Average Error: 0.2 → 0.3
Time: 14.7s
Precision: binary64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
↓
\[\left(\log \left(y + x\right) + \left(\log z + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)\right) - \left(t - \left(a \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + a \cdot \log \left(\sqrt[3]{t}\right)\right)\right) \]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\log \left(y + x\right) + \left(\log z + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)\right) - \left(t - \left(a \cdot \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + a \cdot \log \left(\sqrt[3]{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 (+ y x)) (+ (log z) (* 0.5 (log (/ 1.0 t)))))
  (- t (+ (* a (log (* (cbrt t) (cbrt t)))) (* a (log (cbrt 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(y + x) + (log(z) + (0.5 * log(1.0 / t)))) - (t - ((a * log(cbrt(t) * cbrt(t))) + (a * log(cbrt(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

Original0.2
Target0.2
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 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in t around inf 0.2

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

  4. Applied add-cube-cbrt_binary640.2

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

  5. Applied add-cube-cbrt_binary640.2

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

  6. Applied times-frac_binary640.2

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

  7. Applied log-prod_binary640.3

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

  8. Applied distribute-rgt-in_binary640.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2021215 
(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))))