Average Error: 39.8 → 0.2
Time: 29.8s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{12} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right) \cdot \left(\frac{1}{36} \cdot {x}^{2} + \left(1 + \frac{1}{6} \cdot x\right)\right) \le 1.0285471815570788:\\ \;\;\;\;\left(\frac{1}{12} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right) \cdot \left(\frac{1}{36} \cdot {x}^{2} + \left(1 + \frac{1}{6} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \end{array}\]

Error

Bits error versus x

Target

Original39.8
Target39.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (+ (* 1/12 (pow x 2)) (+ 1 (* 1/3 x))) (+ (* 1/36 (pow x 2)) (+ 1 (* 1/6 x)))) < 1.0285471815570788

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.4

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

    4. Applied add-cube-cbrt0.4

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

    5. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right)} \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(1 + \frac{1}{2} \cdot x\right)}\]

    6. Taylor expanded around 0 0.4

      \[\leadsto \left(\frac{1}{12} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right) \cdot \color{blue}{\left(\frac{1}{36} \cdot {x}^{2} + \left(1 + \frac{1}{6} \cdot x\right)\right)}\]

    if 1.0285471815570788 < (* (+ (* 1/12 (pow x 2)) (+ 1 (* 1/3 x))) (+ (* 1/36 (pow x 2)) (+ 1 (* 1/6 x))))

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

Runtime

Time bar (total: 29.8s)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(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))