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

Average Accuracy: 99.9% → 99.9%

Time: 15.2s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

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 fma(((1.0 - z) + log(z)), y, (x * 0.5));
}

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 fma(Float64(Float64(1.0 - z) + log(z)), y, Float64(x * 0.5))
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[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] * y + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Target

Original	99.9%
Target	99.8%
Herbie	99.9%

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

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   +-commutative [=>] 99.9	\[ \color{blue}{y \cdot \left(\left(1 - z\right) + \log z\right) + x \cdot 0.5} \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(\left(1 - z\right) + \log z\right) \cdot y} + x \cdot 0.5 \]
   fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(\left(1 - z\right) + \log z, y, x \cdot 0.5\right)} \]

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	70.8%
Cost	7708

\[\begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 6.2 \cdot 10^{-266}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-261}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 2
Accuracy	71.0%
Cost	7646

\[\begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ \mathbf{if}\;z \leq 3.7 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-261}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-210} \lor \neg \left(z \leq 2.15 \cdot 10^{-154}\right) \land \left(z \leq 9 \cdot 10^{-96} \lor \neg \left(z \leq 4.9 \cdot 10^{-68}\right) \land z \leq 5.1 \cdot 10^{-33}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 3
Accuracy	71.1%
Cost	7646

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-267}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-261}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-210} \lor \neg \left(z \leq 9.2 \cdot 10^{-156}\right) \land \left(z \leq 5.8 \cdot 10^{-96} \lor \neg \left(z \leq 1.9 \cdot 10^{-68}\right) \land z \leq 10^{-33}\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 4
Accuracy	83.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(1 - z\right) + \log z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 5
Accuracy	98.5%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + \left(1 - z\right) \cdot y\\ \end{array} \]

Alternative 6
Accuracy	99.9%
Cost	7104

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

Alternative 7
Accuracy	70.9%
Cost	448

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

Alternative 8
Accuracy	54.3%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+42}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 9
Accuracy	45.4%
Cost	192

\[x \cdot 0.5 \]

Alternative 10
Accuracy	2.1%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023139 
(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)))))