Average Error: 29.4 → 0.8
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -1.2085430657074867 \cdot 10^{-16}:\\ \;\;\;\;\sqrt[3]{{\left(e^{x \cdot a} - 1\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(a \cdot x\right) \cdot \left(a \cdot x\right)}\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.4
Target0.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (exp (* a x)) 1) < -1.2085430657074867e-16

    1. Initial program 1.0

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

    3. Applied add-cbrt-cube1.0

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

    4. Applied simplify1.0

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

    if -1.2085430657074867e-16 < (- (exp (* a x)) 1)

    1. Initial program 45.0

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

    2. Taylor expanded around 0 13.3

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

    3. Applied simplify0.5

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

    5. Applied add-log-exp0.7

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' 
(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))