Average Error: 0.2 → 0.2
Time: 13.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \mathsf{fma}\left(-t, 1, t\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \mathsf{fma}\left(-t, 1, t\right)\right)
double code(double x, double y, double z, double t, double a) {
	return (((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma(log(t), (a - 0.5), (((log((x + y)) + log(z)) - t) + fma(-t, 1.0, t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.7

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)\]

  5. Applied add-sqr-sqrt21.9

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\sqrt{\log \left(x + y\right) + \log z} \cdot \sqrt{\log \left(x + y\right) + \log z}} - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)\]

  6. Applied prod-diff21.9

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\mathsf{fma}\left(\sqrt{\log \left(x + y\right) + \log z}, \sqrt{\log \left(x + y\right) + \log z}, -\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}\right)\]

  7. Simplified0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\left(\left(\log \left(x + y\right) + \log z\right) - t\right)} + \mathsf{fma}\left(-\sqrt[3]{t}, \sqrt[3]{t} \cdot \sqrt[3]{t}, \sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)\]

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))