Average Error: 28.9 → 0.6
Time: 43.3s
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -8.77956638094267 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} + {1}^{3}\right) \cdot \left({\left(\sqrt{e^{a \cdot x}}\right)}^{3} - {1}^{3}\right)}{\left(\left(1 + e^{x \cdot a}\right) + \sqrt{e^{x \cdot a}}\right) \cdot \left(\left(1 + e^{x \cdot a}\right) - \sqrt{e^{x \cdot a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (exp (* a x)) 1) < -8.77956638094267e-13

    1. Initial program 0.7

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

    3. Applied add-sqr-sqrt0.7

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

    4. Applied difference-of-sqr-10.7

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

    6. Applied flip3--0.7

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

    7. Applied flip3-+0.7

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

    8. Applied frac-times0.7

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

    9. Applied simplify0.7

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

    if -8.77956638094267e-13 < (- (exp (* a x)) 1)

    1. Initial program 44.2

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

    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

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

Runtime

Time bar (total: 43.3s)Debug logProfile

herbie shell --seed 2018214 
(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))