Average Error: 30.0 → 0.8
Time: 12.9s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \left(\sqrt[3]{\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \left(\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right)} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right)\]

Error

Bits error versus a

Bits error versus x

Target

Original30.0
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Initial program 30.0

    \[e^{a \cdot x} - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{(e^{a \cdot x} - 1)^*}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right) \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.8

    \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right) \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}}} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right) \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\]

  7. Final simplification0.8

    \[\leadsto \sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \left(\sqrt[3]{\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \left(\sqrt[3]{(e^{a \cdot x} - 1)^*} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right)} \cdot \sqrt[3]{(e^{a \cdot x} - 1)^*}\right)\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))