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

Percentage Accurate: 99.9% → 99.9%

Time: 7.9s

Alternatives: 10

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 99.8%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. 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. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-188}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-131} \lor \neg \left(z \leq 1.95 \cdot 10^{-53}\right) \land z \leq 7 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 7.8e-212)
     t_0
     (if (<= z 2.35e-188)
       (* x 0.5)
       (if (or (<= z 2.6e-131) (and (not (<= z 1.95e-53)) (<= z 7e-42)))
         t_0
         (- (* x 0.5) (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 7.8e-212) {
		tmp = t_0;
	} else if (z <= 2.35e-188) {
		tmp = x * 0.5;
	} else if ((z <= 2.6e-131) || (!(z <= 1.95e-53) && (z <= 7e-42))) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + log(z))
    if (z <= 7.8d-212) then
        tmp = t_0
    else if (z <= 2.35d-188) then
        tmp = x * 0.5d0
    else if ((z <= 2.6d-131) .or. (.not. (z <= 1.95d-53)) .and. (z <= 7d-42)) then
        tmp = t_0
    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 t_0 = y * (1.0 + Math.log(z));
	double tmp;
	if (z <= 7.8e-212) {
		tmp = t_0;
	} else if (z <= 2.35e-188) {
		tmp = x * 0.5;
	} else if ((z <= 2.6e-131) || (!(z <= 1.95e-53) && (z <= 7e-42))) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	tmp = 0
	if z <= 7.8e-212:
		tmp = t_0
	elif z <= 2.35e-188:
		tmp = x * 0.5
	elif (z <= 2.6e-131) or (not (z <= 1.95e-53) and (z <= 7e-42)):
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 7.8e-212)
		tmp = t_0;
	elseif (z <= 2.35e-188)
		tmp = Float64(x * 0.5);
	elseif ((z <= 2.6e-131) || (!(z <= 1.95e-53) && (z <= 7e-42)))
		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 * (1.0 + log(z));
	tmp = 0.0;
	if (z <= 7.8e-212)
		tmp = t_0;
	elseif (z <= 2.35e-188)
		tmp = x * 0.5;
	elseif ((z <= 2.6e-131) || (~((z <= 1.95e-53)) && (z <= 7e-42)))
		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[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 7.8e-212], t$95$0, If[LessEqual[z, 2.35e-188], N[(x * 0.5), $MachinePrecision], If[Or[LessEqual[z, 2.6e-131], And[N[Not[LessEqual[z, 1.95e-53]], $MachinePrecision], LessEqual[z, 7e-42]]], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 7.8e-212 or 2.34999999999999999e-188 < z < 2.59999999999999996e-131 or 1.9500000000000001e-53 < z < 7.0000000000000004e-42

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 99.6%

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

   3. Taylor expanded in x around 0 68.4%

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

   if 7.8e-212 < z < 2.34999999999999999e-188

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 86.4%

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

   if 2.59999999999999996e-131 < z < 1.9500000000000001e-53 or 7.0000000000000004e-42 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 87.7%

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

      1. associate-*r* 87.7%

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

      2. neg-mul-1 87.7%

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

   4. Simplified 87.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.8 \cdot 10^{-212}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-188}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-131} \lor \neg \left(z \leq 1.95 \cdot 10^{-53}\right) \land z \leq 7 \cdot 10^{-42}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+49} \lor \neg \left(y \leq 8.5 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+49) (not (<= y 8.5e+56)))
   (* y (+ (- 1.0 z) (log z)))
   (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+49) || !(y <= 8.5e+56)) {
		tmp = y * ((1.0 - z) + 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 ((y <= (-1.4d+49)) .or. (.not. (y <= 8.5d+56))) then
        tmp = y * ((1.0d0 - z) + 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 ((y <= -1.4e+49) || !(y <= 8.5e+56)) {
		tmp = y * ((1.0 - z) + Math.log(z));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e+49) or not (y <= 8.5e+56):
		tmp = y * ((1.0 - z) + math.log(z))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+49) || !(y <= 8.5e+56))
		tmp = Float64(y * Float64(Float64(1.0 - z) + 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 ((y <= -1.4e+49) || ~((y <= 8.5e+56)))
		tmp = y * ((1.0 - z) + log(z));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+49], N[Not[LessEqual[y, 8.5e+56]], $MachinePrecision]], N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999e49 or 8.4999999999999998e56 < y

