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

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. sub-neg 99.9%

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

   4. associate-+l+ 99.9%

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

   5. +-commutative 99.9%

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

   6. unsub-neg 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-111} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -1e-111) (not (<= (* x 0.5) 2e-85)))
   (- (* x 0.5) (* y z))
   (* y (+ 1.0 (- (log z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -1e-111) || !((x * 0.5) <= 2e-85)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + (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) <= (-1d-111)) .or. (.not. ((x * 0.5d0) <= 2d-85))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * (1.0d0 + (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) <= -1e-111) || !((x * 0.5) <= 2e-85)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * (1.0 + (Math.log(z) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -1e-111) or not ((x * 0.5) <= 2e-85):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * (1.0 + (math.log(z) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -1e-111) || !(Float64(x * 0.5) <= 2e-85))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(1.0 + Float64(log(z) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -1e-111) || ~(((x * 0.5) <= 2e-85)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * (1.0 + (log(z) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -1e-111], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 2e-85]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 1/2 binary64)) < -1.00000000000000009e-111 or 2e-85 < (*.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 87.2%

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

      1. associate-*r* 87.2%

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

      2. mul-1-neg 87.2%

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

   5. Simplified 87.2%

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

      1. fma-define 87.2%

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

      2. distribute-lft-neg-out 87.2%

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

      3. fmm-undef 87.2%

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

      4. add-sqr-sqrt 48.4%

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

      5. sqrt-unprod 70.5%

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

      6. sqr-neg 70.5%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 56.9%

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

      9. *-commutative 56.9%

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

      10. add-sqr-sqrt 24.7%

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

      11. sqrt-unprod 70.5%

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

      12. sqr-neg 70.5%

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

      13. sqrt-unprod 48.4%

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

      14. add-sqr-sqrt 87.2%

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

   7. Applied egg-rr 87.2%

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

   if -1.00000000000000009e-111 < (*.f64 x #s(literal 1/2 binary64)) < 2e-85

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

      3. sub-neg 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r- 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 93.3%

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

      1. associate-+r- 93.3%

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

      2. +-commutative 93.3%

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

   9. Applied egg-rr 93.3%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-111} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-85}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (y * log(z)) + (y + (x * 0.5));
	} 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) :: tmp
    if (z <= 0.28d0) then
        tmp = (y * log(z)) + (y + (x * 0.5d0))
    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 tmp;
	if (z <= 0.28) {
		tmp = (y * Math.log(z)) + (y + (x * 0.5));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

      3. sub-neg 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. associate-+r- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r- 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in z around 0 99.6%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. mul-1-neg 98.9%

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

   5. Simplified 98.9%

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

      1. fma-define 98.9%

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

      2. distribute-lft-neg-out 98.9%

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

      3. fmm-undef 98.9%

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

      4. add-sqr-sqrt 51.5%

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

      5. sqrt-unprod 61.3%

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

      6. sqr-neg 61.3%

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

      7. sqrt-unprod 12.7%

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

      8. add-sqr-sqrt 30.2%

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

      9. *-commutative 30.2%

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

      10. add-sqr-sqrt 12.7%

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

      11. sqrt-unprod 61.3%

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

      12. sqr-neg 61.3%

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

      13. sqrt-unprod 51.5%

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

      14. add-sqr-sqrt 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.2%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.28) (+ (* x 0.5) (* y (+ 1.0 (log z)))) (- (* x 0.5) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	} 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) :: tmp
    if (z <= 0.28d0) then
        tmp = (x * 0.5d0) + (y * (1.0d0 + log(z)))
    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 tmp;
	if (z <= 0.28) {
		tmp = (x * 0.5) + (y * (1.0 + Math.log(z)));
	} else {
		tmp = (x * 0.5) - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 0.28:
		tmp = (x * 0.5) + (y * (1.0 + math.log(z)))
	else:
		tmp = (x * 0.5) - (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.28)
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(1.0 + log(z))));
	else
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.28)
		tmp = (x * 0.5) + (y * (1.0 + log(z)));
	else
		tmp = (x * 0.5) - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.28], N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(1.0 + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.28000000000000003

   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 0 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 0.28000000000000003 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. mul-1-neg 98.9%

