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

Average Error: 0.1 → 0.1

Time: 2.8s

Precision: binary64

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.1

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

2. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "(log (+ (* 100 (exp (* (sqrt 1e-10) (sqrt (* (- 2) (log x)))))) 1))"
  :precision binary64
  (log (+ (* 100.0 (exp (* (sqrt 1e-10) (sqrt (* (neg 2.0) (log x)))))) 1.0)))