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

Percentage Accurate: 99.9% → 99.9%

Time: 11.6s

Alternatives: 9

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "-6.490074466829139166635784901133232468314e-227")
  2. reg1 = (bf "2.614136470584805442979593654162904073865e299")
  3. reg2 = (bf "4.573960109005089372837962151146615437812e269")

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;x \cdot 0.5 + \left(\left(1 - y \cdot z\right) + -1\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x 0.5) -5e-136)
   (+ (* x 0.5) (+ (- 1.0 (* y z)) -1.0))
   (if (<= (* x 0.5) 5e-11)
     (* y (- (+ 1.0 (log z)) z))
     (- (* x 0.5) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * 0.5) <= -5e-136) {
		tmp = (x * 0.5) + ((1.0 - (y * z)) + -1.0);
	} else if ((x * 0.5) <= 5e-11) {
		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 ((x * 0.5d0) <= (-5d-136)) then
        tmp = (x * 0.5d0) + ((1.0d0 - (y * z)) + (-1.0d0))
    else if ((x * 0.5d0) <= 5d-11) 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 ((x * 0.5) <= -5e-136) {
		tmp = (x * 0.5) + ((1.0 - (y * z)) + -1.0);
	} else if ((x * 0.5) <= 5e-11) {
		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 (x * 0.5) <= -5e-136:
		tmp = (x * 0.5) + ((1.0 - (y * z)) + -1.0)
	elif (x * 0.5) <= 5e-11:
		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 (Float64(x * 0.5) <= -5e-136)
		tmp = Float64(Float64(x * 0.5) + Float64(Float64(1.0 - Float64(y * z)) + -1.0));
	elseif (Float64(x * 0.5) <= 5e-11)
		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 ((x * 0.5) <= -5e-136)
		tmp = (x * 0.5) + ((1.0 - (y * z)) + -1.0);
	elseif ((x * 0.5) <= 5e-11)
		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[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-136], N[(N[(x * 0.5), $MachinePrecision] + N[(N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-11], 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 3 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -5.0000000000000002e-136

   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 83.7%

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

      1. associate-*r* 83.7%

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

      2. neg-mul-1 83.7%

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

   5. Simplified 83.7%

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

      1. expm1-log1p-u 59.0%

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

      2. expm1-undefine 59.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(-y\right) \cdot z\right)} - 1\right)} \]

      3. *-commutative 59.0%

         \[\leadsto x \cdot 0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{z \cdot \left(-y\right)}\right)} - 1\right) \]

   7. Applied egg-rr 59.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left(z \cdot \left(-y\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. sub-neg 59.0%

         \[\leadsto x \cdot 0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left(z \cdot \left(-y\right)\right)} + \left(-1\right)\right)} \]

      2. log1p-undefine 59.0%

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

      3. rem-exp-log 83.7%

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

      4. *-commutative 83.7%

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

      5. distribute-lft-neg-in 83.7%

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

      6. unsub-neg 83.7%

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

      7. metadata-eval 83.7%

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

   9. Simplified 83.7%

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

   if -5.0000000000000002e-136 < (*.f64 x #s(literal 1/2 binary64)) < 5.00000000000000018e-11

   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 74.1%

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

   4. Taylor expanded in x around 0 90.2%

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

   if 5.00000000000000018e-11 < (*.f64 x #s(literal 1/2 binary64))

   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 93.8%

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

      1. associate-*r* 93.8%

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

      2. neg-mul-1 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in x around 0 93.8%

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

      1. +-commutative 93.8%

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

      2. *-commutative 93.8%

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

      3. neg-mul-1 93.8%

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

      4. unsub-neg 93.8%

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

      5. *-commutative 93.8%

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

   8. Simplified 93.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;x \cdot 0.5 + \left(\left(1 - y \cdot z\right) + -1\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-279} \lor \neg \left(z \leq 1.1 \cdot 10^{-213}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 4.2e-279) (not (<= z 1.1e-213)))
   (- (* x 0.5) (* y z))
   (+ y (* y (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 4.2e-279) || !(z <= 1.1e-213)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * log(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 <= 4.2d-279) .or. (.not. (z <= 1.1d-213))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y + (y * log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 4.2e-279) || !(z <= 1.1e-213)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 4.2e-279) or not (z <= 1.1e-213):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y + (y * math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 4.2e-279) || !(z <= 1.1e-213))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y + Float64(y * log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 4.2e-279) || ~((z <= 1.1e-213)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y + (y * log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 4.2e-279], N[Not[LessEqual[z, 1.1e-213]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.20000000000000011e-279 or 1.10000000000000005e-213 < 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 82.3%

