Average Error: 29.1 → 1.0
Time: 2.2m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\frac{\frac{2}{3}}{{x}^{3}} \cdot \left(\frac{\frac{27}{4}}{x} - \frac{9}{2}\right)}{\frac{\frac{x}{\frac{2}{3}} \cdot {x}^{3}}{\frac{\frac{2}{3}}{{x}^{3}}}} + {\left(\frac{\frac{2}{3}}{{x}^{3}}\right)}^{3}}}{2} \le -312466.8822044239:\\ \;\;\;\;\frac{\sqrt[3]{\frac{{\left(\left(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x - 2\right) \cdot \left(x \cdot x - 2\right)\right)}^{3}}{{\left(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + \left(x \cdot x - 2\right)\right)}^{3}}}}{2}\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{\frac{\frac{2}{3}}{{x}^{3}} \cdot \left(\frac{\frac{27}{4}}{x} - \frac{9}{2}\right)}{\frac{\frac{x}{\frac{2}{3}} \cdot {x}^{3}}{\frac{\frac{2}{3}}{{x}^{3}}}} + {\left(\frac{\frac{2}{3}}{{x}^{3}}\right)}^{3}}}{2} \le 3.13902213637179 \cdot 10^{-08}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\log \left(e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{8}^{3} - {\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)}^{3}}}{\sqrt[3]{8 \cdot 8 + \left(\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) + 8 \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)\right)}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2) < -312466.8822044239

    1. Initial program 38.1

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

    2. Taylor expanded around 0 0.7

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

    4. Applied add-cbrt-cube0.7

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

    5. Applied simplify0.7

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

    7. Applied flip--0.7

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

    8. Applied cube-div0.7

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

    if -312466.8822044239 < (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2) < 3.13902213637179e-08

    1. Initial program 1.1

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

    3. Applied add-log-exp1.1

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

    if 3.13902213637179e-08 < (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2)

    1. Initial program 38.3

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cbrt-cube1.3

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

    5. Applied simplify1.3

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 + 8 \cdot {x}^{3}\right) - 12 \cdot {x}^{2}}}}{2}\]

    7. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{8 - \left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)}}{2}}\]
    8. Using strategy rm

    9. Applied flip3--1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{{8}^{3} - {\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)}^{3}}{8 \cdot 8 + \left(\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) + 8 \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)\right)}}}}{2}\]

    10. Applied cbrt-div1.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{{8}^{3} - {\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)}^{3}}}{\sqrt[3]{8 \cdot 8 + \left(\left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right) + 8 \cdot \left(\left(x \cdot x\right) \cdot \left(12 - 8 \cdot x\right)\right)\right)}}}}{2}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.2m)Debug log

herbie shell --seed '#(1151762963 887253659 3096734101 777879090 2714024476 786371635)' 
(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))