Average Error: 29.9 → 0.1
Time: 51.4s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + a \cdot x\right) \le -6.638130364285076 \cdot 10^{-07}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\\ \mathbf{if}\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + a \cdot x\right) \le 0.0001379393693337132:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6}\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{2} + a \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.9
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (* (* (* a x) (* a x)) (* (* a x) 1/6)) (+ (* (* (* a x) (* a x)) 1/2) (* a x))) < -6.638130364285076e-07 or 0.0001379393693337132 < (+ (* (* (* a x) (* a x)) (* (* a x) 1/6)) (+ (* (* (* a x) (* a x)) 1/2) (* a x)))

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1\]

    4. Applied difference-of-sqr-10.3

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

    if -6.638130364285076e-07 < (+ (* (* (* a x) (* a x)) (* (* a x) 1/6)) (+ (* (* (* a x) (* a x)) 1/2) (* a x))) < 0.0001379393693337132

    1. Initial program 45.0

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

    2. Taylor expanded around 0 13.5

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

    3. Applied simplify0.0

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

    5. Applied distribute-lft-in0.0

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

    6. Applied associate-+l+0.0

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

Runtime

Time bar (total: 51.4s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' 
(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))