Average Error: 0.3 → 0.3
Time: 28.3s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\log \left(x + y\right) + \left(\log z + \mathsf{fma}\left(\log t, a, \mathsf{fma}\left(-0.5, \log t, -t\right)\right)\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(x + y\right) + \left(\log z + \mathsf{fma}\left(\log t, a, \mathsf{fma}\left(-0.5, \log t, -t\right)\right)\right)

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ x y)) (+ (log z) (fma (log t) a (fma -0.5 (log t) (- t))))))
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 log(x + y) + (log(z) + fma(log(t), a, fma(-0.5, log(t), -t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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. Applied associate--l+_binary640.3

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

  3. Applied associate-+l+_binary640.3

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

  4. Applied add-cube-cbrt_binary640.3

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

  5. Applied log-prod_binary640.3

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

  6. Applied distribute-rgt-in_binary640.3

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

  7. Applied associate-+r+_binary640.3

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

  8. Simplified0.3

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

  9. Taylor expanded in a around 0 0.3

    \[\leadsto \log \left(x + y\right) + \color{blue}{\left(\left(3 \cdot \left(a \cdot \log \left({t}^{0.3333333333333333}\right)\right) + \log z\right) - \left(t + 1.5 \cdot \log \left({t}^{0.3333333333333333}\right)\right)\right)} \]

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2021275 
(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))))