Average Error: 29.7 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(e^{\log \left(e^{x \cdot a} - 1\right)}\right)}^{3}} \le 0.006577663290784691:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(e^{x \cdot a} - 1\right) \cdot \left(e^{x \cdot a} - 1\right)} \cdot \sqrt[3]{e^{x \cdot a} - 1}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cbrt (pow (exp (log (- (exp (* x a)) 1))) 3)) < 0.006577663290784691

    1. Initial program 44.8

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

    2. Taylor expanded around 0 13.0

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

    3. Applied simplify0.1

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

    if 0.006577663290784691 < (cbrt (pow (exp (log (- (exp (* x a)) 1))) 3))

    1. Initial program 1.0

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

    3. Applied add-cbrt-cube1.1

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

    4. Applied simplify1.1

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

    6. Applied unpow31.1

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

    7. Applied cbrt-prod1.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(e^{x \cdot a} - 1\right) \cdot \left(e^{x \cdot a} - 1\right)} \cdot \sqrt[3]{e^{x \cdot a} - 1}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018170 
(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))