System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log z, y, \mathsf{fma}\left(1 - z, y, x \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (log z) y (fma (- 1.0 z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	return fma(log(z), y, fma((1.0 - z), y, (x * 0.5)));
}

Julia

function code(x, y, z)
	return fma(log(z), y, fma(Float64(1.0 - z), y, Float64(x * 0.5)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 99.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\log z, y, \mathsf{fma}\left(1 - z, y, x \cdot 0.5\right)\right) \]
6. Add Preprocessing

Alternative 2: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z, y, y\right)\\ t_1 := \left(1 - z\right) + \log z\\ \mathbf{if}\;t\_1 \leq -628.12:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \left(-z\right) \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq -417:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -250:\\ \;\;\;\;\left(0.5 - \frac{y}{x}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (log z) y y)) (t_1 (+ (- 1.0 z) (log z))))
   (if (<= t_1 -628.12)
     (fma 0.5 x (* (- z) y))
     (if (<= t_1 -417.0) t_0 (if (<= t_1 -250.0) (* (- 0.5 (/ y x)) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(log(z), y, y);
	double t_1 = (1.0 - z) + log(z);
	double tmp;
	if (t_1 <= -628.12) {
		tmp = fma(0.5, x, (-z * y));
	} else if (t_1 <= -417.0) {
		tmp = t_0;
	} else if (t_1 <= -250.0) {
		tmp = (0.5 - (y / x)) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(log(z), y, y)
	t_1 = Float64(Float64(1.0 - z) + log(z))
	tmp = 0.0
	if (t_1 <= -628.12)
		tmp = fma(0.5, x, Float64(Float64(-z) * y));
	elseif (t_1 <= -417.0)
		tmp = t_0;
	elseif (t_1 <= -250.0)
		tmp = Float64(Float64(0.5 - Float64(y / x)) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -628.12], N[(0.5 * x + N[((-z) * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -417.0], t$95$0, If[LessEqual[t$95$1, -250.0], N[(N[(0.5 - N[(y / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -628.120000000000005

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

      12. distribute-lft-in N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. associate-+r+ N/A

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

      17. mul-1-neg N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, -1 \cdot \left(y \cdot z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 93.0%

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

   if -628.120000000000005 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -417 or -250 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.9%

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

   if -417 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -250

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. flip-+ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, x, \left(1 - z\right) \cdot y\right) \cdot \mathsf{fma}\left(0.5, x, \left(1 - z\right) \cdot y\right) - {\left(\log z \cdot y\right)}^{2}}{\mathsf{fma}\left(0.5, x, \left(1 - z\right) \cdot y\right) - \log z \cdot y}} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(2 \cdot y, \frac{1 - z}{x}, \mathsf{fma}\left(\log z, \frac{y}{x}, 0.5\right)\right) - \frac{\left(1 - z\right) \cdot y}{x}\right) \cdot x} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 74.5%

       \[\leadsto \left(0.5 - \frac{\left(1 - z\right) \cdot y}{x}\right) \cdot x \]

   11. Taylor expanded in z around 0

       \[\leadsto \left(\frac{1}{2} - \frac{y}{x}\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 74.5%

       \[\leadsto \left(0.5 - \frac{y}{x}\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - z\right) + \log z\right) \cdot y\\ t_1 := \left(-z\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ (- 1.0 z) (log z)) y)) (t_1 (* (- z) y)))
   (if (<= t_0 -5e+51) t_1 (if (<= t_0 2e+132) (* x 0.5) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - z) + log(z)) * y;
	double t_1 = -z * y;
	double tmp;
	if (t_0 <= -5e+51) {
		tmp = t_1;
	} else if (t_0 <= 2e+132) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 - z) + log(z)) * y
    t_1 = -z * y
    if (t_0 <= (-5d+51)) then
        tmp = t_1
    else if (t_0 <= 2d+132) then
        tmp = x * 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 - z) + Math.log(z)) * y;
	double t_1 = -z * y;
	double tmp;
	if (t_0 <= -5e+51) {
		tmp = t_1;
	} else if (t_0 <= 2e+132) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 - z) + math.log(z)) * y
	t_1 = -z * y
	tmp = 0
	if t_0 <= -5e+51:
		tmp = t_1
	elif t_0 <= 2e+132:
		tmp = x * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - z) + log(z)) * y)
	t_1 = Float64(Float64(-z) * y)
	tmp = 0.0
	if (t_0 <= -5e+51)
		tmp = t_1;
	elseif (t_0 <= 2e+132)
		tmp = Float64(x * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 - z) + log(z)) * y;
	t_1 = -z * y;
	tmp = 0.0;
	if (t_0 <= -5e+51)
		tmp = t_1;
	elseif (t_0 <= 2e+132)
		tmp = x * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[((-z) * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+51], t$95$1, If[LessEqual[t$95$0, 2e+132], N[(x * 0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -5e51 or 1.99999999999999998e132 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)))

