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, 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. Final simplification 99.9%

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

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+101} \lor \neg \left(y \leq 3.2 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.2e+101) (not (<= y 3.2e-145)))
   (* y (- (+ 1.0 (log z)) z))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.2e+101) || !(y <= 3.2e-145)) {
		tmp = y * ((1.0 + log(z)) - z);
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.2e+101) || !(y <= 3.2e-145))
		tmp = Float64(y * Float64(Float64(1.0 + log(z)) - z));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.2e+101], N[Not[LessEqual[y, 3.2e-145]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000058e101 or 3.20000000000000008e-145 < y

   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. Step-by-step derivation

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.2%

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

      4. associate-+r+ 99.2%

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

      5. *-commutative 99.2%

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

      6. fma-def 99.2%

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

      7. *-commutative 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around inf 87.0%

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

   if -7.20000000000000058e101 < y < 3.20000000000000008e-145

   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-def 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. Taylor expanded in z around inf 88.5%

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

      1. mul-1-neg 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+101} \lor \neg \left(y \leq 3.2 \cdot 10^{-145}\right):\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0038:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0038)
   (+ (* x 0.5) (* y (+ 1.0 (log z))))
   (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0038) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.0038)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.0038], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00379999999999999999

   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 0 99.7%

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

   if 0.00379999999999999999 < 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. Taylor expanded in z around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0038:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0038:\\ \;\;\;\;\left(x \cdot 0.5 + y\right) + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0038) (+ (+ (* x 0.5) y) (* y (log z))) (fma y (- z) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.0038) {
		tmp = ((x * 0.5) + y) + (y * log(z));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.0038)
		tmp = Float64(Float64(Float64(x * 0.5) + y) + Float64(y * log(z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.0038], N[(N[(N[(x * 0.5), $MachinePrecision] + y), $MachinePrecision] + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00379999999999999999

   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. Step-by-step derivation

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-+r+ 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-def 99.9%

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

      7. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 99.8%

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

   if 0.00379999999999999999 < 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. Taylor expanded in z around inf 99.8%

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

      1. mul-1-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0038:\\ \;\;\;\;\left(x \cdot 0.5 + y\right) + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{-268}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.18e-268)
   (- (* x 0.5) (* y z))
   (if (<= z 4.1e-17) (+ y (* y (log z))) (fma y (- z) (* x 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.18e-268) {
		tmp = (x * 0.5) - (y * z);
	} else if (z <= 4.1e-17) {
		tmp = y + (y * log(z));
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.18e-268)
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	elseif (z <= 4.1e-17)
		tmp = Float64(y + Float64(y * log(z)));
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.18e-268], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e-17], N[(y + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.18e-268

   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 71.2%

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

      1. mul-1-neg 71.2%

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

      2. *-commutative 71.2%

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

      3. distribute-rgt-neg-in 71.2%

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

   4. Simplified 71.2%

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

      1. *-commutative 71.2%

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

      2. distribute-rgt-neg-out 71.2%

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

      3. distribute-lft-neg-in 71.2%

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

      4. add-sqr-sqrt 39.8%

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

      5. sqrt-unprod 64.0%

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

      6. sqr-neg 64.0%

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

      7. sqrt-unprod 31.4%

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

      8. add-sqr-sqrt 70.8%

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

      9. cancel-sign-sub-inv 70.8%

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

      10. *-commutative 70.8%

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

      11. *-commutative 70.8%

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

      12. add-sqr-sqrt 31.4%

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

      13. sqrt-unprod 64.0%

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

      14. sqr-neg 64.0%

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

      15. sqrt-unprod 39.8%

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

      16. add-sqr-sqrt 71.2%

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

   6. Applied egg-rr 71.2%

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

   if 1.18e-268 < z < 4.1000000000000001e-17

   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 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 99.8%

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

   7. Taylor expanded in x around 0 63.8%

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

   if 4.1000000000000001e-17 < 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. Taylor expanded in z around inf 99.0%

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

      1. mul-1-neg 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{-268}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-268} \lor \neg \left(z \leq 1.32 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.9e-268) (not (<= z 1.32e-17)))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.9e-268) || !(z <= 1.32e-17)) {
		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 <= 1.9d-268) .or. (.not. (z <= 1.32d-17))) 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 <= 1.9e-268) || !(z <= 1.32e-17)) {
		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 <= 1.9e-268) or not (z <= 1.32e-17):
		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 <= 1.9e-268) || !(z <= 1.32e-17))
		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 <= 1.9e-268) || ~((z <= 1.32e-17)))
		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, 1.9e-268], N[Not[LessEqual[z, 1.32e-17]], $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 < 1.9000000000000001e-268 or 1.3200000000000001e-17 < 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 95.9%

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

      1. mul-1-neg 95.9%

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

      2. *-commutative 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

   4. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. distribute-rgt-neg-out 95.9%

