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

Percentage Accurate: 99.9% → 99.6%

Time: 10.4s

Alternatives: 7

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: 99.6% 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. Step-by-step derivation

   1. cancel-sign-sub 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. unsub-neg 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

   9. associate--r- 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

   10. metadata-eval 99.9%

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

   11. +-commutative 99.9%

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

   12. associate--l+ 99.9%

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

3. Simplified 99.9%

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

   1. +-commutative 99.9%

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

   2. distribute-lft-in 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-32} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(\log z + 1\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x 0.5) -5e-32) (not (<= (* x 0.5) 5e-16)))
   (- (* x 0.5) (* y z))
   (* y (- (+ (log z) 1.0) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -5e-32) || !((x * 0.5) <= 5e-16)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((log(z) + 1.0) - 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) <= (-5d-32)) .or. (.not. ((x * 0.5d0) <= 5d-16))) then
        tmp = (x * 0.5d0) - (y * z)
    else
        tmp = y * ((log(z) + 1.0d0) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * 0.5) <= -5e-32) || !((x * 0.5) <= 5e-16)) {
		tmp = (x * 0.5) - (y * z);
	} else {
		tmp = y * ((Math.log(z) + 1.0) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * 0.5) <= -5e-32) or not ((x * 0.5) <= 5e-16):
		tmp = (x * 0.5) - (y * z)
	else:
		tmp = y * ((math.log(z) + 1.0) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * 0.5) <= -5e-32) || !(Float64(x * 0.5) <= 5e-16))
		tmp = Float64(Float64(x * 0.5) - Float64(y * z));
	else
		tmp = Float64(y * Float64(Float64(log(z) + 1.0) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * 0.5) <= -5e-32) || ~(((x * 0.5) <= 5e-16)))
		tmp = (x * 0.5) - (y * z);
	else
		tmp = y * ((log(z) + 1.0) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * 0.5), $MachinePrecision], -5e-32], N[Not[LessEqual[N[(x * 0.5), $MachinePrecision], 5e-16]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[Log[z], $MachinePrecision] + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 1/2) < -5e-32 or 5.0000000000000004e-16 < (*.f64 x 1/2)

   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. cancel-sign-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. unsub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 100.0%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 89.3%

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

   if -5e-32 < (*.f64 x 1/2) < 5.0000000000000004e-16

   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. cancel-sign-sub 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. unsub-neg 99.8%

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

      8. neg-sub0 99.8%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 99.8%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 90.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-32} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(\log z + 1\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(-1 - \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.04e-10)
   (- (* 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 <= 1.04e-10) {
		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 <= 1.04d-10) 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 <= 1.04e-10) {
		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 <= 1.04e-10:
		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 <= 1.04e-10)
		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 <= 1.04e-10)
		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, 1.04e-10], 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 < 1.04e-10

   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. cancel-sign-sub 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. unsub-neg 99.9%

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

      8. neg-sub0 99.9%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 99.9%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   if 1.04e-10 < 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. cancel-sign-sub 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. unsub-neg 99.9%

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

      8. neg-sub0 99.9%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 99.9%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(-1 - \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 60.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 95000000 \lor \neg \left(z \leq 9 \cdot 10^{+45}\right) \land z \leq 7.6 \cdot 10^{+73}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 95000000.0) (and (not (<= z 9e+45)) (<= z 7.6e+73)))
   (* x 0.5)
   (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 95000000.0) || (!(z <= 9e+45) && (z <= 7.6e+73))) {
		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 <= 95000000.0d0) .or. (.not. (z <= 9d+45)) .and. (z <= 7.6d+73)) 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 <= 95000000.0) || (!(z <= 9e+45) && (z <= 7.6e+73))) {
		tmp = x * 0.5;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 95000000.0) or (not (z <= 9e+45) and (z <= 7.6e+73)):
		tmp = x * 0.5
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 95000000.0) || (!(z <= 9e+45) && (z <= 7.6e+73)))
		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 <= 95000000.0) || (~((z <= 9e+45)) && (z <= 7.6e+73)))
		tmp = x * 0.5;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 95000000.0], And[N[Not[LessEqual[z, 9e+45]], $MachinePrecision], LessEqual[z, 7.6e+73]]], N[(x * 0.5), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.5e7 or 8.9999999999999997e45 < z < 7.60000000000000044e73

   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. cancel-sign-sub 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. unsub-neg 99.8%

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

      8. neg-sub0 99.8%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 99.8%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 57.6%

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

   if 9.5e7 < z < 8.9999999999999997e45 or 7.60000000000000044e73 < 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. cancel-sign-sub 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. unsub-neg 100.0%

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

      8. neg-sub0 100.0%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

      9. associate--r- 100.0%

         \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 75.4%

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

      1. associate-*r* 75.4%

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

      2. neg-mul-1 75.4%

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

   7. Simplified 75.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 95000000 \lor \neg \left(z \leq 9 \cdot 10^{+45}\right) \land z \leq 7.6 \cdot 10^{+73}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% 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. Step-by-step derivation

   1. cancel-sign-sub 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. unsub-neg 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

   9. associate--r- 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

   10. metadata-eval 99.9%

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

   11. +-commutative 99.9%

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

   12. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 78.9%

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

6. Final simplification 78.9%

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

Alternative 7: 39.6% 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. Step-by-step derivation

   1. cancel-sign-sub 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. unsub-neg 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(0 - \left(1 - z\right)\right)} - \log z\right) \]

   9. associate--r- 99.9%

      \[\leadsto x \cdot 0.5 - y \cdot \left(\color{blue}{\left(\left(0 - 1\right) + z\right)} - \log z\right) \]

   10. metadata-eval 99.9%

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

   11. +-commutative 99.9%

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

   12. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 42.1%

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

6. Final simplification 42.1%

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

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 2024048 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

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