Average Error: 0.3 → 0.3
Time: 10.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{3}, \log z, \mathsf{fma}\left(\frac{2}{3}, \log z, \log \left(x + y\right)\right)\right)\right)}^{3}} - t\right)\]
\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, \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{3}, \log z, \mathsf{fma}\left(\frac{2}{3}, \log z, \log \left(x + y\right)\right)\right)\right)}^{3}} - t\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), (cbrt(pow(fma(0.3333333333333333, log(z), fma(0.6666666666666666, log(z), log((x + y)))), 3.0)) - 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

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
  8. Using strategy rm

  9. Applied add-cbrt-cube0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\sqrt[3]{\left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) \cdot \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right)\right) \cdot \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right)}} - t\right)\]

  10. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\frac{1}{3}, \log z, \mathsf{fma}\left(\frac{2}{3}, \log z, \log \left(x + y\right)\right)\right)\right)}^{3}}} - t\right)\]

  11. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{3}, \log z, \mathsf{fma}\left(\frac{2}{3}, \log z, \log \left(x + y\right)\right)\right)\right)}^{3}} - t\right)\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))