Average Error: 29.9 → 0.3
Time: 34.6s
Precision: 64
Internal Precision: 1408
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\sqrt{e^{x \cdot a}} + 1} \cdot \left(\sqrt{\sqrt{e^{a \cdot x}} + 1} \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \left(a \cdot x\right)\right)\right)\right) \le 0.00549780709177216:\\ \;\;\;\;\left(\left|\sqrt[3]{e^{a \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{x \cdot a}}} + 1\right) \cdot \left(\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{8} + a \cdot \left(\frac{1}{2} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)} \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\right) \cdot \sqrt[3]{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus x

Target

Original29.9
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (sqrt (+ (sqrt (exp (* x a))) 1)) (* (sqrt (+ (sqrt (exp (* a x))) 1)) (* (* a x) (+ 1/2 (* 1/8 (* a x)))))) < 0.00549780709177216

    1. Initial program 44.6

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

    3. Applied add-sqr-sqrt44.7

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

    4. Applied difference-of-sqr-144.7

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

    5. Taylor expanded around 0 45.8

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

    6. Applied simplify0.4

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

    8. Applied add-cube-cbrt0.4

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

    9. Applied sqrt-prod0.4

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

    10. Applied simplify0.4

      \[\leadsto \left(\color{blue}{\left|\sqrt[3]{e^{a \cdot x}}\right|} \cdot \sqrt{\sqrt[3]{e^{x \cdot a}}} + 1\right) \cdot \left(\left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right) \cdot \frac{1}{8} + a \cdot \left(\frac{1}{2} \cdot x\right)\right)\]

    if 0.00549780709177216 < (* (sqrt (+ (sqrt (exp (* x a))) 1)) (* (sqrt (+ (sqrt (exp (* a x))) 1)) (* (* a x) (+ 1/2 (* 1/8 (* a x))))))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied difference-of-sqr-10.1

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

    6. Applied add-cube-cbrt0.1

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

Runtime

Time bar (total: 34.6s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))