Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 5.7 → 0.3

Time: 5.6s

Precision: binary64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 12844702815189.3926:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\frac{z \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right) + 0.0833333333333329956}{x} + \left(\left(0.91893853320467001 - x\right) + \sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(\left(x - 0.5\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} + x \cdot \left(\log 1 - \left(1 + \left(-\log x\right)\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.7
Target	1.2
Herbie	0.3

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

1. Split input into 2 regimes

2. if x < 12844702815189.3926

   1. Initial program 0.1

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 0.1

      \[\leadsto \left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)\]

   5. Applied log-prod 0.1

      \[\leadsto \left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)\]

   6. Applied distribute-lft-in 0.1

      \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right)} + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)\]

   7. Applied associate-+l+ 0.2

      \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)\right)}\]

   8. Simplified 0.2

      \[\leadsto \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \color{blue}{\left(\frac{z \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right) + 0.0833333333333329956}{x} + \left(\left(0.91893853320467001 - x\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right)\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 0.2

       \[\leadsto \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\frac{z \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right) + 0.0833333333333329956}{x} + \left(\left(0.91893853320467001 - x\right) + \left(x - 0.5\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right)}\right)\right)\]

   11. Applied associate-*r* 0.2

       \[\leadsto \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\frac{z \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right) + 0.0833333333333329956}{x} + \left(\left(0.91893853320467001 - x\right) + \color{blue}{\left(\left(x - 0.5\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right)\right) \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}}\right)\right)\]

   if 12844702815189.3926 < x

   1. Initial program 9.9

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

   2. Simplified 9.9

      \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 + \left(\frac{z \cdot \left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) + 0.0833333333333329956}{x} - x\right)\right)}\]

   3. Taylor expanded around inf 10.1

      \[\leadsto \color{blue}{\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \left(\frac{{z}^{2} \cdot y}{x} + \log 1 \cdot x\right)\right) - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)}\]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\frac{z}{\frac{x}{z}} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) + x \cdot \left(\log 1 - \left(\left(-\log x\right) + 1\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 12844702815189.3926:\\ \;\;\;\;\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\frac{z \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right) + 0.0833333333333329956}{x} + \left(\left(0.91893853320467001 - x\right) + \sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \left(\left(x - 0.5\right) \cdot \left(\sqrt[3]{\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{\log \left(\sqrt{x}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot \frac{z}{\frac{x}{z}} + x \cdot \left(\log 1 - \left(1 + \left(-\log x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))