(/ (log (+ (* 100 (exp (* (sqrt 1e-10) (sqrt (* (- 2) (log x)))))) 1)) (sqrt (* (- 2) (log x))))

Average Error: 0.4 → 0.4

Time: 3.5s

Precision: binary64

Math

\[\frac{\log \left(100 \cdot e^{\sqrt{1.00000000000000004 \cdot 10^{-10}} \cdot \sqrt{\left(-2\right) \cdot \log x}} + 1\right)}{\sqrt{\left(-2\right) \cdot \log x}}\]

↓

\[\frac{\log \left(100 \cdot e^{\sqrt{1.00000000000000004 \cdot 10^{-10}} \cdot \sqrt{\left(-2\right) \cdot \log x}} + 1\right)}{\sqrt{\left(-2\right) \cdot \log x}}\]

Error

Bits error versus x

Derivation

1. Initial program 0.4

   \[\frac{\log \left(100 \cdot e^{\sqrt{1.00000000000000004 \cdot 10^{-10}} \cdot \sqrt{\left(-2\right) \cdot \log x}} + 1\right)}{\sqrt{\left(-2\right) \cdot \log x}}\]

2. Final simplification 0.4

   \[\leadsto \frac{\log \left(100 \cdot e^{\sqrt{1.00000000000000004 \cdot 10^{-10}} \cdot \sqrt{\left(-2\right) \cdot \log x}} + 1\right)}{\sqrt{\left(-2\right) \cdot \log x}}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "(/ (log (+ (* 100 (exp (* (sqrt 1e-10) (sqrt (* (- 2) (log x)))))) 1)) (sqrt (* (- 2) (log x))))"
  :precision binary64
  (/ (log (+ (* 100.0 (exp (* (sqrt 1e-10) (sqrt (* (neg 2.0) (log x)))))) 1.0)) (sqrt (* (neg 2.0) (log x)))))