Average Error: 29.2 → 1.2
Time: 2.9m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le -3.60230005474265 \cdot 10^{-310} \lor \neg \left(\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le 1.0224609375001041\right):\\ \;\;\;\;\frac{\left(e^{\log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right)\right) + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2}\\ \end{array}\]

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) < -3.60230005474265e-310 or 1.0224609375001041 < (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (expm1 (log1p (/ (- (/ 1 eps) 1) (exp (fma eps x x)))))) 2)

    1. Initial program 57.4

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

    2. Taylor expanded around 0 1.6

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

    4. Applied add-cube-cbrt1.6

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

    6. Applied add-exp-log1.6

      \[\leadsto \frac{\left(2 + \color{blue}{e^{\log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.6

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

    if -3.60230005474265e-310 < (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (expm1 (log1p (/ (- (/ 1 eps) 1) (exp (fma eps x x)))))) 2) < 1.0224609375001041

    1. Initial program 0.2

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

    3. Applied expm1-log1p-u0.8

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

    4. Applied simplify0.8

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

  4. Applied simplify1.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le -3.60230005474265 \cdot 10^{-310} \lor \neg \left(\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2} \le 1.0224609375001041\right):\\ \;\;\;\;\frac{\left(e^{\log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right)\right) + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - (e^{\log_* (1 + \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}})} - 1)^*}{2}\\ \end{array}}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018170 +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))