Average Error: 0.3 → 0.3
Time: 9.9s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(a \cdot \log t + \left(\log \left(x + y\right) - \left(\log \left(\sqrt[3]{t}\right) + \left(t - \log z\right)\right)\right)\right) + \log \left(\sqrt[3]{t}\right) \cdot -0.5\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(a \cdot \log t + \left(\log \left(x + y\right) - \left(\log \left(\sqrt[3]{t}\right) + \left(t - \log z\right)\right)\right)\right) + \log \left(\sqrt[3]{t}\right) \cdot -0.5
(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
 (+
  (+ (* a (log t)) (- (log (+ x y)) (+ (log (cbrt t)) (- t (log z)))))
  (* (log (cbrt t)) -0.5)))
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 ((a * log(t)) + (log(x + y) - (log(cbrt(t)) + (t - log(z))))) + (log(cbrt(t)) * -0.5);
}

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. Using strategy rm

  3. Applied add-cube-cbrt_binary640.3

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

  4. Applied log-prod_binary640.3

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

  5. Applied distribute-rgt-in_binary640.3

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

  6. Applied associate-+r+_binary640.3

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

  7. Simplified0.3

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

  9. Applied sub-neg_binary640.3

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

  10. Applied distribute-rgt-in_binary640.3

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

  11. Applied associate-+r+_binary640.3

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

  12. Simplified0.3

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

  13. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{\left(\left(\log \left(x + y\right) + 3 \cdot \left(a \cdot \log \left({t}^{0.3333333333333333}\right)\right)\right) - \left(\log \left({t}^{0.3333333333333333}\right) + \left(\log \left(\frac{1}{z}\right) + t\right)\right)\right)} + \left(-0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\]

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2021173 
(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))))