Average Error: 29.0 → 0.5
Time: 18.6s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(e^{a \cdot x} - 1\right)} \le 4.3978497306458144 \cdot 10^{-05}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}}} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}}}\right) - 1\right)\\ \end{array}\]

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

Original29.0
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp (log (- (exp (* a x)) 1))) < 4.3978497306458144e-05

    1. Initial program 45.3

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

    2. Taylor expanded around 0 13.5

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

    3. Simplified0.0

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

    if 4.3978497306458144e-05 < (exp (log (- (exp (* a x)) 1)))

    1. Initial program 1.1

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

    3. Applied add-sqr-sqrt1.2

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

    4. Applied difference-of-sqr-11.2

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

    6. Applied add-cube-cbrt1.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(e^{a \cdot x} - 1\right)} \le 4.3978497306458144 \cdot 10^{-05}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(\frac{1}{6} \cdot x\right) \cdot a + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}}} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}}}\right) - 1\right)\\ \end{array}\]

Runtime

Time bar (total: 18.6s)Debug logProfile

herbie shell --seed 2018217 
(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))