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

Percentage Accurate: 99.9% → 99.9%

Time: 13.0s

Alternatives: 7

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + N[(0.5 * x + 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 x around 0

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

   1. sub-neg N/A

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

   2. associate-+r+ N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 N/A

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

   13. lower-fma.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\log z + \left(1 - z\right)\right)\\ t_1 := -y \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (log z) (- 1.0 z)))) (t_1 (- (* y z))))
   (if (<= t_0 -5e+69) t_1 (if (<= t_0 5e+62) (* 0.5 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * (log(z) + (1.0 - z));
	double t_1 = -(y * z);
	double tmp;
	if (t_0 <= -5e+69) {
		tmp = t_1;
	} else if (t_0 <= 5e+62) {
		tmp = 0.5 * x;
	} 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 = y * (log(z) + (1.0d0 - z))
    t_1 = -(y * z)
    if (t_0 <= (-5d+69)) then
        tmp = t_1
    else if (t_0 <= 5d+62) then
        tmp = 0.5d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (Math.log(z) + (1.0 - z));
	double t_1 = -(y * z);
	double tmp;
	if (t_0 <= -5e+69) {
		tmp = t_1;
	} else if (t_0 <= 5e+62) {
		tmp = 0.5 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (math.log(z) + (1.0 - z))
	t_1 = -(y * z)
	tmp = 0
	if t_0 <= -5e+69:
		tmp = t_1
	elif t_0 <= 5e+62:
		tmp = 0.5 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(log(z) + Float64(1.0 - z)))
	t_1 = Float64(-Float64(y * z))
	tmp = 0.0
	if (t_0 <= -5e+69)
		tmp = t_1;
	elseif (t_0 <= 5e+62)
		tmp = Float64(0.5 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (log(z) + (1.0 - z));
	t_1 = -(y * z);
	tmp = 0.0;
	if (t_0 <= -5e+69)
		tmp = t_1;
	elseif (t_0 <= 5e+62)
		tmp = 0.5 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[Log[z], $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(y * z), $MachinePrecision])}, If[LessEqual[t$95$0, -5e+69], t$95$1, If[LessEqual[t$95$0, 5e+62], N[(0.5 * x), $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))) < -5.00000000000000036e69 or 5.00000000000000029e62 < (*.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. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 62.2

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

   5. Applied rewrites 62.2%

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

   if -5.00000000000000036e69 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 5.00000000000000029e62

   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 x around inf

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

      1. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq -5 \cdot 10^{+69}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq 5 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z + \left(1 - z\right) \leq -2000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z} - y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (log z) (- 1.0 z)) -2000.0)
   (fma z (- (/ y z) y) (* 0.5 x))
   (fma y (log z) (fma 0.5 x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(z) + (1.0 - z)) <= -2000.0) {
		tmp = fma(z, ((y / z) - y), (0.5 * x));
	} else {
		tmp = fma(y, log(z), fma(0.5, x, y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(z) + Float64(1.0 - z)) <= -2000.0)
		tmp = fma(z, Float64(Float64(y / z) - y), Float64(0.5 * x));
	else
		tmp = fma(y, log(z), fma(0.5, x, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[z], $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], -2000.0], N[(z * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(y * N[Log[z], $MachinePrecision] + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -2e3

   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. lift--.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-+l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 97.8

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r/ N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 97.8

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

   10. Applied rewrites 97.8%

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

   if -2e3 < (+.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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \left(1 - z\right) \leq -2000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{z} - y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (log z) (- 1.0 z)) -150.0)
   (fma (- z) y (* 0.5 x))
   (fma y (log z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(z) + (1.0 - z)) <= -150.0) {
		tmp = fma(-z, y, (0.5 * x));
	} else {
		tmp = fma(y, log(z), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(z) + Float64(1.0 - z)) <= -150.0)
		tmp = fma(Float64(-z), y, Float64(0.5 * x));
	else
		tmp = fma(y, log(z), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[z], $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $MachinePrecision], -150.0], N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)) < -150

   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

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

      2. lift-log.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 81.2

         \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   7. Applied rewrites 81.2%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-div N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 81.4

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

   9. Applied rewrites 81.4%

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

   if -150 < (+.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. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{y \cdot \log z + y \cdot 1} \]

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 65.0

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

   8. Applied rewrites 65.0%

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

4. Final simplification 79.8%

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

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- (log z) z) y)))
   (if (<= y -3.5e+87)
     t_0
     (if (<= y 520000000000.0) (fma (- z) y (* 0.5 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (log(z) - z), y);
	double tmp;
	if (y <= -3.5e+87) {
		tmp = t_0;
	} else if (y <= 520000000000.0) {
		tmp = fma(-z, y, (0.5 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(log(z) - z), y)
	tmp = 0.0
	if (y <= -3.5e+87)
		tmp = t_0;
	elseif (y <= 520000000000.0)
		tmp = fma(Float64(-z), y, Float64(0.5 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -3.5e+87], t$95$0, If[LessEqual[y, 520000000000.0], N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999986e87 or 5.2e11 < 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 x around 0

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   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-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.49999999999999986e87 < y < 5.2e11

   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

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

      2. lift-log.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 89.2

         \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

   7. Applied rewrites 89.2%

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-div N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 89.4

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

   9. Applied rewrites 89.4%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{elif}\;y \leq 520000000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-log.f64 N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. flip-+ N/A

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

   9. lift-+.f64 N/A

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

   10. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 76.8

      \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]

7. Applied rewrites 76.8%

   \[\leadsto x \cdot 0.5 + \frac{y}{\color{blue}{\frac{-1}{z}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. remove-double-div N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 76.9

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

9. Applied rewrites 76.9%

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

10. Final simplification 76.9%

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

Alternative 7: 40.6% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-*.f64 38.7

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

5. Applied rewrites 38.7%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. 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 2024219 
(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)))))