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

Percentage Accurate: 99.9% → 99.9%

Time: 8.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-def 99.9%

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

   4. sub-neg 99.9%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. sub-neg 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-237} \lor \neg \left(z \leq 1.55 \cdot 10^{-124}\right) \land z \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.4e-237) (and (not (<= z 1.55e-124)) (<= z 1.9e-8)))
   (* y (+ (log z) 1.0))
   (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.4e-237) || (!(z <= 1.55e-124) && (z <= 1.9e-8))) {
		tmp = y * (log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= 2.4d-237) .or. (.not. (z <= 1.55d-124)) .and. (z <= 1.9d-8)) then
        tmp = y * (log(z) + 1.0d0)
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.4e-237) || (!(z <= 1.55e-124) && (z <= 1.9e-8))) {
		tmp = y * (Math.log(z) + 1.0);
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2.4e-237) or (not (z <= 1.55e-124) and (z <= 1.9e-8)):
		tmp = y * (math.log(z) + 1.0)
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2.4e-237) || (!(z <= 1.55e-124) && (z <= 1.9e-8)))
		tmp = Float64(y * Float64(log(z) + 1.0));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 2.4e-237) || (~((z <= 1.55e-124)) && (z <= 1.9e-8)))
		tmp = y * (log(z) + 1.0);
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2.4e-237], And[N[Not[LessEqual[z, 1.55e-124]], $MachinePrecision], LessEqual[z, 1.9e-8]]], N[(y * N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e-237 or 1.5499999999999999e-124 < z < 1.90000000000000014e-8

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-+l+ 99.6%

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

      3. distribute-lft-in 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. associate-+r+ 99.5%

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

      6. fma-def 99.5%

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

      7. +-commutative 99.5%

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

      8. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around 0 98.9%

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

   5. Taylor expanded in y around inf 64.0%

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

   if 2.4e-237 < z < 1.5499999999999999e-124 or 1.90000000000000014e-8 < 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 0 98.9%

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

   3. Taylor expanded in z around inf 88.5%

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

      1. associate-*r* 88.5%

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

      2. mul-1-neg 88.5%

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

   5. Simplified 88.5%

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

   6. Taylor expanded in x around 0 88.5%

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

      1. fma-def 88.5%

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

      2. mul-1-neg 88.5%

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

      3. fma-neg 88.5%

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

   8. Simplified 88.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-237} \lor \neg \left(z \leq 1.55 \cdot 10^{-124}\right) \land z \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 3: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+65} \lor \neg \left(y \leq 9.2 \cdot 10^{+57}\right):\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.05e+65) (not (<= y 9.2e+57)))
   (+ y (* y (- (log z) z)))
   (- (* x 0.5) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.05e+65) || !(y <= 9.2e+57)) {
		tmp = y + (y * (log(z) - z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.05d+65)) .or. (.not. (y <= 9.2d+57))) then
        tmp = y + (y * (log(z) - z))
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.05e+65) || !(y <= 9.2e+57)) {
		tmp = y + (y * (Math.log(z) - z));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.05e+65) or not (y <= 9.2e+57):
		tmp = y + (y * (math.log(z) - z))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.05e+65) || !(y <= 9.2e+57))
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.05e+65) || ~((y <= 9.2e+57)))
		tmp = y + (y * (log(z) - z));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.05e+65], N[Not[LessEqual[y, 9.2e+57]], $MachinePrecision]], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0500000000000001e65 or 9.1999999999999995e57 < 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-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 93.8%

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

   if -2.0500000000000001e65 < y < 9.1999999999999995e57

   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 0 99.9%

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

   3. Taylor expanded in z around inf 90.9%

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

      1. associate-*r* 90.9%

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

      2. mul-1-neg 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0 90.9%

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

      1. fma-def 90.9%

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

      2. mul-1-neg 90.9%

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

      3. fma-neg 90.9%

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

   8. Simplified 90.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+65} \lor \neg \left(y \leq 9.2 \cdot 10^{+57}\right):\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log z - z\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(1 + t_0\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log z) z)))
   (if (<= y -1.02e+65)
     (* y (+ 1.0 t_0))
     (if (<= y 1.9e+62) (- (* x 0.5) (* z y)) (+ y (* y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = log(z) - z;
	double tmp;
	if (y <= -1.02e+65) {
		tmp = y * (1.0 + t_0);
	} else if (y <= 1.9e+62) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y + (y * t_0);
	}
	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 = log(z) - z
    if (y <= (-1.02d+65)) then
        tmp = y * (1.0d0 + t_0)
    else if (y <= 1.9d+62) then
        tmp = (x * 0.5d0) - (z * y)
    else
        tmp = y + (y * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(z) - z;
	double tmp;
	if (y <= -1.02e+65) {
		tmp = y * (1.0 + t_0);
	} else if (y <= 1.9e+62) {
		tmp = (x * 0.5) - (z * y);
	} else {
		tmp = y + (y * t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(z) - z
	tmp = 0
	if y <= -1.02e+65:
		tmp = y * (1.0 + t_0)
	elif y <= 1.9e+62:
		tmp = (x * 0.5) - (z * y)
	else:
		tmp = y + (y * t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(z) - z)
	tmp = 0.0
	if (y <= -1.02e+65)
		tmp = Float64(y * Float64(1.0 + t_0));
	elseif (y <= 1.9e+62)
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	else
		tmp = Float64(y + Float64(y * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(z) - z;
	tmp = 0.0;
	if (y <= -1.02e+65)
		tmp = y * (1.0 + t_0);
	elseif (y <= 1.9e+62)
		tmp = (x * 0.5) - (z * y);
	else
		tmp = y + (y * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, -1.02e+65], N[(y * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+62], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02000000000000005e65

