Average Error: 29.6 → 25.9
Time: 5.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 0.9068563409748999:\\ \;\;\;\;\frac{(\left(e^{-x}\right) \cdot \left(e^{x \cdot \varepsilon}\right) + \left((\left(e^{(\varepsilon \cdot \left(-x\right) + \left(-x\right))_*}\right) \cdot \left(\frac{\sqrt[3]{-1}}{\varepsilon} + \sqrt[3]{-1}\right) + \left(\frac{e^{x \cdot \varepsilon}}{e^{x} \cdot \varepsilon}\right))_*\right))_*}{2}\\ \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 1.995381120183222 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (/ (/ (fma (/ 1 eps) (exp (fma eps x x)) (fma (- 1 (/ 1 eps)) (pow (exp x) (- 1 eps)) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2) < 0.9068563409748999

    1. Initial program 61.6

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

    3. Applied add-cube-cbrt61.5

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

    4. Applied simplify61.5

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}\right)} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\]

    5. Applied simplify61.5

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}}{2}\]

    6. Taylor expanded around inf 62.0

      \[\leadsto \frac{\color{blue}{\frac{1}{e^{(\varepsilon \cdot x + x)_*}} \cdot {-1}^{\frac{1}{3}} + \left(e^{\varepsilon \cdot x - x} + \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + \frac{1}{e^{(\varepsilon \cdot x + x)_*} \cdot \varepsilon} \cdot {-1}^{\frac{1}{3}}\right)\right)}}{2}\]

    7. Applied simplify52.4

      \[\leadsto \color{blue}{\frac{(\left(e^{-x}\right) \cdot \left(e^{x \cdot \varepsilon}\right) + \left((\left(e^{(\varepsilon \cdot \left(-x\right) + \left(-x\right))_*}\right) \cdot \left(\frac{\sqrt[3]{-1}}{\varepsilon} + \sqrt[3]{-1}\right) + \left(\frac{e^{x \cdot \varepsilon}}{\varepsilon \cdot e^{x}}\right))_*\right))_*}{2}}\]

    if 0.9068563409748999 < (/ (/ (fma (/ 1 eps) (exp (fma eps x x)) (fma (- 1 (/ 1 eps)) (pow (exp x) (- 1 eps)) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2) < 1.995381120183222e+223

    1. Initial program 2.4

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

    3. Applied exp-neg2.4

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

    4. Applied un-div-inv2.4

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

    5. Applied exp-neg2.4

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

    6. Applied associate-*r/2.4

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

    7. Applied frac-sub2.5

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

    8. Applied simplify4.6

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

    9. Applied simplify4.6

      \[\leadsto \frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{\color{blue}{e^{(\left(1 - 0\right) \cdot x + x)_*}}}}{2}\]

    if 1.995381120183222e+223 < (/ (/ (fma (/ 1 eps) (exp (fma eps x x)) (fma (- 1 (/ 1 eps)) (pow (exp x) (- 1 eps)) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2)

    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-sqr-sqrt1.1

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

    4. Applied simplify1.1

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

    5. Applied simplify1.1

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

  4. Applied simplify25.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 0.9068563409748999:\\ \;\;\;\;\frac{(\left(e^{-x}\right) \cdot \left(e^{x \cdot \varepsilon}\right) + \left((\left(e^{(\varepsilon \cdot \left(-x\right) + \left(-x\right))_*}\right) \cdot \left(\frac{\sqrt[3]{-1}}{\varepsilon} + \sqrt[3]{-1}\right) + \left(\frac{e^{x \cdot \varepsilon}}{e^{x} \cdot \varepsilon}\right))_*\right))_*}{2}\\ \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 1.995381120183222 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018195 +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))