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

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
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. Final simplification 99.9%

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

Alternative 2: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 3.9 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-175}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-86} \lor \neg \left(z \leq 1.7 \cdot 10^{-66}\right) \land z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 3.9e-199)
     t_0
     (if (<= z 9.2e-175)
       (* x 0.5)
       (if (or (<= z 6.6e-86) (and (not (<= z 1.7e-66)) (<= z 2e-37)))
         t_0
         (- (* x 0.5) (* y z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 3.9e-199) {
		tmp = t_0;
	} else if (z <= 9.2e-175) {
		tmp = x * 0.5;
	} else if ((z <= 6.6e-86) || (!(z <= 1.7e-66) && (z <= 2e-37))) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + log(z))
    if (z <= 3.9d-199) then
        tmp = t_0
    else if (z <= 9.2d-175) then
        tmp = x * 0.5d0
    else if ((z <= 6.6d-86) .or. (.not. (z <= 1.7d-66)) .and. (z <= 2d-37)) then
        tmp = t_0
    else
        tmp = (x * 0.5d0) - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + Math.log(z));
	double tmp;
	if (z <= 3.9e-199) {
		tmp = t_0;
	} else if (z <= 9.2e-175) {
		tmp = x * 0.5;
	} else if ((z <= 6.6e-86) || (!(z <= 1.7e-66) && (z <= 2e-37))) {
		tmp = t_0;
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	tmp = 0
	if z <= 3.9e-199:
		tmp = t_0
	elif z <= 9.2e-175:
		tmp = x * 0.5
	elif (z <= 6.6e-86) or (not (z <= 1.7e-66) and (z <= 2e-37)):
		tmp = t_0
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 3.9e-199)
		tmp = t_0;
	elseif (z <= 9.2e-175)
		tmp = Float64(x * 0.5);
	elseif ((z <= 6.6e-86) || (!(z <= 1.7e-66) && (z <= 2e-37)))
		tmp = t_0;
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + log(z));
	tmp = 0.0;
	if (z <= 3.9e-199)
		tmp = t_0;
	elseif (z <= 9.2e-175)
		tmp = x * 0.5;
	elseif ((z <= 6.6e-86) || (~((z <= 1.7e-66)) && (z <= 2e-37)))
		tmp = t_0;
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 3.9e-199], t$95$0, If[LessEqual[z, 9.2e-175], N[(x * 0.5), $MachinePrecision], If[Or[LessEqual[z, 6.6e-86], And[N[Not[LessEqual[z, 1.7e-66]], $MachinePrecision], LessEqual[z, 2e-37]]], t$95$0, N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 3.9000000000000001e-199 or 9.2e-175 < z < 6.59999999999999974e-86 or 1.69999999999999999e-66 < z < 2.00000000000000013e-37

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-rgt-in 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. neg-sub0 99.6%

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

      7. associate-+l- 99.6%

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

      8. neg-sub0 99.6%

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

      9. distribute-lft-neg-out 99.6%

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

      10. unsub-neg 99.6%

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

      11. fma-def 99.6%

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

      12. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around -inf 62.4%

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

   5. Taylor expanded in z around 0 62.4%

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

   if 3.9000000000000001e-199 < z < 9.2e-175

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 80.5%

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

   if 6.59999999999999974e-86 < z < 1.69999999999999999e-66 or 2.00000000000000013e-37 < 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 94.4%

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

      1. associate-*r* 94.4%

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

      2. neg-mul-1 94.4%

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

   4. Simplified 94.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.9 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-175}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-86} \lor \neg \left(z \leq 1.7 \cdot 10^{-66}\right) \land z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 2 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-176}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-66} \lor \neg \left(z \leq 7.2 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))))
   (if (<= z 2e-199)
     t_0
     (if (<= z 6e-176)
       (* x 0.5)
       (if (<= z 8.5e-86)
         t_0
         (if (or (<= z 2.5e-66) (not (<= z 7.2e-39)))
           (- (* x 0.5) (* y z))
           (+ y (* y (log z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double tmp;
	if (z <= 2e-199) {
		tmp = t_0;
	} else if (z <= 6e-176) {
		tmp = x * 0.5;
	} else if (z <= 8.5e-86) {
		tmp = t_0;
	} else if ((z <= 2.5e-66) || !(z <= 7.2e-39)) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + log(z))
    if (z <= 2d-199) then
        tmp = t_0
    else if (z <= 6d-176) then
        tmp = x * 0.5d0
    else if (z <= 8.5d-86) then
        tmp = t_0
    else if ((z <= 2.5d-66) .or. (.not. (z <= 7.2d-39))) 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 t_0 = y * (1.0 + Math.log(z));
	double tmp;
	if (z <= 2e-199) {
		tmp = t_0;
	} else if (z <= 6e-176) {
		tmp = x * 0.5;
	} else if (z <= 8.5e-86) {
		tmp = t_0;
	} else if ((z <= 2.5e-66) || !(z <= 7.2e-39)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 + math.log(z))
	tmp = 0
	if z <= 2e-199:
		tmp = t_0
	elif z <= 6e-176:
		tmp = x * 0.5
	elif z <= 8.5e-86:
		tmp = t_0
	elif (z <= 2.5e-66) or not (z <= 7.2e-39):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y + (y * math.log(z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	tmp = 0.0
	if (z <= 2e-199)
		tmp = t_0;
	elseif (z <= 6e-176)
		tmp = Float64(x * 0.5);
	elseif (z <= 8.5e-86)
		tmp = t_0;
	elseif ((z <= 2.5e-66) || !(z <= 7.2e-39))
		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)
	t_0 = y * (1.0 + log(z));
	tmp = 0.0;
	if (z <= 2e-199)
		tmp = t_0;
	elseif (z <= 6e-176)
		tmp = x * 0.5;
	elseif (z <= 8.5e-86)
		tmp = t_0;
	elseif ((z <= 2.5e-66) || ~((z <= 7.2e-39)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y + (y * log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2e-199], t$95$0, If[LessEqual[z, 6e-176], N[(x * 0.5), $MachinePrecision], If[LessEqual[z, 8.5e-86], t$95$0, If[Or[LessEqual[z, 2.5e-66], N[Not[LessEqual[z, 7.2e-39]], $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 4 regimes

