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

Percentage Accurate: 99.9% → 99.7%

Time: 8.4s

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $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 0 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+58} \lor \neg \left(x \cdot 0.5 \leq -1 \cdot 10^{-19} \lor \neg \left(x \cdot 0.5 \leq -2 \cdot 10^{-75}\right) \land x \cdot 0.5 \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -2e+58)
         (not
          (or (<= (* x 0.5) -1e-19)
              (and (not (<= (* x 0.5) -2e-75)) (<= (* x 0.5) 5e+32)))))
   (- (* x 0.5) (* y z))
   (+ y (* y (- (log z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -2e+58) || !(((x * 0.5) <= -1e-19) || (!((x * 0.5) <= -2e-75) && ((x * 0.5) <= 5e+32)))) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * (log(z) - 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+58)) .or. (.not. ((x * 0.5d0) <= (-1d-19)) .or. (.not. ((x * 0.5d0) <= (-2d-75))) .and. ((x * 0.5d0) <= 5d+32))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y + (y * (log(z) - 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+58) || !(((x * 0.5) <= -1e-19) || (!((x * 0.5) <= -2e-75) && ((x * 0.5) <= 5e+32)))) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y + (y * (Math.log(z) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -2e+58) or not (((x * 0.5) <= -1e-19) or (not ((x * 0.5) <= -2e-75) and ((x * 0.5) <= 5e+32))):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y + (y * (math.log(z) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -2e+58) || !((Float64(x * 0.5) <= -1e-19) || (!(Float64(x * 0.5) <= -2e-75) && (Float64(x * 0.5) <= 5e+32))))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -2e+58) || ~((((x * 0.5) <= -1e-19) || (~(((x * 0.5) <= -2e-75)) && ((x * 0.5) <= 5e+32)))))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y + (y * (log(z) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -2e+58], N[Not[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-19], And[N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], -2e-75]], $MachinePrecision], LessEqual[N[(x * 0.5), $MachinePrecision], 5e+32]]]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -1.99999999999999989e58 or -9.9999999999999998e-20 < (*.f64 x 1/2) < -1.9999999999999999e-75 or 4.9999999999999997e32 < (*.f64 x 1/2)

   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 92.6%

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

      1. associate-*r* 92.6%

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

      2. neg-mul-1 92.6%

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

   4. Simplified 92.6%

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

      1. distribute-lft-neg-out 92.6%

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

      2. unsub-neg 92.6%

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

      3. add-sqr-sqrt 39.9%

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

      4. sqrt-prod 79.0%

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

      5. sqr-neg 79.0%

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

      6. sqrt-unprod 42.3%

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

      7. add-sqr-sqrt 70.3%

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

      8. *-commutative 70.3%

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

      9. add-sqr-sqrt 42.3%

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

      10. sqrt-unprod 79.0%

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

      11. sqr-neg 79.0%

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

      12. sqrt-prod 39.9%

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

      13. add-sqr-sqrt 92.6%

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

   6. Applied egg-rr 92.6%

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

   if -1.99999999999999989e58 < (*.f64 x 1/2) < -9.9999999999999998e-20 or -1.9999999999999999e-75 < (*.f64 x 1/2) < 4.9999999999999997e32

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-+l+ 99.7%

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

      3. distribute-rgt-in 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+r+ 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. distribute-lft-neg-out 99.7%

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

      10. unsub-neg 99.7%

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

      11. fma-def 99.7%

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

      12. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 85.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{+58} \lor \neg \left(x \cdot 0.5 \leq -1 \cdot 10^{-19} \lor \neg \left(x \cdot 0.5 \leq -2 \cdot 10^{-75}\right) \land x \cdot 0.5 \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \]