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 59.5

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

   5. Applied rewrites 59.5%

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

   if -5e51 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 1.99999999999999998e132

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 65.6

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   5. Applied rewrites 65.6%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - z\right) + \log z\right) \cdot y \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{elif}\;\left(\left(1 - z\right) + \log z\right) \cdot y \leq 2 \cdot 10^{+132}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - z\right) + \log z \leq -100000000:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(-z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (- 1.0 z) (log z)) -100000000.0)
   (fma 0.5 x (fma (- z) y y))
   (fma 0.5 x (fma (log z) y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) + log(z)) <= -100000000.0) {
		tmp = fma(0.5, x, fma(-z, y, y));
	} else {
		tmp = fma(0.5, x, fma(log(z), y, y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(1.0 - z) + log(z)) <= -100000000.0)
		tmp = fma(0.5, x, fma(Float64(-z), y, y));
	else
		tmp = fma(0.5, x, fma(log(z), y, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], -100000000.0], N[(0.5 * x + N[((-z) * y + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * x + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -1e8

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

      12. distribute-lft-in N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. associate-+r+ N/A

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

      17. mul-1-neg N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{fma}\left(-1 \cdot z, y, y\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

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

   if -1e8 < (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 5: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \left(-z\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log z) z) y y)))
   (if (<= y -3.8e+93) t_0 (if (<= y 3.8e+49) (fma 0.5 x (* (- z) y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((log(z) - z), y, y);
	double tmp;
	if (y <= -3.8e+93) {
		tmp = t_0;
	} else if (y <= 3.8e+49) {
		tmp = fma(0.5, x, (-z * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(log(z) - z), y, y)
	tmp = 0.0
	if (y <= -3.8e+93)
		tmp = t_0;
	elseif (y <= 3.8e+49)
		tmp = fma(0.5, x, Float64(Float64(-z) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[y, -3.8e+93], t$95$0, If[LessEqual[y, 3.8e+49], N[(0.5 * x + N[((-z) * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e93 or 3.7999999999999999e49 < y

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -3.7999999999999998e93 < y < 3.7999999999999999e49

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

      12. distribute-lft-in N/A

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

      13. sub-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. sub-neg N/A

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

      16. associate-+r+ N/A

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

      17. mul-1-neg N/A

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

      18. +-commutative N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, -1 \cdot \left(y \cdot z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 87.0%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(1 + \log z\right) - z, y, x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- (+ 1.0 (log z)) z) y (* x 0.5)))

C

double code(double x, double y, double z) {
	return fma(((1.0 + log(z)) - z), y, (x * 0.5));
}

Julia

function code(x, y, z)
	return fma(Float64(Float64(1.0 + log(z)) - z), y, Float64(x * 0.5))
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 99.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift--.f64 N/A

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

   9. associate-+r- N/A

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

   10. lower--.f64 N/A

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

   11. lower-+.f64 99.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left(1 + \log z\right) - z, y, x \cdot 0.5\right) \]
6. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. associate-+r+ N/A

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

   9. mul-1-neg N/A

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

   10. +-commutative N/A

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

   11. distribute-rgt-in N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 8: 75.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 99.9

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

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

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

   12. distribute-lft-in N/A

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

   13. sub-neg N/A

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

   14. lower-fma.f64 N/A

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

   15. sub-neg N/A

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

   16. associate-+r+ N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, -1 \cdot \left(y \cdot z\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 73.5%

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

Alternative 9: 40.7% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 0.5))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return x * 0.5;
}

Python

def code(x, y, z):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 37.0

      \[\leadsto \color{blue}{0.5 \cdot x} \]

5. Applied rewrites 37.0%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

6. Final simplification 37.0%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (* 1/2 x)) (* y (- z (log z)))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))