Average Error: 29.3 → 0.5
Time: 33.1s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -5.697442517768994 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(e^{a \cdot x} - 1\right) + \left(\left(e^{a \cdot x} - 1\right) \cdot e^{a \cdot x}\right) \cdot \sqrt{e^{a \cdot x}}}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(1 + e^{a \cdot x}\right) - \sqrt{e^{a \cdot x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{8} \cdot x\right)\right) + a \cdot \left(\frac{1}{2} \cdot x\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} + 1\right)\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.3
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (exp (* a x)) 1) < -5.697442517768994e-12

    1. Initial program 0.5

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

    3. Applied add-sqr-sqrt0.5

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

    4. Applied difference-of-sqr-10.5

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

    6. Applied flip--0.5

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

    7. Applied flip3-+0.5

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

    8. Applied frac-times0.5

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

    9. Applied simplify0.5

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

    10. Applied simplify0.5

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

    if -5.697442517768994e-12 < (- (exp (* a x)) 1)

    1. Initial program 44.4

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

    3. Applied add-sqr-sqrt44.4

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

    4. Applied difference-of-sqr-144.4

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

    5. Taylor expanded around 0 45.4

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\color{blue}{\left(1 + \left(\frac{1}{8} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)} - 1\right)\]

    6. Applied simplify0.5

      \[\leadsto \color{blue}{\left(a \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{8} \cdot x\right) + \frac{1}{2} \cdot x\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} + 1\right)}\]
    7. Using strategy rm

    8. Applied distribute-lft-in0.5

      \[\leadsto \color{blue}{\left(a \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{8} \cdot x\right)\right) + a \cdot \left(\frac{1}{2} \cdot x\right)\right)} \cdot \left(\sqrt{e^{a \cdot x}} + 1\right)\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 33.1s)Debug log

herbie shell --seed '#(212267722 3993171362 1093346726 3605783651 2106536041 3335990851)' 
(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))