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto \left(x \cdot 0.5 + y\right) + \left(\color{blue}{\left(0 - z\right)} + \log z\right) \cdot y \]

      7. associate-+l- 99.7%

         \[\leadsto \left(x \cdot 0.5 + y\right) + \color{blue}{\left(0 - \left(z - \log z\right)\right)} \cdot y \]

      8. neg-sub0 99.7%

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

      9. distribute-lft-neg-out 99.7%

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

      10. unsub-neg 99.7%

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

      11. fma-def 99.7%

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

      12. *-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 97.4%

         \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z - \log z\right)\right)} \]

   5. Applied egg-rr 97.4%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(z - \log z\right)\right)} \]

   6. Taylor expanded in x around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. cancel-sign-sub-inv 91.0%

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

      3. *-lft-identity 91.0%

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

      4. distribute-rgt-in 91.0%

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

      5. +-commutative 91.0%

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

      6. sub-neg 91.0%

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

      7. +-commutative 91.0%

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

      8. distribute-neg-in 91.0%

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

      9. remove-double-neg 91.0%

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

      10. associate-+r+ 91.0%

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

      11. +-commutative 91.0%

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

      12. sub-neg 91.0%

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

   8. Simplified 91.0%

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

   if -1.3999999999999999e49 < y < 8.4999999999999998e56

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 86.1%

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

      1. associate-*r* 86.1%

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

      2. neg-mul-1 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+49} \lor \neg \left(y \leq 8.5 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 4: 98.8% 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. Taylor expanded in z around 0 99.2%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 97.4%

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

      1. associate-*r* 97.4%

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

      2. neg-mul-1 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 98.3%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* x 0.5) (+ y (* y (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 + (y * 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 + (y * 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 + (y * 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 + (y * 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(y * 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 + (y * 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[(y * 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. Taylor expanded in z around 0 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 97.4%

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

      1. associate-*r* 97.4%

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

      2. neg-mul-1 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 98.3%

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

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 99.8%

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

2. Final simplification 99.8%

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

Alternative 7: 74.4% 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 99.8%

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

2. Taylor expanded in z around inf 72.0%

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

   1. associate-*r* 72.0%

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

   2. neg-mul-1 72.0%

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

4. Simplified 72.0%

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

5. Final simplification 72.0%

   \[\leadsto x \cdot 0.5 - y \cdot z \]

Alternative 8: 59.1% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.5e+47) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e+47) {
		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 <= 2.5d+47) 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 <= 2.5e+47) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.5e+47:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.5e+47)
		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 <= 2.5e+47)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.5e+47], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.50000000000000011e47

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 49.7%

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

   if 2.50000000000000011e47 < z

   1. Initial program 100.0%

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

      1. fma-udef 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. associate-+r+ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in z around inf 76.1%

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

      1. mul-1-neg 76.1%

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

      2. distribute-rgt-neg-in 76.1%

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

   8. Simplified 76.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 9: 40.6% 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. Taylor expanded in x around inf 40.4%

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

3. Final simplification 40.4%

   \[\leadsto x \cdot 0.5 \]

Alternative 10: 1.9% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 y)

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

def code(x, y, z):
	return y

Julia

function code(x, y, z)
	return y
end

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. associate-+l+ 99.8%

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

   3. distribute-rgt-in 99.8%

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

   4. *-lft-identity 99.8%

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

   5. associate-+r+ 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto \left(x \cdot 0.5 + y\right) + \left(\color{blue}{\left(0 - z\right)} + \log z\right) \cdot y \]

   7. associate-+l- 99.8%

      \[\leadsto \left(x \cdot 0.5 + y\right) + \color{blue}{\left(0 - \left(z - \log z\right)\right)} \cdot y \]

   8. neg-sub0 99.8%

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

   9. distribute-lft-neg-out 99.8%

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

   10. unsub-neg 99.8%

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

   11. fma-def 99.8%

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

   12. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 70.9%

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

5. Taylor expanded in y around inf 33.0%

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

6. Taylor expanded in z around 0 2.0%

   \[\leadsto \color{blue}{y} \]

7. Final simplification 2.0%

   \[\leadsto y \]

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 2023290 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

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