Average Error: 39.9 → 0.5
Time: 26.9s
Precision: 64
Internal Precision: 1408
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right) + x}}{x} \cdot \left(\frac{1}{12} \cdot \sqrt[3]{{x}^{8}} + \left(\log \left(e^{\sqrt[3]{x \cdot x}}\right) + \frac{1}{3} \cdot \sqrt[3]{{x}^{5}}\right)\right) \le 1.014607935005169:\\ \;\;\;\;\frac{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} - 1}{e^{x + x} + \left(e^{x} + 1\right)}}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original39.9
Target39.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (cbrt (+ (* (+ 1/2 (* 1/6 x)) (* x x)) x)) x) (+ (* 1/12 (cbrt (pow x 8))) (+ (log (exp (cbrt (* x x)))) (* 1/3 (cbrt (pow x 5)))))) < 1.014607935005169

    1. Initial program 60.1

      \[\frac{e^{x} - 1}{x}\]

    2. Taylor expanded around 0 0.3

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

    3. Applied simplify0.3

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

    if 1.014607935005169 < (* (/ (cbrt (+ (* (+ 1/2 (* 1/6 x)) (* x x)) x)) x) (+ (* 1/12 (cbrt (pow x 8))) (+ (log (exp (cbrt (* x x)))) (* 1/3 (cbrt (pow x 5))))))

    1. Initial program 0.6

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

    3. Applied flip3--0.7

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

    4. Applied simplify0.7

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

    5. Applied simplify0.7

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

Runtime

Time bar (total: 26.9s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))