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

Percentage Accurate: 99.9% → 99.7%

Time: 12.8s

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. lift--.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\color{blue}{\left(1 - z\right)} + \log z\right) \]

   3. lift-log.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\left(1 - z\right) + \color{blue}{\log z}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} \]

   6. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot \frac{1}{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   8. lift-+.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   9. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(1 - z\right) \cdot y + \log z \cdot y\right)} + x \cdot \frac{1}{2} \]

   10. associate-+l+ N/A

       \[\leadsto \color{blue}{\left(1 - z\right) \cdot y + \left(\log z \cdot y + x \cdot \frac{1}{2}\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(1 - z, y, \log z \cdot y + x \cdot \frac{1}{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{y \cdot \log z} + x \cdot \frac{1}{2}\right) \]

   13. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(1 - z\right) + \log z\right)\\ t_1 := z \cdot \left(-y\right)\\ \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (- 1.0 z) (log z)))) (t_1 (* z (- y))))
   (if (<= t_0 -500000.0) t_1 (if (<= t_0 2e+41) (* x 0.5) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((1.0 - z) + log(z));
	double t_1 = z * -y;
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+41) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * ((1.0d0 - z) + log(z))
    t_1 = z * -y
    if (t_0 <= (-500000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d+41) then
        tmp = x * 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * ((1.0 - z) + Math.log(z));
	double t_1 = z * -y;
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+41) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((1.0 - z) + math.log(z))
	t_1 = z * -y
	tmp = 0
	if t_0 <= -500000.0:
		tmp = t_1
	elif t_0 <= 2e+41:
		tmp = x * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(1.0 - z) + log(z)))
	t_1 = Float64(z * Float64(-y))
	tmp = 0.0
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 2e+41)
		tmp = Float64(x * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((1.0 - z) + log(z));
	t_1 = z * -y;
	tmp = 0.0;
	if (t_0 <= -500000.0)
		tmp = t_1;
	elseif (t_0 <= 2e+41)
		tmp = x * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, -500000.0], t$95$1, If[LessEqual[t$95$0, 2e+41], N[(x * 0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -5e5 or 2.00000000000000001e41 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 62.1

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

   5. Simplified 62.1%

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

   if -5e5 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 2.00000000000000001e41

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 72.8

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

   5. Simplified 72.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\left(1 - z\right) + \log z\right) \leq -500000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;y \cdot \left(\left(1 - z\right) + \log z\right) \leq 2 \cdot 10^{+41}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x 0.5) (* z y))))
   (if (<= (* x 0.5) -2e-138)
     t_0
     (if (<= (* x 0.5) 5e-103) (+ y (* y (- (log z) z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 0.5) - (z * y);
	double tmp;
	if ((x * 0.5) <= -2e-138) {
		tmp = t_0;
	} else if ((x * 0.5) <= 5e-103) {
		tmp = y + (y * (log(z) - z));
	} else {
		tmp = 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 = (x * 0.5d0) - (z * y)
    if ((x * 0.5d0) <= (-2d-138)) then
        tmp = t_0
    else if ((x * 0.5d0) <= 5d-103) then
        tmp = y + (y * (log(z) - z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 0.5) - (z * y);
	double tmp;
	if ((x * 0.5) <= -2e-138) {
		tmp = t_0;
	} else if ((x * 0.5) <= 5e-103) {
		tmp = y + (y * (Math.log(z) - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 0.5) - (z * y)
	tmp = 0
	if (x * 0.5) <= -2e-138:
		tmp = t_0
	elif (x * 0.5) <= 5e-103:
		tmp = y + (y * (math.log(z) - z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (Float64(x * 0.5) <= -2e-138)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 5e-103)
		tmp = Float64(y + Float64(y * Float64(log(z) - z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 0.5) - (z * y);
	tmp = 0.0;
	if ((x * 0.5) <= -2e-138)
		tmp = t_0;
	elseif ((x * 0.5) <= 5e-103)
		tmp = y + (y * (log(z) - z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -2e-138], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-103], N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -2.00000000000000013e-138 or 4.99999999999999966e-103 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      4. lower-neg.f64 86.4

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

   5. Simplified 86.4%

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

   if -2.00000000000000013e-138 < (*.f64 x #s(literal 1/2 binary64)) < 4.99999999999999966e-103

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 93.9

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

   5. Simplified 93.9%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 93.9

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

   7. Applied egg-rr 93.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x 0.5) (* z y))))
   (if (<= (* x 0.5) -2e-138)
     t_0
     (if (<= (* x 0.5) 5e-103) (fma y (- (log z) z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 0.5) - (z * y);
	double tmp;
	if ((x * 0.5) <= -2e-138) {
		tmp = t_0;
	} else if ((x * 0.5) <= 5e-103) {
		tmp = fma(y, (log(z) - z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (Float64(x * 0.5) <= -2e-138)
		tmp = t_0;
	elseif (Float64(x * 0.5) <= 5e-103)
		tmp = fma(y, Float64(log(z) - z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * 0.5), $MachinePrecision], -2e-138], t$95$0, If[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-103], N[(y * N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -2.00000000000000013e-138 or 4.99999999999999966e-103 < (*.f64 x #s(literal 1/2 binary64))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      4. lower-neg.f64 86.4

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

   5. Simplified 86.4%

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

   if -2.00000000000000013e-138 < (*.f64 x #s(literal 1/2 binary64)) < 4.99999999999999966e-103

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-+r+ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z - z, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.8%

