Average Error: 33.7 → 0.4
Time: 49.8s
Precision: 64
Ground Truth: 128
\[e^{a \cdot x} - 1\]
⬇
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.62029496121211 \cdot 10^{-15}:\\ \;\;\;\;{\left(\sqrt[3]{e^{a \cdot x} - 1}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;x \cdot a + {\left(x \cdot a\right)}^2 \cdot \frac{1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original33.7
Comparison8.1
Herbie0.4
\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

  1. Split input into 2 regimes.

  2. if (* a x) < -2.62029496121211e-15

    1. Initial program 1.0

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

    3. Applied add-cube-cbrt 1.0

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

    if -2.62029496121211e-15 < (* a x)

    1. Initial program 48.2

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

    2. Applied taylor 22.5

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

    3. Taylor expanded around 0 22.5

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

    4. Applied simplify 22.5

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

    5. Applied taylor 0.1

      \[\leadsto \left(0 \cdot \frac{1}{6} + a \cdot x\right) + \frac{1}{2} \cdot {\left(a \cdot x\right)}^2\]

    6. Taylor expanded around inf 0.1

      \[\leadsto \left(\color{blue}{0} \cdot \frac{1}{6} + a \cdot x\right) + \frac{1}{2} \cdot {\left(a \cdot x\right)}^2\]

    7. Applied simplify 0.1

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

Runtime

Total time: 49.8s Debug log

Please include this information when filing a bug report:

herbie --seed '#(522858590 1256824439 3603190050 1131246291 690254383 1599979850)'
(FPCore (a x)
  :name "NMSE section 3.5"

  :target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))