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

Percentage Accurate: 99.9% → 99.9%

Time: 12.3s

Alternatives: 11

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

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((log(z) + 1.0) - 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 * ((log(z) + 1.0d0) - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. +-commutative N/A

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

   4. lift--.f64 N/A

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

   5. associate-+r- N/A

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

   6. lower--.f64 N/A

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

   7. lower-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\log z + \left(1 - z\right)\right)\\ t_1 := y \cdot \left(-z\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+34}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (log z) (- 1.0 z)))) (t_1 (* y (- z))))
   (if (<= t_0 -1e+144) t_1 (if (<= t_0 1e+34) (* x 0.5) 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+144) {
		tmp = t_1;
	} else if (t_0 <= 1e+34) {
		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 = y * (log(z) + (1.0d0 - z))
    t_1 = y * -z
    if (t_0 <= (-1d+144)) then
        tmp = t_1
    else if (t_0 <= 1d+34) 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 = y * (Math.log(z) + (1.0 - z));
	double t_1 = y * -z;
	double tmp;
	if (t_0 <= -1e+144) {
		tmp = t_1;
	} else if (t_0 <= 1e+34) {
		tmp = x * 0.5;
	} 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+144:
		tmp = t_1
	elif t_0 <= 1e+34:
		tmp = x * 0.5
	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(y * Float64(-z))
	tmp = 0.0
	if (t_0 <= -1e+144)
		tmp = t_1;
	elseif (t_0 <= 1e+34)
		tmp = Float64(x * 0.5);
	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+144)
		tmp = t_1;
	elseif (t_0 <= 1e+34)
		tmp = x * 0.5;
	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+144], t$95$1, If[LessEqual[t$95$0, 1e+34], 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))) < -1.00000000000000002e144 or 9.99999999999999946e33 < (*.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 53.6

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

   5. Simplified 53.6%

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

   if -1.00000000000000002e144 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 9.99999999999999946e33

   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 71.3

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

   5. Simplified 71.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq -1 \cdot 10^{+144}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;y \cdot \left(\log z + \left(1 - z\right)\right) \leq 10^{+34}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-143)
   (fma x 0.5 (* y (- z)))
   (if (<= (* x 0.5) 1e-76) (fma y (- (log z) z) y) (fma (- z) y (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-143) {
		tmp = fma(x, 0.5, (y * -z));
	} else if ((x * 0.5) <= 1e-76) {
		tmp = fma(y, (log(z) - z), y);
	} else {
		tmp = fma(-z, y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * 0.5) <= -5e-143)
		tmp = fma(x, 0.5, Float64(y * Float64(-z)));
	elseif (Float64(x * 0.5) <= 1e-76)
		tmp = fma(y, Float64(log(z) - z), y);
	else
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-143], N[(x * 0.5 + N[(y * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-76], N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -5.0000000000000002e-143

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 86.3

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

   5. Simplified 86.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.3

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

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

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 86.3

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

   7. Applied egg-rr 86.3%

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

   if -5.0000000000000002e-143 < (*.f64 x #s(literal 1/2 binary64)) < 9.99999999999999927e-77

   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 94.7

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

   5. Simplified 94.7%

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

   if 9.99999999999999927e-77 < (*.f64 x #s(literal 1/2 binary64))

   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

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 87.9

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

   5. Simplified 87.9%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 87.9

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, y \cdot \left(-z\right)\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+209)
   (fma y (log z) y)
   (if (<= y 4.4e+250) (fma (- z) y (* x 0.5)) (+ y (* y (log z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+209) {
		tmp = fma(y, log(z), y);
	} else if (y <= 4.4e+250) {
		tmp = fma(-z, y, (x * 0.5));
	} else {
		tmp = y + (y * log(z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+209)
		tmp = fma(y, log(z), y);
	elseif (y <= 4.4e+250)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	else
		tmp = Float64(y + Float64(y * log(z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e+209], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision], If[LessEqual[y, 4.4e+250], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.3999999999999997e209

   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 94.8

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

   5. Simplified 94.8%

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

   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 69.8

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

   8. Simplified 69.8%

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

   if -3.3999999999999997e209 < y < 4.40000000000000029e250

   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

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 82.6

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

   5. Simplified 82.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 82.6

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

   7. Applied egg-rr 82.6%

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

   if 4.40000000000000029e250 < y

   1. Initial program 99.6%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   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 99.6

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

   5. Simplified 99.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in z around 0

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

      1. lower-log.f64 72.6

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

   10. Simplified 72.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{+209}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (log z) y)))
   (if (<= y -3.4e+209) t_0 (if (<= y 4.4e+250) (fma (- z) y (* x 0.5)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, log(z), y);
	double tmp;
	if (y <= -3.4e+209) {
		tmp = t_0;
	} else if (y <= 4.4e+250) {
		tmp = fma(-z, y, (x * 0.5));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, log(z), y)
	tmp = 0.0
	if (y <= -3.4e+209)
		tmp = t_0;
	elseif (y <= 4.4e+250)
		tmp = fma(Float64(-z), y, Float64(x * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -3.4e+209], t$95$0, If[LessEqual[y, 4.4e+250], N[((-z) * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3999999999999997e209 or 4.40000000000000029e250 < y

   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 96.5

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

   5. Simplified 96.5%

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

   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 70.7

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

   8. Simplified 70.7%

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

   if -3.3999999999999997e209 < y < 4.40000000000000029e250

   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

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 82.6

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

   5. Simplified 82.6%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 82.6

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

   7. Applied egg-rr 82.6%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   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

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(\left(1 + \log z\right) - z\right)} \]
   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 99.8

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z - z, \mathsf{fma}\left(0.5, x, y\right)\right)} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 99.2

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

   10. Simplified 99.2%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.3

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

   7. Applied egg-rr 99.3%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Log[z], $MachinePrecision] + N[(1.0 - z), $MachinePrecision]), $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. 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. Final simplification 99.9%

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

Alternative 8: 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. Simplified 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 9: 74.9% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[((-z) * 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. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 76.5

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

5. Simplified 76.5%

   \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 76.5

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

7. Applied egg-rr 76.5%

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

Alternative 10: 74.9% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 76.5

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

5. Simplified 76.5%

   \[\leadsto x \cdot 0.5 + \color{blue}{y \cdot \left(-z\right)} \]
6. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 76.5

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lower-neg.f64 N/A

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

   8. lower-*.f64 76.5

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

7. Applied egg-rr 76.5%

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

8. Final simplification 76.5%

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

Alternative 11: 40.5% 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 x around inf

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

   1. lower-*.f64 43.0

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

5. Simplified 43.0%

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

6. Final simplification 43.0%

   \[\leadsto x \cdot 0.5 \]
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 2024208 
(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)))))