Average Error: 29.6 → 0.6
Time: 17.9s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x} - 1}}\right) \le 0.008949034111164792:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(\frac{1}{6} \cdot x\right) + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}}}\right) \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right)\\ \end{array}\]

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

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

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (- (exp (* a x)) 1)) (cbrt (- (exp (* a x)) 1))) (* (cbrt (sqrt (- (exp (* a x)) 1))) (cbrt (sqrt (- (exp (* a x)) 1))))) < 0.008949034111164792

    1. Initial program 44.8

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

    2. Taylor expanded around 0 12.6

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

    3. Simplified0.1

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

    if 0.008949034111164792 < (* (* (cbrt (- (exp (* a x)) 1)) (cbrt (- (exp (* a x)) 1))) (* (cbrt (sqrt (- (exp (* a x)) 1))) (cbrt (sqrt (- (exp (* a x)) 1)))))

    1. Initial program 1.4

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

    3. Applied add-cube-cbrt1.4

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

    5. Applied add-cube-cbrt1.4

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

    7. Applied flip3--1.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt{e^{a \cdot x} - 1}}\right) \le 0.008949034111164792:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \left(\frac{1}{6} \cdot x\right) + \frac{1}{2}\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}}}\right) \cdot \left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right)\\ \end{array}\]

Runtime

Time bar (total: 17.9s)Debug logProfile

herbie shell --seed 2018216 
(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))