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

   5. Simplified 98.9%

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

      1. fma-define 98.9%

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

      2. distribute-lft-neg-out 98.9%

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

      3. fmm-undef 98.9%

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

      4. add-sqr-sqrt 51.5%

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

      5. sqrt-unprod 61.3%

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

      6. sqr-neg 61.3%

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

      7. sqrt-unprod 12.7%

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

      8. add-sqr-sqrt 30.2%

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

      9. *-commutative 30.2%

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

      10. add-sqr-sqrt 12.7%

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

      11. sqrt-unprod 61.3%

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

      12. sqr-neg 61.3%

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

      13. sqrt-unprod 51.5%

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

      14. add-sqr-sqrt 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.28:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. sub-neg 99.9%

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

   4. associate-+l+ 99.9%

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

   5. +-commutative 99.9%

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

   6. unsub-neg 99.9%

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

3. Simplified 99.9%

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

   1. associate-+r- 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r- 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-define 99.9%

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

   6. +-commutative 99.9%

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

   7. distribute-rgt-in 99.9%

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

   8. associate-+r+ 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y + N[(y * N[(N[Log[z], $MachinePrecision] - 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. associate-+r- 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+r- 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-rgt-in 99.9%

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

   7. *-un-lft-identity 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 8: 58.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+95}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.2e+95) (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.2e+95) {
		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 <= 2.2d+95) 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 <= 2.2e+95) {
		tmp = x * 0.5;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.2e+95:
		tmp = x * 0.5
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.2e+95)
		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 <= 2.2e+95)
		tmp = x * 0.5;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.2e+95], N[(x * 0.5), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.1999999999999999e95

   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 92.0%

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

      1. +-commutative 92.0%

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

      2. associate-+r- 92.0%

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

      3. associate-/l* 91.9%

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

      4. fma-define 91.9%

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

      5. associate-+r- 91.9%

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

      6. +-commutative 91.9%

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

      7. associate-+r- 91.9%

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

   5. Simplified 91.9%

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

   6. Taylor expanded in y around 0 57.7%

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

   if 2.1999999999999999e95 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. associate-+r- 93.7%

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

      3. associate-/l* 80.8%

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

      4. fma-define 80.8%

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

      5. associate-+r- 80.8%

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

      6. +-commutative 80.8%

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

      7. associate-+r- 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in z around inf 76.5%

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

      1. associate-*r* 76.5%

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

      2. clear-num 76.4%

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

      3. un-div-inv 77.0%

         \[\leadsto \color{blue}{\frac{x \cdot -1}{\frac{x}{y \cdot z}}} \]

      4. *-commutative 77.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\frac{x}{y \cdot z}} \]

      5. neg-mul-1 77.0%

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

      6. *-commutative 77.0%

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

   8. Applied egg-rr 77.0%

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

   9. Taylor expanded in x around 0 82.8%

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

       1. mul-1-neg 82.8%

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

       2. distribute-rgt-neg-out 82.8%

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

   11. Simplified 82.8%

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

Alternative 9: 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. Add Preprocessing

3. Taylor expanded in z around inf 78.7%

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

   1. associate-*r* 78.7%

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

   2. mul-1-neg 78.7%

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

5. Simplified 78.7%

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

   1. fma-define 78.7%

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

   2. distribute-lft-neg-out 78.7%

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

   3. fmm-undef 78.7%

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

   4. add-sqr-sqrt 40.9%

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

   5. sqrt-unprod 57.9%

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

   6. sqr-neg 57.9%

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

   7. sqrt-unprod 19.6%

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

   8. add-sqr-sqrt 42.9%

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

   9. *-commutative 42.9%

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

   10. add-sqr-sqrt 19.6%

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

   11. sqrt-unprod 57.9%

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

   12. sqr-neg 57.9%

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

   13. sqrt-unprod 40.9%

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

   14. add-sqr-sqrt 78.7%

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

7. Applied egg-rr 78.7%

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

8. Final simplification 78.7%

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

Alternative 10: 40.4% 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. Add Preprocessing

3. Taylor expanded in x around inf 92.6%

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

   1. +-commutative 92.6%

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

   2. associate-+r- 92.6%

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

   3. associate-/l* 88.0%

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

   4. fma-define 88.0%

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

   5. associate-+r- 88.0%

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

   6. +-commutative 88.0%

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

   7. associate-+r- 88.0%

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

5. Simplified 88.0%

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

6. Taylor expanded in y around 0 43.9%

   \[\leadsto x \cdot \color{blue}{0.5} \]
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 2024170 
(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)))))