Average Error: 29.7 → 9.2
Time: 22.6s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -7.235975056987809 \cdot 10^{-13}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{\log \left(e^{e^{a \cdot x} - 1}\right)}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.7
Target0.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (exp (* a x)) 1) < -7.235975056987809e-13

    1. Initial program 0.6

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

    3. Applied add-cube-cbrt0.6

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

    5. Applied add-sqr-sqrt0.6

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

    6. Applied difference-of-sqr-10.6

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

    7. Applied cbrt-prod0.6

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

    9. Applied add-log-exp0.6

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

    if -7.235975056987809e-13 < (- (exp (* a x)) 1)

    1. Initial program 44.9

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

    2. Taylor expanded around 0 13.7

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

Runtime

Time bar (total: 22.6s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' 
(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))