Alternative 3: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - y \cdot z\\ \mathbf{if}\;z \leq 1.15 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (log z)))) (t_1 (- (* x 0.5) (* y z))))
   (if (<= z 1.15e-211)
     t_1
     (if (<= z 2.5e-103)
       t_0
       (if (<= z 8e-50)
         t_1
         (if (<= z 1.35e-13) t_0 (fma y (- z) (* x 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 + log(z));
	double t_1 = (x * 0.5) - (y * z);
	double tmp;
	if (z <= 1.15e-211) {
		tmp = t_1;
	} else if (z <= 2.5e-103) {
		tmp = t_0;
	} else if (z <= 8e-50) {
		tmp = t_1;
	} else if (z <= 1.35e-13) {
		tmp = t_0;
	} else {
		tmp = fma(y, -z, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + log(z)))
	t_1 = Float64(Float64(x * 0.5) - Float64(y * z))
	tmp = 0.0
	if (z <= 1.15e-211)
		tmp = t_1;
	elseif (z <= 2.5e-103)
		tmp = t_0;
	elseif (z <= 8e-50)
		tmp = t_1;
	elseif (z <= 1.35e-13)
		tmp = t_0;
	else
		tmp = fma(y, Float64(-z), Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.15e-211], t$95$1, If[LessEqual[z, 2.5e-103], t$95$0, If[LessEqual[z, 8e-50], t$95$1, If[LessEqual[z, 1.35e-13], t$95$0, N[(y * (-z) + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.14999999999999994e-211 or 2.49999999999999983e-103 < z < 8.00000000000000006e-50

   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 72.3%

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

      1. associate-*r* 72.3%

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

      2. neg-mul-1 72.3%

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

   4. Simplified 72.3%

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

      1. distribute-lft-neg-out 72.3%

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

      2. unsub-neg 72.3%

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

      3. add-sqr-sqrt 36.0%

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

      4. sqrt-prod 70.4%

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

      5. sqr-neg 70.4%

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

      6. sqrt-unprod 35.9%

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

      7. add-sqr-sqrt 71.7%

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

      8. *-commutative 71.7%

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

      9. add-sqr-sqrt 35.9%

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

      10. sqrt-unprod 70.4%

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

      11. sqr-neg 70.4%

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

      12. sqrt-prod 36.0%

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

      13. add-sqr-sqrt 72.3%

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

   6. Applied egg-rr 72.3%

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

   if 1.14999999999999994e-211 < z < 2.49999999999999983e-103 or 8.00000000000000006e-50 < z < 1.35000000000000005e-13

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in x around 0 67.7%

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

   4. Taylor expanded in z around 0 67.6%

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

   if 1.35000000000000005e-13 < z

   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 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 98.6%

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

      1. mul-1-neg 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-211}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-103}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-50}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-211} \lor \neg \left(z \leq 8.4 \cdot 10^{-103}\right) \land \left(z \leq 1.02 \cdot 10^{-50} \lor \neg \left(z \leq 5.7 \cdot 10^{-14}\right)\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 4.6e-211)
         (and (not (<= z 8.4e-103)) (or (<= z 1.02e-50) (not (<= z 5.7e-14)))))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (log z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 4.6e-211) || (!(z <= 8.4e-103) && ((z <= 1.02e-50) || !(z <= 5.7e-14)))) {
		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 <= 4.6d-211) .or. (.not. (z <= 8.4d-103)) .and. (z <= 1.02d-50) .or. (.not. (z <= 5.7d-14))) 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 <= 4.6e-211) || (!(z <= 8.4e-103) && ((z <= 1.02e-50) || !(z <= 5.7e-14)))) {
		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 <= 4.6e-211) or (not (z <= 8.4e-103) and ((z <= 1.02e-50) or not (z <= 5.7e-14))):
		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 <= 4.6e-211) || (!(z <= 8.4e-103) && ((z <= 1.02e-50) || !(z <= 5.7e-14))))
		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 <= 4.6e-211) || (~((z <= 8.4e-103)) && ((z <= 1.02e-50) || ~((z <= 5.7e-14)))))
		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, 4.6e-211], And[N[Not[LessEqual[z, 8.4e-103]], $MachinePrecision], Or[LessEqual[z, 1.02e-50], N[Not[LessEqual[z, 5.7e-14]], $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 < 4.59999999999999976e-211 or 8.40000000000000019e-103 < z < 1.0199999999999999e-50 or 5.69999999999999969e-14 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 89.2%

