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

Percentage Accurate: 99.9% → 99.6%

Time: 11.2s

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(Float64(y * log(Float64(z * exp(1)))) - Float64(y * z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + ((y * log((z * 2.71828182845904523536))) - (y * z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[(y * N[Log[N[(z * E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y * 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. Taylor expanded in z around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. add-log-exp 99.9%

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

   3. exp-sum 99.9%

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

   4. add-exp-log 99.9%

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

5. Applied egg-rr 99.9%

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

   1. exp-1-e 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log \left(z \cdot e\right)\\ \mathbf{if}\;z \leq 4.1 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log (* z E)))))
   (if (<= z 4.1e-188)
     t_0
     (if (<= z 7.2e-168)
       (* x 0.5)
       (if (<= z 8.5e-71) t_0 (fma y (- z) (* x 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log((z * ((double) M_E)));
	double tmp;
	if (z <= 4.1e-188) {
		tmp = t_0;
	} else if (z <= 7.2e-168) {
		tmp = x * 0.5;
	} else if (z <= 8.5e-71) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * log(Float64(z * exp(1))))
	tmp = 0.0
	if (z <= 4.1e-188)
		tmp = t_0;
	elseif (z <= 7.2e-168)
		tmp = Float64(x * 0.5);
	elseif (z <= 8.5e-71)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[N[(z * E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 4.1e-188], t$95$0, If[LessEqual[z, 7.2e-168], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 8.5e-71], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 4.09999999999999982e-188 or 7.1999999999999998e-168 < z < 8.49999999999999988e-71

   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 99.7%

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

      1. +-commutative 99.7%

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

      2. add-log-exp 99.7%

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

      3. exp-sum 99.7%

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

      4. add-exp-log 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. exp-1-e 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 60.6%

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

   if 4.09999999999999982e-188 < z < 7.1999999999999998e-168

   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 95.8%

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

   if 8.49999999999999988e-71 < z

   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. Taylor expanded in z around inf 86.5%

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

      1. mul-1-neg 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{-188}:\\ \;\;\;\;y \cdot \log \left(z \cdot e\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-168}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-71}:\\ \;\;\;\;y \cdot \log \left(z \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log \left(z \cdot e\right)\\ \mathbf{if}\;z \leq 5.5 \cdot 10^{-187}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log (* z E)))))
   (if (<= z 5.5e-187)
     t_0
     (if (<= z 3.3e-167)
       (* x 0.5)
       (if (<= z 1.35e-75) t_0 (- (* x 0.5) (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log((z * ((double) M_E)));
	double tmp;
	if (z <= 5.5e-187) {
		tmp = t_0;
	} else if (z <= 3.3e-167) {
		tmp = x * 0.5;
	} else if (z <= 1.35e-75) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log((z * Math.E));
	double tmp;
	if (z <= 5.5e-187) {
		tmp = t_0;
	} else if (z <= 3.3e-167) {
		tmp = x * 0.5;
	} else if (z <= 1.35e-75) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log((z * math.e))
	tmp = 0
	if z <= 5.5e-187:
		tmp = t_0
	elif z <= 3.3e-167:
		tmp = x * 0.5
	elif z <= 1.35e-75:
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(Float64(z * exp(1))))
	tmp = 0.0
	if (z <= 5.5e-187)
		tmp = t_0;
	elseif (z <= 3.3e-167)
		tmp = Float64(x * 0.5);
	elseif (z <= 1.35e-75)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log((z * 2.71828182845904523536));
	tmp = 0.0;
	if (z <= 5.5e-187)
		tmp = t_0;
	elseif (z <= 3.3e-167)
		tmp = x * 0.5;
	elseif (z <= 1.35e-75)
		tmp = t_0;
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[N[(z * E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 5.5e-187], t$95$0, If[LessEqual[z, 3.3e-167], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 1.35e-75], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.50000000000000033e-187 or 3.29999999999999995e-167 < z < 1.3499999999999999e-75

   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 99.7%

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

      1. +-commutative 99.7%

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

      2. add-log-exp 99.7%

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

      3. exp-sum 99.7%

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

      4. add-exp-log 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. exp-1-e 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 60.6%

