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

Percentage Accurate: 99.9% → 99.9%

Time: 12.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   9. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(\frac{1}{2} \cdot x + y \cdot \left(1 + \log z\right)\right)} \]
6. 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. distribute-rgt-neg-in N/A

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

   5. distribute-lft-in N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-log.f64 99.9

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

7. Simplified 99.9%

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

Alternative 2: 60.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\log z + \left(1 - z\right)\right)\\ t_1 := -y \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (log z) (- 1.0 z)))) (t_1 (- (* y z))))
   (if (<= t_0 -1e+189) t_1 (if (<= t_0 4e+90) (* 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 <= -1e+189) {
		tmp = t_1;
	} else if (t_0 <= 4e+90) {
		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 <= (-1d+189)) then
        tmp = t_1
    else if (t_0 <= 4d+90) 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 <= -1e+189) {
		tmp = t_1;
	} else if (t_0 <= 4e+90) {
		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 <= -1e+189:
		tmp = t_1
	elif t_0 <= 4e+90:
		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 <= -1e+189)
		tmp = t_1;
	elseif (t_0 <= 4e+90)
		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 <= -1e+189)
		tmp = t_1;
	elseif (t_0 <= 4e+90)
		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, -1e+189], t$95$1, If[LessEqual[t$95$0, 4e+90], 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))) < -1e189 or 3.99999999999999987e90 < (*.f64 y (+.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 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 65.7

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

   5. Simplified 65.7%

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

   if -1e189 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 3.99999999999999987e90

   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 64.1

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

   5. Simplified 64.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq -1 \cdot 10^{+189}:\\ \;\;\;\;-y \cdot z\\ \mathbf{elif}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq 4 \cdot 10^{+90}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- (log z) z) y)))
   (if (<= y -1.42e+122) t_0 (if (<= y 2.2e+34) (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 <= -1.42e+122) {
		tmp = t_0;
	} else if (y <= 2.2e+34) {
		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 <= -1.42e+122)
		tmp = t_0;
	elseif (y <= 2.2e+34)
		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, -1.42e+122], t$95$0, If[LessEqual[y, 2.2e+34], N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.42000000000000005e122 or 2.2000000000000002e34 < 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 87.9

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

   5. Simplified 87.9%

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

   if -1.42000000000000005e122 < y < 2.2000000000000002e34

   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. 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. *-commutative N/A

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

      9. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.3

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

   7. Simplified 89.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(0.5 - \frac{y \cdot z}{x}\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5.8e-142)
   (* x (- 0.5 (/ (* y z) x)))
   (if (<= z 2.4e-28) (fma y (log z) y) (fma (- z) y (* 0.5 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.8e-142) {
		tmp = x * (0.5 - ((y * z) / x));
	} else if (z <= 2.4e-28) {
		tmp = fma(y, log(z), y);
	} else {
		tmp = fma(-z, y, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.8e-142)
		tmp = Float64(x * Float64(0.5 - Float64(Float64(y * z) / x)));
	elseif (z <= 2.4e-28)
		tmp = fma(y, log(z), y);
	else
		tmp = fma(Float64(-z), y, Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.8e-142], N[(x * N[(0.5 - N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-28], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision], N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.7999999999999998e-142

   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

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

      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. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 57.8

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 57.9

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

   10. Simplified 57.9%

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

   if 5.7999999999999998e-142 < z < 2.4000000000000002e-28

   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 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 64.3

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

   5. Simplified 64.3%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 64.3

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

   8. Simplified 64.3%

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

   if 2.4000000000000002e-28 < z

   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. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.9

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

   7. Simplified 95.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(0.5 - \frac{y \cdot z}{x}\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.275:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.275) (fma y (log z) (fma 0.5 x y)) (fma (- z) y (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.275) {
		tmp = fma(y, log(z), fma(0.5, x, y));
	} else {
		tmp = fma(-z, y, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.275)
		tmp = fma(y, log(z), fma(0.5, x, y));
	else
		tmp = fma(Float64(-z), y, Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.275], N[(y * N[Log[z], $MachinePrecision] + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision], N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.27500000000000002

   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. Simplified 99.0%

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

   if 0.27500000000000002 < z

   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. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.1

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

   7. Simplified 98.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.275:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, \mathsf{fma}\left(0.5, x, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% 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

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

   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. *-commutative N/A

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

   9. lower-fma.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 76.6

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

7. Simplified 76.6%

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

8. Final simplification 76.6%

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

Alternative 7: 40.5% 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 40.5

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

5. Simplified 40.5%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
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 2024212 
(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)))))