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

Alternative 6: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 6 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x 0.5) (* z y))))
   (if (<= z 6e-232) t_0 (if (<= z 9.5e-56) (fma y (log z) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 0.5) - (z * y);
	double tmp;
	if (z <= 6e-232) {
		tmp = t_0;
	} else if (z <= 9.5e-56) {
		tmp = fma(y, log(z), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 0.5) - Float64(z * y))
	tmp = 0.0
	if (z <= 6e-232)
		tmp = t_0;
	elseif (z <= 9.5e-56)
		tmp = fma(y, log(z), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 0.5), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 6e-232], t$95$0, If[LessEqual[z, 9.5e-56], N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < 5.99999999999999979e-232 or 9.4999999999999991e-56 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      4. lower-neg.f64 89.7

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

   5. Simplified 89.7%

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

   if 5.99999999999999979e-232 < z < 9.4999999999999991e-56

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(1 + \log z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(y \cdot 1 + y \cdot \log z\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot x + \left(\color{blue}{y} + y \cdot \log z\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + y\right) + y \cdot \log z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \log z + \left(\frac{1}{2} \cdot x + y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z, \frac{1}{2} \cdot x + y\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z}, \frac{1}{2} \cdot x + y\right) \]

      7. lower-fma.f64 99.6

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 63.3

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

   8. Simplified 63.3%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-232}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(y, \log z, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.56) (fma x 0.5 (fma y (log z) y)) (fma (- 1.0 z) y (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.56) {
		tmp = fma(x, 0.5, fma(y, log(z), y));
	} else {
		tmp = fma((1.0 - z), y, (x * 0.5));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.56)
		tmp = fma(x, 0.5, fma(y, log(z), y));
	else
		tmp = fma(Float64(1.0 - z), y, Float64(x * 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.56], N[(x * 0.5 + N[(y * N[Log[z], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.56000000000000005

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + y \cdot \left(1 + \log z\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(y \cdot 1 + y \cdot \log z\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot x + \left(\color{blue}{y} + y \cdot \log z\right) \]

      3. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + y\right) + y \cdot \log z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \log z + \left(\frac{1}{2} \cdot x + y\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \log z, \frac{1}{2} \cdot x + y\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\log z}, \frac{1}{2} \cdot x + y\right) \]

      7. lower-fma.f64 98.9

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

   5. Simplified 98.9%

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

      1. lift-log.f64 N/A

         \[\leadsto y \cdot \color{blue}{\log z} + \left(\frac{1}{2} \cdot x + y\right) \]

      2. *-commutative N/A

         \[\leadsto y \cdot \log z + \left(\color{blue}{x \cdot \frac{1}{2}} + y\right) \]

      3. lift-*.f64 N/A

         \[\leadsto y \cdot \log z + \left(\color{blue}{x \cdot \frac{1}{2}} + y\right) \]

      4. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(y \cdot \log z + x \cdot \frac{1}{2}\right) + y} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} + y \cdot \log z\right)} + y \]

      6. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2} + \left(y \cdot \log z + y\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + \left(y \cdot \log z + y\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, y \cdot \log z + y\right)} \]

      9. lower-fma.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   if 0.56000000000000005 < z

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + y \cdot \left(\left(1 - z\right) + \log z\right) \]

      2. lift--.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\color{blue}{\left(1 - z\right)} + \log z\right) \]

      3. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\left(1 - z\right) + \color{blue}{\log z}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot \frac{1}{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

      9. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(1 - z\right) \cdot y + \log z \cdot y\right)} + x \cdot \frac{1}{2} \]

      10. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(1 - z\right) \cdot y + \left(\log z \cdot y + x \cdot \frac{1}{2}\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 - z, y, \log z \cdot y + x \cdot \frac{1}{2}\right)} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{y \cdot \log z} + x \cdot \frac{1}{2}\right) \]

      13. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{\frac{1}{2} \cdot x}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 98.6

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

   7. Simplified 98.6%

      \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{0.5 \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.56:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 - z), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + y \cdot \left(\left(1 - z\right) + \log z\right) \]

   2. lift--.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\color{blue}{\left(1 - z\right)} + \log z\right) \]

   3. lift-log.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \left(\left(1 - z\right) + \color{blue}{\log z}\right) \]

   4. lift-+.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} \]

   6. +-commutative N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot \frac{1}{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   8. lift-+.f64 N/A

      \[\leadsto y \cdot \color{blue}{\left(\left(1 - z\right) + \log z\right)} + x \cdot \frac{1}{2} \]

   9. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(1 - z\right) \cdot y + \log z \cdot y\right)} + x \cdot \frac{1}{2} \]

   10. associate-+l+ N/A

       \[\leadsto \color{blue}{\left(1 - z\right) \cdot y + \left(\log z \cdot y + x \cdot \frac{1}{2}\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(1 - z, y, \log z \cdot y + x \cdot \frac{1}{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{y \cdot \log z} + x \cdot \frac{1}{2}\right) \]

   13. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(1 - z, y, \color{blue}{\frac{1}{2} \cdot x}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 73.9

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

7. Simplified 73.9%

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

8. Final simplification 73.9%

   \[\leadsto \mathsf{fma}\left(1 - z, y, x \cdot 0.5\right) \]
9. Add Preprocessing

Alternative 9: 74.5% accurate, 8.6× 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. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto x \cdot \frac{1}{2} + \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

   4. lower-neg.f64 74.9

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

5. Simplified 74.9%

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

6. Final simplification 74.9%

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

Alternative 10: 40.0% accurate, 20.0× 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. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 38.3

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

5. Simplified 38.3%

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

6. Final simplification 38.3%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

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