2. if z < 1.99999999999999996e-199 or 6e-176 < z < 8.499999999999999e-86

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-rgt-in 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+r+ 99.6%

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

      6. neg-sub0 99.6%

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

      7. associate-+l- 99.6%

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

      8. neg-sub0 99.6%

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

      9. distribute-lft-neg-out 99.6%

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

      10. unsub-neg 99.6%

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

      11. fma-def 99.6%

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

      12. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around -inf 61.3%

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

   5. Taylor expanded in z around 0 61.3%

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

   if 1.99999999999999996e-199 < z < 6e-176

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 80.5%

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

   if 8.499999999999999e-86 < z < 2.49999999999999981e-66 or 7.2000000000000001e-39 < 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 94.4%

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

      1. associate-*r* 94.4%

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

      2. neg-mul-1 94.4%

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

   4. Simplified 94.4%

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

   if 2.49999999999999981e-66 < z < 7.2000000000000001e-39

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. distribute-lft-in 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. associate-+l+ 99.7%

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

      7. fma-def 99.8%

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

      8. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{y \cdot \log z + y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-176}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-66} \lor \neg \left(z \leq 7.2 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \log z\\ \end{array} \]

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+23} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{-93}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -2e+23) (not (<= (* x 0.5) 5e-93)))
   (- (* x 0.5) (* y z))
   (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e+23) || !((x * 0.5) <= 5e-93)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 - z) + 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 (((x * 0.5d0) <= (-2d+23)) .or. (.not. ((x * 0.5d0) <= 5d-93))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * ((1.0d0 - z) + log(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e+23) || !((x * 0.5) <= 5e-93)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((1.0 - z) + Math.log(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -2e+23) or not ((x * 0.5) <= 5e-93):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * ((1.0 - z) + math.log(z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -2e+23) || !(Float64(x * 0.5) <= 5e-93))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(Float64(1.0 - z) + log(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -2e+23) || ~(((x * 0.5) <= 5e-93)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * ((1.0 - z) + log(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -2e+23], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-93]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -1.9999999999999998e23 or 4.99999999999999994e-93 < (*.f64 x 1/2)

   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 84.1%

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

      1. associate-*r* 84.1%

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

      2. neg-mul-1 84.1%

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

   4. Simplified 84.1%

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

   if -1.9999999999999998e23 < (*.f64 x 1/2) < 4.99999999999999994e-93

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-rgt-in 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. distribute-lft-neg-out 99.7%

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

      10. unsub-neg 99.7%

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

      11. fma-def 99.7%

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

      12. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around -inf 90.7%

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

   5. Taylor expanded in z around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. associate-+r+ 90.7%

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

      4. mul-1-neg 90.7%

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

      5. sub-neg 90.7%

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

   7. Simplified 90.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+23} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{-93}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(1 - z\right) + \log z\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, y\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.7e-6)
   (+ (* x 0.5) (* y (+ 1.0 (log z))))
   (- (fma x 0.5 y) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.7e-6) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} else {
		tmp = fma(x, 0.5, y) - (y * z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.7e-6)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = Float64(fma(x, 0.5, y) - Float64(y * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.7e-6], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5 + y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.7000000000000002e-6

   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 99.6%

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

   if 3.7000000000000002e-6 < 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. sub-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+r+ 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. unsub-neg 100.0%

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

      11. fma-def 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, y\right) - y \cdot z\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 7: 75.2% 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 71.7%

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

   1. associate-*r* 71.7%

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

   2. neg-mul-1 71.7%

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

4. Simplified 71.7%

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

5. Final simplification 71.7%

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

Alternative 8: 60.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.8e+14) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.8e+14) {
		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 <= 4.8d+14) 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 <= 4.8e+14) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.8e+14:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.8e+14)
		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 <= 4.8e+14)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.8e+14], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.8e14

   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 47.6%

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

   if 4.8e14 < 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. sub-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. *-lft-identity 100.0%

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

      5. associate-+r+ 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. distribute-lft-neg-out 100.0%

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

      10. unsub-neg 100.0%

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

      11. fma-def 100.0%

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

      12. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around -inf 73.2%

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

   5. Taylor expanded in z around inf 73.2%

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

      1. associate-*r* 73.2%

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

      2. neg-mul-1 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 9: 40.5% 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 39.5%

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

3. Final simplification 39.5%

   \[\leadsto x \cdot 0.5 \]

Alternative 10: 1.8% 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 99.9%

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

   1. sub-neg 99.9%

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

   2. associate-+l+ 99.9%

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

   3. distribute-rgt-in 99.8%

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

   4. *-lft-identity 99.8%

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

   5. associate-+r+ 99.8%

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

   6. neg-sub0 99.8%

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

   7. associate-+l- 99.8%

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

   8. neg-sub0 99.8%

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

   9. distribute-lft-neg-out 99.8%

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

   10. unsub-neg 99.8%

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

   11. fma-def 99.8%

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

   12. *-commutative 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.0%

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

   2. pow3 99.0%

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

5. Applied egg-rr 99.0%

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

6. Taylor expanded in y around inf 2.0%

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

7. Final simplification 2.0%

   \[\leadsto y \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023268 
(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)))))