Average Error: 29.1 → 0.2
Time: 3.0m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -5.4502547482117476 \cdot 10^{-05}:\\ \;\;\;\;(\left(\frac{-\left(1 + x\right)}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\right) \cdot \left((\left(1 + x\right) \cdot x + 1)_*\right) + \left(\frac{x}{1 + x}\right))_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 5.932170016532016 \cdot 10^{-13}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{(\left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) \cdot \left((x \cdot x + \left(1 - x\right))_*\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*} \cdot \left(\sqrt[3]{(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*} \cdot \sqrt[3]{(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < -5.4502547482117476e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied div-inv0.1

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

    6. Applied prod-diff0.1

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

    7. Applied simplify0.1

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

    8. Applied simplify0.1

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

    if -5.4502547482117476e-05 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < 5.932170016532016e-13

    1. Initial program 59.8

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

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

    if 5.932170016532016e-13 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x)))

    1. Initial program 0.6

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

    3. Applied flip3-+0.6

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

    4. Applied associate-/r/0.6

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

    5. Applied fma-neg0.6

      \[\leadsto \color{blue}{(\left(\frac{x}{{x}^{3} + {1}^{3}}\right) \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) + \left(-\frac{x + 1}{x - 1}\right))_*}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.7

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

    8. Applied simplify0.7

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

    9. Applied simplify0.7

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -5.4502547482117476 \cdot 10^{-05}:\\ \;\;\;\;(\left(\frac{-\left(1 + x\right)}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\right) \cdot \left((\left(1 + x\right) \cdot x + 1)_*\right) + \left(\frac{x}{1 + x}\right))_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 5.932170016532016 \cdot 10^{-13}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{(\left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) \cdot \left((x \cdot x + \left(1 - x\right))_*\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*} \cdot \left(\sqrt[3]{(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*} \cdot \sqrt[3]{(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*}\right)\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))