Average Error: 29.7 → 13.7
Time: 24.1s
Precision: 64
Internal Precision: 128
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.249896967385658 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} \cdot e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{e^{\left(x + x\right) \cdot \left(a + a\right)} - -1}}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;x \le 6.630488935242125 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + e^{\left(x + x\right) \cdot \left(a + a\right)}}{\left(\left(\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \end{array}\]

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

Original29.7
Target0.2
Herbie13.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5.249896967385658e+123

    1. Initial program 15.1

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

    3. Applied flip--15.1

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
    4. Using strategy rm

    5. Applied flip--15.1

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

    6. Applied associate-/l/15.1

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

    7. Simplified15.0

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

    9. Applied flip-+15.1

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

    if -5.249896967385658e+123 < x < 6.630488935242125e+90

    1. Initial program 34.4

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

    2. Taylor expanded around 0 20.0

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

    3. Simplified13.1

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

    if 6.630488935242125e+90 < x

    1. Initial program 16.4

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

    3. Applied flip--16.5

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]
    4. Using strategy rm

    5. Applied flip--16.5

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

    6. Applied associate-/l/16.5

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

    7. Simplified16.4

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

    9. Applied add-cube-cbrt16.4

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.249896967385658 \cdot 10^{+123}:\\ \;\;\;\;\frac{\frac{e^{\left(x + x\right) \cdot \left(a + a\right)} \cdot e^{\left(x + x\right) \cdot \left(a + a\right)} - 1}{e^{\left(x + x\right) \cdot \left(a + a\right)} - -1}}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right) \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;x \le 6.630488935242125 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\frac{1}{2} + \left(x \cdot \frac{1}{6}\right) \cdot a\right) + a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + e^{\left(x + x\right) \cdot \left(a + a\right)}}{\left(\left(\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 24.1s)Debug logProfile

herbie shell --seed 2018304 
(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))