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

Percentage Accurate: 99.9% → 99.7%

Time: 10.6s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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)) - (y * 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)));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Target

Original	99.9%
Target	99.8%
Herbie	99.7%

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0 99.9%

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7232

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

Alternative 2
Accuracy	85.3%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-149} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot \left(\log z - z\right)\\ \end{array} \]

Alternative 3
Accuracy	85.3%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -5 \cdot 10^{-149} \lor \neg \left(x \cdot 0.5 \leq 2 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \left(\log z - z\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	98.8%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq 0.0011:\\ \;\;\;\;y \cdot \log z + \left(x \cdot 0.5 + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	7104

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

Alternative 6
Accuracy	76.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+262}:\\ \;\;\;\;y \cdot \left(\log z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 7
Accuracy	76.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+262}:\\ \;\;\;\;y + y \cdot \log z\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 8
Accuracy	76.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq 0.0011:\\ \;\;\;\;x \cdot 0.5 - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - y \cdot z\\ \end{array} \]

Alternative 9
Accuracy	76.7%
Cost	576

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

Alternative 10
Accuracy	60.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 72000000000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 11
Accuracy	63.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq 4.3 \cdot 10^{+14}:\\ \;\;\;\;x \cdot 0.5 - y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 12
Accuracy	60.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq 29000000000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 13
Accuracy	39.4%
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

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