Average Error: 29.8 → 0.3
Time: 23.1s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00011577235552458078:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{a \cdot x} \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x}\right) - \left(e^{a \cdot x} \cdot e^{a \cdot x} + -1\right)\right) \cdot \left(-1 + e^{a \cdot x}\right)}}{\sqrt[3]{\left(\left(e^{a \cdot x} + 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.8
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00011577235552458078

    1. Initial program 0.0

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.0

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

    5. Applied flip--0.0

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

    6. Applied flip3--0.0

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

    7. Applied frac-times0.0

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

    8. Applied associate-*l/0.0

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

    9. Applied cbrt-div0.0

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

    10. Simplified0.0

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

    if -0.00011577235552458078 < (* a x)

    1. Initial program 44.6

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

    2. Taylor expanded around 0 14.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]

    3. Simplified0.5

      \[\leadsto \color{blue}{\left(a \cdot x + \frac{1}{6} \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right)\right) + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)}\]
    4. Using strategy rm

    5. Applied associate-+l+0.5

      \[\leadsto \color{blue}{a \cdot x + \left(\frac{1}{6} \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00011577235552458078:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{a \cdot x} \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x}\right) - \left(e^{a \cdot x} \cdot e^{a \cdot x} + -1\right)\right) \cdot \left(-1 + e^{a \cdot x}\right)}}{\sqrt[3]{\left(\left(e^{a \cdot x} + 1\right) + e^{a \cdot x} \cdot e^{a \cdot x}\right) \cdot \left(e^{a \cdot x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) + \frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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))