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

      1. associate-*r* 89.2%

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

      2. neg-mul-1 89.2%

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

   4. Simplified 89.2%

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

      1. distribute-lft-neg-out 89.2%

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

      2. unsub-neg 89.2%

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

      3. add-sqr-sqrt 43.0%

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

      4. sqrt-prod 61.9%

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

      5. sqr-neg 61.9%

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

      6. sqrt-unprod 22.4%

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

      7. add-sqr-sqrt 41.6%

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

      8. *-commutative 41.6%

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

      9. add-sqr-sqrt 22.4%

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

      10. sqrt-unprod 61.9%

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

      11. sqr-neg 61.9%

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

      12. sqrt-prod 43.0%

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

      13. add-sqr-sqrt 89.2%

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

   6. Applied egg-rr 89.2%

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

   if 4.59999999999999976e-211 < z < 8.40000000000000019e-103 or 1.0199999999999999e-50 < z < 5.69999999999999969e-14

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 99.7%

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

   3. Taylor expanded in x around 0 67.7%

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

   4. Taylor expanded in z around 0 67.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.6 \cdot 10^{-211} \lor \neg \left(z \leq 8.4 \cdot 10^{-103}\right) \land \left(z \leq 1.02 \cdot 10^{-50} \lor \neg \left(z \leq 5.7 \cdot 10^{-14}\right)\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0045:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.0045)
   (+ (* 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 <= 0.0045) {
		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 <= 0.0045)
		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, 0.0045], 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 < 0.00449999999999999966

   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 99.4%

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

   if 0.00449999999999999966 < 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 99.3%

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

      1. mul-1-neg 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.0045:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot 0.5\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 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 76.0%

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

   1. associate-*r* 76.0%

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

   2. neg-mul-1 76.0%

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

4. Simplified 76.0%

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

   1. distribute-lft-neg-out 76.0%

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

   2. unsub-neg 76.0%

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

   3. add-sqr-sqrt 36.0%

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

   4. sqrt-prod 54.5%

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

   5. sqr-neg 54.5%

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

   6. sqrt-unprod 21.8%

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

   7. add-sqr-sqrt 39.6%

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

   8. *-commutative 39.6%

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

   9. add-sqr-sqrt 21.8%

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

   10. sqrt-unprod 54.5%

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

   11. sqr-neg 54.5%

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

   12. sqrt-prod 36.0%

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

   13. add-sqr-sqrt 76.0%

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

6. Applied egg-rr 76.0%

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

7. Final simplification 76.0%

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

Alternative 8: 60.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.02e+16) (* x 0.5) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.02e+16) {
		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 <= 1.02d+16) 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 <= 1.02e+16) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.02e+16:
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.02e+16)
		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 <= 1.02e+16)
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.02e+16], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.02e16

   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 55.3%

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

   if 1.02e16 < 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 0 100.0%

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

   3. Taylor expanded in x around 0 78.8%

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

   4. Taylor expanded in z around inf 78.7%

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

      1. associate-*r* 78.7%

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

      2. neg-mul-1 78.7%

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

   6. Simplified 78.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.02 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 9: 40.9% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf 40.5%

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

3. Final simplification 40.5%

   \[\leadsto x \cdot 0.5 \]

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-sqr-sqrt 99.5%

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

   2. pow2 99.5%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - y \cdot \color{blue}{{\left(\sqrt{z - \log z}\right)}^{2}} \]

5. Applied egg-rr 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - y \cdot \color{blue}{{\left(\sqrt{z - \log z}\right)}^{2}} \]
6. Step-by-step derivation

   1. add-cube-cbrt 99.0%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - \color{blue}{\left(\sqrt[3]{y \cdot {\left(\sqrt{z - \log z}\right)}^{2}} \cdot \sqrt[3]{y \cdot {\left(\sqrt{z - \log z}\right)}^{2}}\right) \cdot \sqrt[3]{y \cdot {\left(\sqrt{z - \log z}\right)}^{2}}} \]

   2. pow3 99.0%

      \[\leadsto \mathsf{fma}\left(x, 0.5, y\right) - \color{blue}{{\left(\sqrt[3]{y \cdot {\left(\sqrt{z - \log z}\right)}^{2}}\right)}^{3}} \]

   3. unpow2 99.0%

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

   4. add-sqr-sqrt 99.0%

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

   5. *-commutative 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in y around inf 2.0%

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

9. 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 2023293 
(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)))))