Average Error: 0.3 → 0.3
Time: 14.7s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log \left(x + y\right)\right) - t \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log \left(x + y\right)\right) - t

(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
 (- (+ (fma (log t) (- a 0.5) (log z)) (log (+ x y))) 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 (fma(log(t), (a - 0.5), log(z)) + log(x + y)) - 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. Simplified0.3

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

  3. Taylor expanded in a around 0 0.3

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

  4. Simplified0.3

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

  5. Applied *-un-lft-identity_binary640.3

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

  6. Applied cancel-sign-sub-inv_binary640.3

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

  7. Applied associate-+r+_binary640.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2022068 
(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))))