Average Error: 29.4 → 0.1
Time: 46.0s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*}} \cdot \left(\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*} \cdot \sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*}\right)\right) \cdot \sqrt[3]{\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*} \cdot (\left(\frac{\sqrt[3]{-3}}{x} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\frac{\frac{26}{81}}{x} + \frac{1}{9}\right) + \left(\sqrt[3]{-3 \cdot \frac{1}{x}}\right))_*} \le -0.0042552816706777725:\\ \;\;\;\;(x \cdot \left(\frac{1}{x + 1}\right) + \left(-\frac{x + 1}{x - 1}\right))_*\\ \mathbf{if}\;\left(\sqrt[3]{\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*}} \cdot \left(\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*} \cdot \sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*}\right)\right) \cdot \sqrt[3]{\sqrt[3]{(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*} \cdot (\left(\frac{\sqrt[3]{-3}}{x} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \left(\frac{\frac{26}{81}}{x} + \frac{1}{9}\right) + \left(\sqrt[3]{-3 \cdot \frac{1}{x}}\right))_*} \le 0.00030879751590440435:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (* (* (cbrt (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)))) (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))))) (cbrt (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (fma (* (/ (cbrt -3) x) (cbrt (/ 1 x))) (+ (/ 26/81 x) 1/9) (cbrt (* -3 (/ 1 x))))))) < -0.0042552816706777725

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto \color{blue}{x \cdot \frac{1}{x + 1}} - \frac{x + 1}{x - 1}\]

    4. Applied fma-neg0.1

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

    if -0.0042552816706777725 < (* (* (cbrt (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)))) (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))))) (cbrt (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (fma (* (/ (cbrt -3) x) (cbrt (/ 1 x))) (+ (/ 26/81 x) 1/9) (cbrt (* -3 (/ 1 x))))))) < 0.00030879751590440435

    1. Initial program 59.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.4

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

    3. Applied simplify0.1

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

    if 0.00030879751590440435 < (* (* (cbrt (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)))) (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))))) (cbrt (* (cbrt (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x))) (fma (* (/ (cbrt -3) x) (cbrt (/ 1 x))) (+ (/ 26/81 x) 1/9) (cbrt (* -3 (/ 1 x)))))))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.2

      \[\leadsto \color{blue}{\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 46.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))