Average Error: 0.3 → 0.4
Time: 38.7s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\begin{array}{l} t_1 := \log \left(\sqrt[3]{y}\right)\\ t_1 + \left(\left(\left(\mathsf{fma}\left(a, \log t, \log z\right) + \frac{x}{y}\right) - \mathsf{fma}\left(0.5, \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \log t\right), t\right)\right) + 2 \cdot t_1\right) \end{array} \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\begin{array}{l}
t_1 := \log \left(\sqrt[3]{y}\right)\\
t_1 + \left(\left(\left(\mathsf{fma}\left(a, \log t, \log z\right) + \frac{x}{y}\right) - \mathsf{fma}\left(0.5, \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \log t\right), t\right)\right) + 2 \cdot t_1\right)
\end{array}

(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
 (let* ((t_1 (log (cbrt y))))
   (+
    t_1
    (+
     (-
      (+ (fma a (log t) (log z)) (/ x y))
      (fma 0.5 (fma (/ x y) (/ x y) (log t)) t))
     (* 2.0 t_1)))))
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) {
	double t_1 = log(cbrt(y));
	return t_1 + (((fma(a, log(t), log(z)) + (x / y)) - fma(0.5, fma((x / y), (x / y), log(t)), t)) + (2.0 * t_1));
}

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
Herbie0.4
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

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. Taylor expanded in y around inf 7.9

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

  3. Simplified0.4

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

  4. Applied associate-+r-_binary640.4

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

  5. Applied associate--r-_binary640.4

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

  6. Simplified0.4

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

  7. Applied add-cube-cbrt_binary640.4

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

  8. Applied log-prod_binary640.4

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

  9. Applied associate-+r+_binary640.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

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))))