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

      1. associate-*r* 82.3%

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

      2. neg-mul-1 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in x around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. *-commutative 82.3%

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

      3. neg-mul-1 82.3%

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

      4. unsub-neg 82.3%

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

      5. *-commutative 82.3%

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

   8. Simplified 82.3%

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

   if 4.20000000000000011e-279 < z < 1.10000000000000005e-213

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 92.0%

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

   4. Taylor expanded in x around 0 76.7%

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

   5. Taylor expanded in z around 0 76.7%

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

      1. distribute-rgt-in 76.8%

         \[\leadsto \color{blue}{1 \cdot y + \log z \cdot y} \]

      2. *-un-lft-identity 76.8%

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

   7. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{y + \log z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-279} \lor \neg \left(z \leq 1.1 \cdot 10^{-213}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-280} \lor \neg \left(z \leq 3 \cdot 10^{-214}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 3.6e-280) (not (<= z 3e-214)))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 3.6e-280) || !(z <= 3e-214)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + log(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 <= 3.6d-280) .or. (.not. (z <= 3d-214))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * (1.0d0 + log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 3.6e-280) || !(z <= 3e-214)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 3.6e-280) or not (z <= 3e-214):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * (1.0 + math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 3.6e-280) || !(z <= 3e-214))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(1.0 + log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 3.6e-280) || ~((z <= 3e-214)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * (1.0 + log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 3.6e-280], N[Not[LessEqual[z, 3e-214]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999994e-280 or 2.99999999999999994e-214 < 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 82.3%

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

      1. associate-*r* 82.3%

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

      2. neg-mul-1 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in x around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. *-commutative 82.3%

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

      3. neg-mul-1 82.3%

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

      4. unsub-neg 82.3%

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

      5. *-commutative 82.3%

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

   8. Simplified 82.3%

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

   if 3.59999999999999994e-280 < z < 2.99999999999999994e-214

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 92.0%

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

   4. Taylor expanded in x around 0 76.7%

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

   5. Taylor expanded in z around 0 76.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-280} \lor \neg \left(z \leq 3 \cdot 10^{-214}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.033:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.033) (+ (* 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 <= 0.033) {
		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 <= 0.033d0) 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 <= 0.033) {
		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 <= 0.033:
		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 <= 0.033)
		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 <= 0.033)
		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, 0.033], 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 < 0.033000000000000002

   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 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 0.033000000000000002 < 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 99.5%

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

      1. associate-*r* 99.5%

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

      2. neg-mul-1 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. neg-mul-1 99.5%

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

      4. unsub-neg 99.5%

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

      5. *-commutative 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.033:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 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. Add Preprocessing

Alternative 7: 60.6% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.65e+14) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.65e+14) {
		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 <= 2.65d+14) 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 <= 2.65e+14) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.65e+14:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.65e+14)
		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 <= 2.65e+14)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.65e+14], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.65e14

   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 92.3%

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

   4. Taylor expanded in y around 0 54.5%

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

   if 2.65e14 < 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 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. neg-mul-1 100.0%

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

      4. unsub-neg 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 79.2%

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

       1. neg-mul-1 79.2%

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

       2. distribute-rgt-neg-out 79.2%

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

   11. Simplified 79.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% 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 99.9%

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

3. Taylor expanded in z around inf 76.5%

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

   1. associate-*r* 76.5%

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

   2. neg-mul-1 76.5%

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

5. Simplified 76.5%

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

6. Taylor expanded in x around 0 76.5%

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

   1. +-commutative 76.5%

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

   2. *-commutative 76.5%

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

   3. neg-mul-1 76.5%

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

   4. unsub-neg 76.5%

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

   5. *-commutative 76.5%

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

8. Simplified 76.5%

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

9. Final simplification 76.5%

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

Alternative 9: 40.9% 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 89.7%

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

4. Taylor expanded in y around 0 39.8%

   \[\leadsto x \cdot \color{blue}{0.5} \]
5. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (* 1/2 x)) (* y (- z (log z)))))

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