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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-define 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 1.7 \cdot 10^{-247}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-221}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(0.5 - y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))) (t_1 (- (* x 0.5) (* z y))))
   (if (<= z 1.7e-247)
     (+ y (* (log z) y))
     (if (<= z 4.1e-221)
       t_1
       (if (<= z 2.7e-159)
         t_0
         (if (<= z 2.3e-131)
           (* x (- 0.5 (* y (/ z x))))
           (if (<= z 2.3e-17) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double t_1 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 1.7e-247) {
		tmp = y + (log(z) * y);
	} else if (z <= 4.1e-221) {
		tmp = t_1;
	} else if (z <= 2.7e-159) {
		tmp = t_0;
	} else if (z <= 2.3e-131) {
		tmp = x * (0.5 - (y * (z / x)));
	} else if (z <= 2.3e-17) {
		tmp = t_0;
	} 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 * (1.0d0 + log(z))
    t_1 = (x * 0.5d0) - (z * y)
    if (z <= 1.7d-247) then
        tmp = y + (log(z) * y)
    else if (z <= 4.1d-221) then
        tmp = t_1
    else if (z <= 2.7d-159) then
        tmp = t_0
    else if (z <= 2.3d-131) then
        tmp = x * (0.5d0 - (y * (z / x)))
    else if (z <= 2.3d-17) then
        tmp = t_0
    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 * (1.0 + Math.log(z));
	double t_1 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 1.7e-247) {
		tmp = y + (Math.log(z) * y);
	} else if (z <= 4.1e-221) {
		tmp = t_1;
	} else if (z <= 2.7e-159) {
		tmp = t_0;
	} else if (z <= 2.3e-131) {
		tmp = x * (0.5 - (y * (z / x)));
	} else if (z <= 2.3e-17) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	t_1 = (x * 0.5) - (z * y)
	tmp = 0
	if z <= 1.7e-247:
		tmp = y + (math.log(z) * y)
	elif z <= 4.1e-221:
		tmp = t_1
	elif z <= 2.7e-159:
		tmp = t_0
	elif z <= 2.3e-131:
		tmp = x * (0.5 - (y * (z / x)))
	elif z <= 2.3e-17:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	t_1 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (z <= 1.7e-247)
		tmp = Float64(y + Float64(log(z) * y));
	elseif (z <= 4.1e-221)
		tmp = t_1;
	elseif (z <= 2.7e-159)
		tmp = t_0;
	elseif (z <= 2.3e-131)
		tmp = Float64(x * Float64(0.5 - Float64(y * Float64(z / x))));
	elseif (z <= 2.3e-17)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + log(z));
	t_1 = (x * 0.5) - (z * y);
	tmp = 0.0;
	if (z <= 1.7e-247)
		tmp = y + (log(z) * y);
	elseif (z <= 4.1e-221)
		tmp = t_1;
	elseif (z <= 2.7e-159)
		tmp = t_0;
	elseif (z <= 2.3e-131)
		tmp = x * (0.5 - (y * (z / x)));
	elseif (z <= 2.3e-17)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.7e-247], N[(y + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e-221], t$95$1, If[LessEqual[z, 2.7e-159], t$95$0, If[LessEqual[z, 2.3e-131], N[(x * N[(0.5 - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e-17], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < 1.7000000000000001e-247

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. fma-define 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around inf 59.3%

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

      1. associate--l+ 59.3%

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

      2. distribute-lft-in 59.3%

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

      3. *-rgt-identity 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in z around 0 59.3%

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

   if 1.7000000000000001e-247 < z < 4.09999999999999981e-221 or 2.30000000000000009e-17 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. distribute-rgt-neg-in 94.4%

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

   5. Simplified 94.4%

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

   if 4.09999999999999981e-221 < z < 2.7e-159 or 2.30000000000000022e-131 < z < 2.30000000000000009e-17

   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. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 64.9%

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

      1. associate--l+ 64.9%

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

      2. distribute-lft-in 64.8%

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

      3. *-rgt-identity 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in z around 0 64.8%

