Average Error: 29.6 → 0.1
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) \cdot \left(\left(\frac{-1}{x \cdot x} - \frac{\frac{3}{x}}{x \cdot x}\right) - \frac{3}{x}\right)}{{\left(\sqrt[3]{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\right)}^{3}} \le -0.002072280133890457:\\ \;\;\;\;\frac{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}{\sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}} \cdot \frac{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}{\sqrt[3]{\left(\sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}}\\ \mathbf{if}\;\frac{\left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) \cdot \left(\left(\frac{-1}{x \cdot x} - \frac{\frac{3}{x}}{x \cdot x}\right) - \frac{3}{x}\right)}{{\left(\sqrt[3]{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\right)}^{3}} \le 0.00026690773176357957:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right) - \left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}{\left(\frac{x}{x + 1} + \frac{x + 1}{x - 1}\right) \cdot \left(\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(\left(x - 1\right) \cdot \left(x - 1\right)\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (/ (* (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (- (- (/ (- 1) (* x x)) (/ (/ 3 x) (* x x))) (/ 3 x))) (pow (cbrt (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x)))) 3)) < -0.002072280133890457

    1. Initial program 0.0

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

    3. Applied flip--0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.1

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

    6. Applied difference-of-squares0.1

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

    7. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}{\sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}} \cdot \frac{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}{\sqrt[3]{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.1

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

    if -0.002072280133890457 < (/ (* (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (- (- (/ (- 1) (* x x)) (/ (/ 3 x) (* x x))) (/ 3 x))) (pow (cbrt (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x)))) 3)) < 0.00026690773176357957

    1. Initial program 59.3

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

    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 simplify0.0

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]

    if 0.00026690773176357957 < (/ (* (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (- (- (/ (- 1) (* x x)) (/ (/ 3 x) (* x x))) (/ 3 x))) (pow (cbrt (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x)))) 3))

    1. Initial program 0.1

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

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied frac-times0.1

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

    6. Applied frac-times0.1

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

    7. Applied frac-sub0.1

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

    8. Applied associate-/l/0.1

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))