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

      3. distribute-lft-neg-in 95.9%

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

      4. add-sqr-sqrt 42.4%

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

      5. sqrt-unprod 51.6%

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

      6. sqr-neg 51.6%

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

      7. sqrt-unprod 18.3%

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

      8. add-sqr-sqrt 30.4%

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

      9. cancel-sign-sub-inv 30.4%

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

      10. *-commutative 30.4%

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

      11. *-commutative 30.4%

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

      12. add-sqr-sqrt 18.3%

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

      13. sqrt-unprod 51.6%

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

      14. sqr-neg 51.6%

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

      15. sqrt-unprod 42.4%

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

      16. add-sqr-sqrt 95.9%

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

   6. Applied egg-rr 95.9%

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

   if 1.9000000000000001e-268 < z < 1.3200000000000001e-17

   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 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 63.7%

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

   7. Taylor expanded in z around 0 63.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-268} \lor \neg \left(z \leq 1.32 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]

Alternative 7: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{-269} \lor \neg \left(z \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 8.6e-269) (not (<= z 1.5e-16)))
   (- (* x 0.5) (* y z))
   (+ y (* y (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 8.6e-269) || !(z <= 1.5e-16)) {
		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 <= 8.6d-269) .or. (.not. (z <= 1.5d-16))) 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 <= 8.6e-269) || !(z <= 1.5e-16)) {
		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 <= 8.6e-269) or not (z <= 1.5e-16):
		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 <= 8.6e-269) || !(z <= 1.5e-16))
		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 <= 8.6e-269) || ~((z <= 1.5e-16)))
		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, 8.6e-269], N[Not[LessEqual[z, 1.5e-16]], $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 < 8.59999999999999977e-269 or 1.49999999999999997e-16 < 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 95.9%

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

      1. mul-1-neg 95.9%

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

      2. *-commutative 95.9%

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

      3. distribute-rgt-neg-in 95.9%

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

   4. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. distribute-rgt-neg-out 95.9%

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

      3. distribute-lft-neg-in 95.9%

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

      4. add-sqr-sqrt 42.4%

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

      5. sqrt-unprod 51.6%

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

      6. sqr-neg 51.6%

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

      7. sqrt-unprod 18.3%

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

      8. add-sqr-sqrt 30.4%

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

      9. cancel-sign-sub-inv 30.4%

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

      10. *-commutative 30.4%

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

      11. *-commutative 30.4%

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

      12. add-sqr-sqrt 18.3%

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

      13. sqrt-unprod 51.6%

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

      14. sqr-neg 51.6%

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

      15. sqrt-unprod 42.4%

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

      16. add-sqr-sqrt 95.9%

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

   6. Applied egg-rr 95.9%

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

   if 8.59999999999999977e-269 < z < 1.49999999999999997e-16

   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 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-+r+ 99.8%

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

      5. *-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 99.8%

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

   7. Taylor expanded in x around 0 63.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{-269} \lor \neg \left(z \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]

Alternative 8: 74.9% 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. Taylor expanded in z around inf 70.4%

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

   1. mul-1-neg 70.4%

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

   2. *-commutative 70.4%

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

   3. distribute-rgt-neg-in 70.4%

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

4. Simplified 70.4%

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

   1. *-commutative 70.4%

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

   2. distribute-rgt-neg-out 70.4%

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

   3. distribute-lft-neg-in 70.4%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 43.4%

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

   6. sqr-neg 43.4%

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

   7. sqrt-unprod 19.9%

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

   8. add-sqr-sqrt 33.4%

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

   9. cancel-sign-sub-inv 33.4%

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

   10. *-commutative 33.4%

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

   11. *-commutative 33.4%

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

   12. add-sqr-sqrt 19.9%

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

   13. sqrt-unprod 43.4%

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

   14. sqr-neg 43.4%

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

   15. sqrt-unprod 30.8%

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

   16. add-sqr-sqrt 70.4%

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

6. Applied egg-rr 70.4%

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

7. Final simplification 70.4%

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

Alternative 9: 60.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.9e+33) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.9e+33) {
		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 <= 4.9d+33) 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 <= 4.9e+33) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.9e+33:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.9e+33)
		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 <= 4.9e+33)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.9e+33], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.90000000000000014e33

   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 44.6%

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

   if 4.90000000000000014e33 < 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 99.1%

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

      4. associate-+r+ 99.1%

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

      5. *-commutative 99.1%

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

      6. fma-def 99.1%

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

      7. *-commutative 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in z around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. distribute-rgt-neg-out 80.9%

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

   8. Simplified 80.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.9 \cdot 10^{+33}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 10: 40.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf 34.6%

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

3. Final simplification 34.6%

   \[\leadsto x \cdot 0.5 \]

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 2023318 
(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)))))