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

Average Error: 0.1 → 0.1

Time: 12.1s

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	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	18.6
Cost	7708

\[\begin{array}{l} t_0 := y \cdot \left(1 + \log z\right)\\ t_1 := x \cdot 0.5 - z \cdot y\\ \mathbf{if}\;z \leq 1.05 \cdot 10^{-293}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 2
Error	18.6
Cost	7647

\[\begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-293} \lor \neg \left(z \leq 2.3 \cdot 10^{-264}\right) \land \left(z \leq 1.52 \cdot 10^{-223} \lor \neg \left(z \leq 6.8 \cdot 10^{-178}\right) \land \left(z \leq 1.5 \cdot 10^{-173} \lor \neg \left(z \leq 1.35 \cdot 10^{-73}\right) \land z \leq 7.5 \cdot 10^{-37}\right)\right):\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \end{array} \]

Alternative 3
Error	18.6
Cost	7647

\[\begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{-293}:\\ \;\;\;\;y + \log z \cdot y\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-264} \lor \neg \left(z \leq 1.5 \cdot 10^{-223} \lor \neg \left(z \leq 5.7 \cdot 10^{-179}\right) \land \left(z \leq 2.35 \cdot 10^{-173} \lor \neg \left(z \leq 1.1 \cdot 10^{-73}\right) \land z \leq 5.5 \cdot 10^{-38}\right)\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log z\right)\\ \end{array} \]

Alternative 4
Error	10.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{elif}\;x \cdot 0.5 \leq 5 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(\left(1 + \log z\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 5
Error	18.2
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \cdot 0.5 \leq -4 \cdot 10^{-119} \lor \neg \left(x \cdot 0.5 \leq 5 \cdot 10^{-108}\right):\\ \;\;\;\;x \cdot 0.5 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\log z - z\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq 0.008:\\ \;\;\;\;\log z \cdot y + \left(y + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, x \cdot 0.5\right)\\ \end{array} \]

Alternative 7
Error	0.1
Cost	7104

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

Alternative 8
Error	29.2
Cost	785

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-119}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-29} \lor \neg \left(x \leq 0.245\right) \land x \leq 2 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \]

Alternative 9
Error	18.6
Cost	448

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

Alternative 10
Error	35.2
Cost	192

\[x \cdot 0.5 \]

Alternative 11
Error	62.7
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023066 
(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)))))