Average Error: 61.0 → 0.4
Time: 37.2s
Precision: 64
Internal Precision: 1344
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\left(\left(\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}} \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \left(\left(-\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}} \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right)\right) + \left(-x\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{-\left(x + \left(\frac{1}{2} \cdot {x}^{2} + 1\right)\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.4

    \[\leadsto -\left(x + \color{blue}{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1} \cdot \sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.4

    \[\leadsto -\left(x + \sqrt{\frac{1}{2} \cdot {x}^{2} + 1} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right)}\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.4

    \[\leadsto -\left(x + \color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{2} \cdot {x}^{2} + 1}}\right)\right)\]

  9. Final simplification0.4

    \[\leadsto \left(\left(\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}} \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \left(\left(-\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}} \cdot \sqrt[3]{\sqrt{1 + {x}^{2} \cdot \frac{1}{2}}}\right)\right) + \left(-x\right)\]

Runtime

Time bar (total: 37.2s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))