Average Error: 33.5 → 3.6
Time: 1.2m
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.4384882105786329 \cdot 10^{-06}:\\ \;\;\;\;\frac{e^{\left(2 + 1\right) \cdot \left(x \cdot a\right)} - 1}{\left(1 + e^{x \cdot a}\right) + {\left(e^{x}\right)}^{\left(a + a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original33.5
Target7.9
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -1.4384882105786329e-06

    1. Initial program 0.5

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

    3. Applied add-cube-cbrt0.6

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

    5. Applied flip3--0.6

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

    6. Applied simplify0.5

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

    7. Applied simplify0.5

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

    8. Applied simplify11.7

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

    if -1.4384882105786329e-06 < (* a x)

    1. Initial program 47.8

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

    2. Taylor expanded around 0 43.8

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

    3. Applied simplify0.1

      \[\leadsto \color{blue}{\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{2} + x \cdot a}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.2m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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))