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

   9. Taylor expanded in y around 0 64.9%

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

   if 2.7e-159 < z < 2.30000000000000022e-131

   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 99.8%

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

      2. associate-+r+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 93.1%

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

      1. associate-/l* 93.2%

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

      2. associate-/l* 93.2%

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

      3. distribute-lft-out 93.7%

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

   7. Simplified 93.7%

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

   8. Taylor expanded in z around inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. associate-/l* 74.7%

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

      3. distribute-rgt-neg-in 74.7%

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

      4. distribute-frac-neg 74.7%

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

   10. Simplified 74.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-247}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-221}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(0.5 - y \cdot \frac{z}{x}\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-50} \lor \neg \left(x \cdot 0.5 \leq 10^{-49}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e-50) (not (<= (* x 0.5) 1e-49)))
   (- (* x 0.5) (* z y))
   (* y (- (+ 1.0 (log z)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-50) || !((x * 0.5) <= 1e-49)) {
		tmp = (x * 0.5) - (z * y);
	} 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) <= (-1d-50)) .or. (.not. ((x * 0.5d0) <= 1d-49))) then
        tmp = (x * 0.5d0) - (z * y)
    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) <= -1e-50) || !((x * 0.5) <= 1e-49)) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y * ((1.0 + Math.log(z)) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -1e-50) or not ((x * 0.5) <= 1e-49):
		tmp = (x * 0.5) - (z * y)
	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) <= -1e-50) || !(Float64(x * 0.5) <= 1e-49))
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	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) <= -1e-50) || ~(((x * 0.5) <= 1e-49)))
		tmp = (x * 0.5) - (z * y);
	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], -1e-50], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 1e-49]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $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)) < -1.00000000000000001e-50 or 9.99999999999999936e-50 < (*.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 86.6%

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

      1. mul-1-neg 86.6%

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

      2. distribute-rgt-neg-in 86.6%

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

   5. Simplified 86.6%

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

   if -1.00000000000000001e-50 < (*.f64 x #s(literal 1/2 binary64)) < 9.99999999999999936e-50

   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 90.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-50} \lor \neg \left(x \cdot 0.5 \leq 10^{-49}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-248} \lor \neg \left(z \leq 4.6 \cdot 10^{-222}\right) \land z \leq 10^{-17}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 5.4e-248) (and (not (<= z 4.6e-222)) (<= z 1e-17)))
   (* y (+ 1.0 (log z)))
   (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 5.4e-248) || (!(z <= 4.6e-222) && (z <= 1e-17))) {
		tmp = y * (1.0 + log(z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	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 <= 5.4d-248) .or. (.not. (z <= 4.6d-222)) .and. (z <= 1d-17)) then
        tmp = y * (1.0d0 + log(z))
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 5.4e-248) || (!(z <= 4.6e-222) && (z <= 1e-17))) {
		tmp = y * (1.0 + Math.log(z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 5.4e-248) or (not (z <= 4.6e-222) and (z <= 1e-17)):
		tmp = y * (1.0 + math.log(z))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 5.4e-248) || (!(z <= 4.6e-222) && (z <= 1e-17)))
		tmp = Float64(y * Float64(1.0 + log(z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 5.4e-248) || (~((z <= 4.6e-222)) && (z <= 1e-17)))
		tmp = y * (1.0 + log(z));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 5.4e-248], And[N[Not[LessEqual[z, 4.6e-222]], $MachinePrecision], LessEqual[z, 1e-17]]], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.4000000000000002e-248 or 4.6000000000000003e-222 < z < 1.00000000000000007e-17

   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. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 59.0%

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

      1. associate--l+ 59.0%

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

      2. distribute-lft-in 58.9%

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

      3. *-rgt-identity 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in z around 0 58.9%

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

   9. Taylor expanded in y around 0 59.0%

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

   if 5.4000000000000002e-248 < z < 4.6000000000000003e-222 or 1.00000000000000007e-17 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 94.4%

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

      1. mul-1-neg 94.4%

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

      2. distribute-rgt-neg-in 94.4%

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

   5. Simplified 94.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-248} \lor \neg \left(z \leq 4.6 \cdot 10^{-222}\right) \land z \leq 10^{-17}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

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) - (z * y);
	}
	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) - (z * y)
    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) - (z * y);
	}
	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) - (z * y)
	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(z * y));
	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) - (z * y);
	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[(z * y), $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 98.8%

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

   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 z around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-in 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 98.8%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + (((1.0d0 - z) + log(z)) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 60.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+28}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 3.1e+28) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.1e+28) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	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 <= 3.1d+28) then
        tmp = x * 0.5d0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.1e+28) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.1e+28:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.1e+28)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.1e+28)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.1e+28], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.1000000000000001e28

   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 47.2%

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

   if 3.1000000000000001e28 < 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. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-define 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 75.5%

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

      1. mul-1-neg 75.5%

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

      2. distribute-rgt-neg-in 75.5%

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

   7. Simplified 75.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{+28}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

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) - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 72.1%

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

   1. mul-1-neg 72.1%

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

   2. distribute-rgt-neg-in 72.1%

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

5. Simplified 72.1%

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

6. Final simplification 72.1%

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

Alternative 9: 40.1% 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 99.8%

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

3. Taylor expanded in x around inf 37.3%

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

4. Final simplification 37.3%

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

Developer target: 99.9% 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 2024097 
(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)))))