   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-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around -inf 92.6%

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

      1. mul-1-neg 92.6%

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

      2. distribute-rgt-neg-in 92.6%

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

      3. sub-neg 92.6%

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

      4. mul-1-neg 92.6%

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

      5. sub-neg 92.6%

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

      6. +-commutative 92.6%

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

      7. distribute-neg-in 92.6%

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

      8. remove-double-neg 92.6%

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

      9. sub-neg 92.6%

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

      10. metadata-eval 92.6%

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

      11. +-commutative 92.6%

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

   6. Simplified 92.6%

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

   if -1.02000000000000005e65 < y < 1.89999999999999992e62

   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 0 99.9%

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

   3. Taylor expanded in z around inf 90.9%

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

      1. associate-*r* 90.9%

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

      2. mul-1-neg 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in x around 0 90.9%

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

      1. fma-def 90.9%

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

      2. mul-1-neg 90.9%

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

      3. fma-neg 90.9%

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

   8. Simplified 90.9%

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

   if 1.89999999999999992e62 < 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-lft-in 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 94.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+65}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \]

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.28d0) then
        tmp = (log(z) * y) + (y + (x * 0.5d0))
    else
        tmp = (x * 0.5d0) - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (Math.log(z) * y) + (y + (x * 0.5));
	} else {
		tmp = (x * 0.5) - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.28:
		tmp = (math.log(z) * y) + (y + (x * 0.5))
	else:
		tmp = (x * 0.5) - (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(log(z) * y) + Float64(y + Float64(x * 0.5)));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.28)
		tmp = (log(z) * y) + (y + (x * 0.5));
	else
		tmp = (x * 0.5) - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + N[(y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.7%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. 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-lft-in 99.6%

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

      4. *-rgt-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 98.8%

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

   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 0 98.5%

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

   3. Taylor expanded in z around inf 99.6%

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

      1. associate-*r* 99.6%

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

      2. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 99.6%

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

      1. fma-def 99.6%

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

      2. mul-1-neg 99.6%

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

      3. fma-neg 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.2%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x * 0.5) - (y * ((z + -1.0) - 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 * ((z + (-1.0d0)) - log(z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 75.4% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in z around 0 99.1%

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

3. Taylor expanded in z around inf 75.7%

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

   1. associate-*r* 75.7%

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

   2. mul-1-neg 75.7%

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

5. Simplified 75.7%

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

6. Taylor expanded in x around 0 75.7%

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

   1. fma-def 75.7%

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

   2. mul-1-neg 75.7%

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

   3. fma-neg 75.7%

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

8. Simplified 75.7%

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

9. Final simplification 75.7%

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

Alternative 8: 61.0% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.8e+26) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e+26) {
		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 <= 1.8d+26) 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 <= 1.8e+26) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.8e+26:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e+26)
		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 <= 1.8e+26)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.8e+26], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000012e26

   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 48.4%

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

   if 1.80000000000000012e26 < 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. *-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 76.8%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

      3. neg-mul-1 76.8%

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

   6. Simplified 76.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 9: 40.7% 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 36.7%

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

3. Final simplification 36.7%

   \[\leadsto x \cdot 0.5 \]

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-lft-in 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. associate-+r+ 99.8%

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

   6. fma-def 99.8%

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

   7. +-commutative 99.8%

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

   8. unsub-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + \color{blue}{\left(\sqrt[3]{y \cdot \left(\log z - z\right)} \cdot \sqrt[3]{y \cdot \left(\log z - z\right)}\right) \cdot \sqrt[3]{y \cdot \left(\log z - z\right)}} \]

   2. pow3 99.0%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \left(\log z - z\right)}\right)}^{3}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \left(\log z - z\right)}\right)}^{3}} \]

6. Taylor expanded in y around inf 1.7%

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

7. Final simplification 1.7%

   \[\leadsto y \]

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 2023252 
(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)))))