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

   if 5.50000000000000033e-187 < z < 3.29999999999999995e-167

   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 95.8%

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

   if 1.3499999999999999e-75 < 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 86.5%

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

      1. associate-*r* 86.5%

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

      2. mul-1-neg 86.5%

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

   5. Simplified 86.5%

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

      1. distribute-lft-neg-out 86.5%

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

      2. unsub-neg 86.5%

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

      3. *-commutative 86.5%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-187}:\\ \;\;\;\;y \cdot \log \left(z \cdot e\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-167}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;y \cdot \log \left(z \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2400 \lor \neg \left(y \leq 6.8 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2400.0) (not (<= y 6.8e+48)))
   (* y (- (+ 1.0 (log z)) z))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2400.0) || !(y <= 6.8e+48)) {
		tmp = y * ((1.0 + log(z)) - 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 ((y <= (-2400.0d0)) .or. (.not. (y <= 6.8d+48))) then
        tmp = y * ((1.0d0 + log(z)) - 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 ((y <= -2400.0) || !(y <= 6.8e+48)) {
		tmp = y * ((1.0 + Math.log(z)) - z);
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2400.0) or not (y <= 6.8e+48):
		tmp = y * ((1.0 + math.log(z)) - z)
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2400.0) || !(y <= 6.8e+48))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - 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 ((y <= -2400.0) || ~((y <= 6.8e+48)))
		tmp = y * ((1.0 + log(z)) - z);
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2400.0], N[Not[LessEqual[y, 6.8e+48]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2400 or 6.8000000000000006e48 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 80.1%

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

      1. associate-/l* 80.0%

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

      2. +-commutative 80.0%

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

      3. associate--l+ 80.0%

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

   5. Simplified 80.0%

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

   6. Taylor expanded in x around 0 85.6%

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

   if -2400 < y < 6.8000000000000006e48

   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 83.0%

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

      1. associate-*r* 83.0%

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

      2. mul-1-neg 83.0%

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

   5. Simplified 83.0%

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

      1. distribute-lft-neg-out 83.0%

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

      2. unsub-neg 83.0%

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

      3. *-commutative 83.0%

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

   7. Applied egg-rr 83.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2400 \lor \neg \left(y \leq 6.8 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.235:\\ \;\;\;\;x \cdot 0.5 + y \cdot \log \left(z \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.235) (+ (* x 0.5) (* y (log (* z E)))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.235) {
		tmp = (x * 0.5) + (y * log((z * ((double) M_E))));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.235)
		tmp = Float64(Float64(x * 0.5) + Float64(y * log(Float64(z * exp(1)))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.235], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[Log[N[(z * E), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.23499999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.5%

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

      1. +-commutative 99.8%

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

      2. add-log-exp 99.8%

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

      3. exp-sum 99.8%

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

      4. add-exp-log 99.8%

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

   5. Applied egg-rr 98.5%

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

      1. exp-1-e 99.8%

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

   7. Simplified 98.5%

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

   if 0.23499999999999999 < z

   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 97.2%

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

      1. mul-1-neg 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.235:\\ \;\;\;\;x \cdot 0.5 + y \cdot \log \left(z \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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. Final simplification 99.9%

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

Alternative 7: 60.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 22000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 22000.0) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 22000.0) {
		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 <= 22000.0d0) 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 <= 22000.0) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 22000.0:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 22000.0)
		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 <= 22000.0)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 22000.0], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 22000

   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 49.4%

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

   if 22000 < 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 0 100.0%

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

   4. Taylor expanded in x around 0 76.3%

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

   5. Taylor expanded in z around inf 73.5%

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

      1. associate-*r* 73.5%

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

      2. mul-1-neg 73.5%

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

   7. Simplified 73.5%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 22000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 15.9× speedup

Math

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

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 z around inf 72.2%

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

   1. associate-*r* 72.2%

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

   2. mul-1-neg 72.2%

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

5. Simplified 72.2%

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

   1. distribute-lft-neg-out 72.2%

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

   2. unsub-neg 72.2%

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

   3. *-commutative 72.2%

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

7. Applied egg-rr 72.2%

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

8. Final simplification 72.2%

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

Alternative 9: 39.9% 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 38.7%

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

4. Final simplification 38.7%

   \[\leadsto x \cdot 0.5 \]
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 2024053 
(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)))))