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

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 8

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-92} \lor \neg \left(z \leq 2.9 \cdot 10^{-62}\right) \land z \leq 1.18 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.05e-256)
   (* x 0.5)
   (if (or (<= z 8.8e-92) (and (not (<= z 2.9e-62)) (<= z 1.18e-26)))
     (* y (+ 1.0 (log z)))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e-256) {
		tmp = x * 0.5;
	} else if ((z <= 8.8e-92) || (!(z <= 2.9e-62) && (z <= 1.18e-26))) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	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) :: tmp
    if (z <= 1.05d-256) then
        tmp = x * 0.5d0
    else if ((z <= 8.8d-92) .or. (.not. (z <= 2.9d-62)) .and. (z <= 1.18d-26)) then
        tmp = y * (1.0d0 + log(z))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e-256) {
		tmp = x * 0.5;
	} else if ((z <= 8.8e-92) || (!(z <= 2.9e-62) && (z <= 1.18e-26))) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.05e-256:
		tmp = x * 0.5
	elif (z <= 8.8e-92) or (not (z <= 2.9e-62) and (z <= 1.18e-26)):
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.05e-256)
		tmp = Float64(x * 0.5);
	elseif ((z <= 8.8e-92) || (!(z <= 2.9e-62) && (z <= 1.18e-26)))
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.05e-256)
		tmp = x * 0.5;
	elseif ((z <= 8.8e-92) || (~((z <= 2.9e-62)) && (z <= 1.18e-26)))
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.05e-256], N[(x * 0.5), $MachinePrecision], If[Or[LessEqual[z, 8.8e-92], And[N[Not[LessEqual[z, 2.9e-62]], $MachinePrecision], LessEqual[z, 1.18e-26]]], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.05000000000000001e-256

   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 x around inf 70.3%

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

   if 1.05000000000000001e-256 < z < 8.79999999999999949e-92 or 2.89999999999999986e-62 < z < 1.17999999999999996e-26

   1. Initial program 99.7%

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

      1. distribute-lft-in 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0 73.0%

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

   6. Taylor expanded in z around 0 73.0%

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

      1. *-commutative 73.0%

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

      2. distribute-lft1-in 73.2%

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

      3. +-commutative 73.2%

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

   8. Applied egg-rr 73.2%

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

   if 8.79999999999999949e-92 < z < 2.89999999999999986e-62 or 1.17999999999999996e-26 < 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 x around inf 86.3%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 79.9%

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

      2. +-commutative 79.9%

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

      3. associate--l+ 79.9%

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

   5. Simplified 79.9%

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

   6. Taylor expanded in z around inf 81.8%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 81.8%

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

      2. neg-mul-1 81.8%

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

      3. distribute-rgt-neg-in 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in x around 0 93.4%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
   10. Step-by-step derivation

       1. +-commutative 93.4%

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

       2. *-commutative 93.4%

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

       3. mul-1-neg 93.4%

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

       4. unsub-neg 93.4%

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

       5. *-commutative 93.4%

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

   11. Simplified 93.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-256}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-92} \lor \neg \left(z \leq 2.9 \cdot 10^{-62}\right) \land z \leq 1.18 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-34} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -2e-34) (not (<= (* x 0.5) 2e+23)))
   (- (* x 0.5) (* y z))
   (* y (- (+ 1.0 (log z)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e-34) || !((x * 0.5) <= 2e+23)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 + log(z)) - z);
	}
	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) :: tmp
    if (((x * 0.5d0) <= (-2d-34)) .or. (.not. ((x * 0.5d0) <= 2d+23))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * ((1.0d0 + log(z)) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e-34) || !((x * 0.5) <= 2e+23)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 + Math.log(z)) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -2e-34) or not ((x * 0.5) <= 2e+23):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * ((1.0 + math.log(z)) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -2e-34) || !(Float64(x * 0.5) <= 2e+23))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -2e-34) || ~(((x * 0.5) <= 2e+23)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * ((1.0 + log(z)) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -2e-34], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e+23]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -1.99999999999999986e-34 or 1.9999999999999998e23 < (*.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 x around inf 99.2%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 99.1%

