -6.99999999999999997e-175 or -5.9999999999999999e-298 < x < 1.3999999999999999e50

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.0

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

   5. Applied rewrites 43.0%

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

   if -6.99999999999999997e-175 < x < -5.9999999999999999e-298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 83.7

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-log.f64 55.8

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

   11. Applied rewrites 55.8%

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

Alternative 9: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+18}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+18) (- z) (if (<= z 1.26e+83) (fma (log y) -0.5 x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+18) {
		tmp = -z;
	} else if (z <= 1.26e+83) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+18)
		tmp = Float64(-z);
	elseif (z <= 1.26e+83)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+18], (-z), If[LessEqual[z, 1.26e+83], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1e18 or 1.26000000000000001e83 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 61.6

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

   5. Applied rewrites 61.6%

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

   if -1e18 < z < 1.26000000000000001e83

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 60.7

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

   8. Applied rewrites 60.7%

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

Alternative 10: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.7e+122) (- (fma (log y) -0.5 x) z) (- y (fma y (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.7e+122) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(y, log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.7e+122)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(y, log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.7e+122], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e122

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 2.6999999999999998e122 < 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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   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. lower-fma.f64 N/A

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

      12. lower-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. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. log-rec N/A

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-out N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-log.f64 89.7

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

   8. Applied rewrites 89.7%

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

Alternative 11: 47.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= x -4.5e+123) t_0 (if (<= x 1.4e+50) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (x <= -4.5e+123) {
		tmp = t_0;
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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 = 1.0d0 / (1.0d0 / x)
    if (x <= (-4.5d+123)) then
        tmp = t_0
    else if (x <= 1.4d+50) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (x <= -4.5e+123) {
		tmp = t_0;
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 / (1.0 / x)
	tmp = 0
	if x <= -4.5e+123:
		tmp = t_0
	elif x <= 1.4e+50:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (x <= -4.5e+123)
		tmp = t_0;
	elseif (x <= 1.4e+50)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 / (1.0 / x);
	tmp = 0.0;
	if (x <= -4.5e+123)
		tmp = t_0;
	elseif (x <= 1.4e+50)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+123], t$95$0, If[LessEqual[x, 1.4e+50], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.49999999999999983e123 or 1.3999999999999999e50 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 73.0

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 73.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -4.49999999999999983e123 < x < 1.3999999999999999e50

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.3

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

   5. Applied rewrites 39.3%

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

Alternative 12: 30.4% accurate, 39.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.0

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

5. Applied rewrites 30.0%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))