Average Error: 29.3 → 1.4
Time: 3.4m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le -171798691840.0:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le 1.0001530382339117:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (expm1 (log1p (/ (- (/ 1 eps) 1) (exp (fma eps x x)))))) 2) < -171798691840.0 or 1.0001530382339117 < (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (expm1 (log1p (/ (- (/ 1 eps) 1) (exp (fma eps x x)))))) 2)

    1. Initial program 56.8

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.7

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

    if -171798691840.0 < (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (expm1 (log1p (/ (- (/ 1 eps) 1) (exp (fma eps x x)))))) 2) < 1.0001530382339117

    1. Initial program 0.6

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u1.0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{(e^{\log_* (1 + \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x})} - 1)^*}}{2}\]

    4. Applied simplify1.0

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))