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

      2. +-commutative 99.1%

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

      3. associate--l+ 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in z around inf 83.9%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in x around 0 83.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
   10. Step-by-step derivation

       1. +-commutative 83.9%

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

       2. *-commutative 83.9%

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

       3. mul-1-neg 83.9%

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

       4. unsub-neg 83.9%

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

       5. *-commutative 83.9%

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

   11. Simplified 83.9%

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

   if -1.99999999999999986e-34 < (*.f64 x #s(literal 1/2 binary64)) < 1.9999999999999998e23

   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. distribute-lft-in 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 89.6%

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

   6. Taylor expanded in y around 0 89.7%

      \[\leadsto \color{blue}{y \cdot \left(\left(1 + \log z\right) - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-34} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	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) :: tmp
    if (z <= 0.28d0) then
        tmp = (x * 0.5d0) + (y * (1.0d0 + log(z)))
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + Math.log(z)));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.28:
		tmp = (x * 0.5) + (y * (1.0 + math.log(z)))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.28)
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   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 98.0%

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

   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 x around inf 85.7%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 77.8%

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

      2. +-commutative 77.8%

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

      3. associate--l+ 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in z around inf 85.2%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 85.2%

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

      2. neg-mul-1 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
   10. Step-by-step derivation

       1. +-commutative 99.2%

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

       2. *-commutative 99.2%

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

       3. mul-1-neg 99.2%

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

       4. unsub-neg 99.2%

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

       5. *-commutative 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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]

Derivation

1. Initial program 99.9%

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

Alternative 6: 60.1% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 5.5e+22) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.5e+22) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	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) :: tmp
    if (z <= 5.5d+22) then
        tmp = x * 0.5d0
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.5e+22) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5.5e+22:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5.5e+22)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5.5e+22)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5.5e+22], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.50000000000000021e22

   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 inf 41.5%

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

   if 5.50000000000000021e22 < 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 x around inf 85.7%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 77.5%

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

      2. +-commutative 77.5%

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

      3. associate--l+ 77.5%

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

   5. Simplified 77.5%

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

   6. Taylor expanded in z around inf 82.1%

      \[\leadsto x \cdot \left(0.5 + \color{blue}{z \cdot \left(-1 \cdot \frac{y}{x} + \frac{y \cdot \left(1 + -1 \cdot \log \left(\frac{1}{z}\right)\right)}{x \cdot z}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative 82.1%

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

      2. times-frac 74.3%

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

      3. distribute-lft-out 85.6%

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

      4. +-commutative 85.6%

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

      5. mul-1-neg 85.6%

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

      6. log-rec 85.6%

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

      7. remove-double-neg 85.6%

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

      8. *-lft-identity 85.6%

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

      9. *-lft-identity 85.6%

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

      10. +-commutative 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in z around inf 72.9%

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

       1. associate-*r* 72.9%

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

       2. neg-mul-1 72.9%

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

   11. Simplified 72.9%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{+22}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] - 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 x around inf 86.7%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + \frac{y \cdot \left(\left(1 + \log z\right) - z\right)}{x}\right)} \]
4. Step-by-step derivation

   1. associate-/l* 82.8%

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

   2. +-commutative 82.8%

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

   3. associate--l+ 82.8%

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

5. Simplified 82.8%

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

6. Taylor expanded in z around inf 63.4%

   \[\leadsto x \cdot \left(0.5 + \color{blue}{-1 \cdot \frac{y \cdot z}{x}}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 63.4%

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

   2. neg-mul-1 63.4%

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

   3. distribute-rgt-neg-in 63.4%

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

8. Simplified 63.4%

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

9. Taylor expanded in x around 0 70.3%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) + 0.5 \cdot x} \]
10. Step-by-step derivation

    1. +-commutative 70.3%

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

    2. *-commutative 70.3%

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

    3. mul-1-neg 70.3%

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

    4. unsub-neg 70.3%

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

    5. *-commutative 70.3%

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

11. Simplified 70.3%

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

12. Final simplification 70.3%

    \[\leadsto x \cdot 0.5 - y \cdot z \]
13. Add Preprocessing

Alternative 8: 40.2% accurate, 37.0× 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 35.6%

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

4. Final simplification 35.6%

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

Developer target: 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 2024100 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

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