Average Error: 28.9 → 0.2
Time: 3.9m
Precision: 64
Internal Precision: 1344
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(6 \cdot a\right) \cdot x + {\left(x \cdot a\right)}^{3} \cdot 36\right) + 18 \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} \le -0.0012473541376008107:\\ \;\;\;\;\frac{e^{\left(x \cdot a\right) \cdot \left(3 + 3\right)}}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} - \frac{1}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)}\\ \mathbf{elif}\;\frac{\left(\left(6 \cdot a\right) \cdot x + {\left(x \cdot a\right)}^{3} \cdot 36\right) + 18 \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} \le 0.007531445725249147:\\ \;\;\;\;\left(x \cdot a + \frac{1}{2} \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)\right) + {\left(x \cdot a\right)}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(x \cdot a\right) \cdot \left(3 + 3\right)}}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} - \frac{1}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* 18 (* (* a x) (* a x))) (+ (* (* 6 a) x) (* 36 (pow (* a x) 3)))) (* (+ (* (exp (* a x)) (exp (* a x))) (+ (* 1 1) (* (exp (* a x)) 1))) (+ (pow (exp (* a x)) 3) (pow 1 3)))) < -0.0012473541376008107 or 0.007531445725249147 < (/ (+ (* 18 (* (* a x) (* a x))) (+ (* (* 6 a) x) (* 36 (pow (* a x) 3)))) (* (+ (* (exp (* a x)) (exp (* a x))) (+ (* 1 1) (* (exp (* a x)) 1))) (+ (pow (exp (* a x)) 3) (pow 1 3))))

    1. Initial program 0.0

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

    3. Applied flip3--0.0

      \[\leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\]
    4. Using strategy rm

    5. Applied flip--0.0

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

    6. Applied associate-/l/0.0

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

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{\left(x \cdot a\right) \cdot \left(3 + 3\right)} - 1}}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}\]
    8. Using strategy rm

    9. Applied div-sub0.0

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

    if -0.0012473541376008107 < (/ (+ (* 18 (* (* a x) (* a x))) (+ (* (* 6 a) x) (* 36 (pow (* a x) 3)))) (* (+ (* (exp (* a x)) (exp (* a x))) (+ (* 1 1) (* (exp (* a x)) 1))) (+ (pow (exp (* a x)) 3) (pow 1 3)))) < 0.007531445725249147

    1. Initial program 44.1

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

    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. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(6 \cdot a\right) \cdot x + {\left(x \cdot a\right)}^{3} \cdot 36\right) + 18 \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} \le -0.0012473541376008107:\\ \;\;\;\;\frac{e^{\left(x \cdot a\right) \cdot \left(3 + 3\right)}}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} - \frac{1}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)}\\ \mathbf{elif}\;\frac{\left(\left(6 \cdot a\right) \cdot x + {\left(x \cdot a\right)}^{3} \cdot 36\right) + 18 \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} \le 0.007531445725249147:\\ \;\;\;\;\left(x \cdot a + \frac{1}{2} \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)\right) + {\left(x \cdot a\right)}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(x \cdot a\right) \cdot \left(3 + 3\right)}}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)} - \frac{1}{\left({1}^{3} + {\left(e^{x \cdot a}\right)}^{3}\right) \cdot \left(e^{x \cdot a} \cdot e^{x \cdot a} + \left(1 + e^{x \cdot a}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 3.9m)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))