Average Error: 58.2 → 7.7
Time: 2.3m
Precision: 64
Internal Precision: 2368
\[\frac{\sqrt{y1 \cdot y1 + x1 \cdot x1} \cdot \sin \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\pi - 2 \cdot \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}\]
\[\begin{array}{l} \mathbf{if}\;x1 \le -7.560066695063207 \cdot 10^{+105}:\\ \;\;\;\;\frac{\left(-x1\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right) \cdot 2\right)}\\ \mathbf{if}\;x1 \le 1.4695493763731748 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{x1 \cdot x1 + y1 \cdot y1} \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\left(\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot x1}{\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right) \cdot 2\right)}\\ \end{array}\]

Error

Bits error versus y1

Bits error versus x1

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if x1 < -7.560066695063207e+105

    1. Initial program 58.7

      \[\frac{\sqrt{y1 \cdot y1 + x1 \cdot x1} \cdot \sin \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\pi - 2 \cdot \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}\]
    2. Applied simplify58.6

      \[\leadsto \color{blue}{\frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\sin \left(\pi - 2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    3. Using strategy rm
    4. Applied sub-neg58.6

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\sin \color{blue}{\left(\pi + \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}}\]
    5. Applied sin-sum58.6

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\color{blue}{\sin \pi \cdot \cos \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right) + \cos \pi \cdot \sin \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    6. Applied simplify48.2

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\color{blue}{0} + \cos \pi \cdot \sin \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}\]
    7. Applied simplify48.2

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{0 + \color{blue}{\sin \left(2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    8. Taylor expanded around -inf 5.4

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot x1}{\sin \left(2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    9. Applied simplify5.4

      \[\leadsto \color{blue}{\frac{\left(-x1\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]

    if -7.560066695063207e+105 < x1 < 1.4695493763731748e+127

    1. Initial program 57.9

      \[\frac{\sqrt{y1 \cdot y1 + x1 \cdot x1} \cdot \sin \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\pi - 2 \cdot \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}\]
    2. Applied simplify57.6

      \[\leadsto \color{blue}{\frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\sin \left(\pi - 2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    3. Using strategy rm
    4. Applied sub-neg57.6

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\sin \color{blue}{\left(\pi + \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}}\]
    5. Applied sin-sum57.6

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\color{blue}{\sin \pi \cdot \cos \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right) + \cos \pi \cdot \sin \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    6. Applied simplify9.4

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\color{blue}{0} + \cos \pi \cdot \sin \left(-2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}\]
    7. Applied simplify9.4

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{0 + \color{blue}{\sin \left(2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    8. Using strategy rm
    9. Applied sin-29.4

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{0 + \color{blue}{2 \cdot \left(\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}}\]

    if 1.4695493763731748e+127 < x1

    1. Initial program 58.9

      \[\frac{\sqrt{y1 \cdot y1 + x1 \cdot x1} \cdot \sin \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\pi - 2 \cdot \left(\frac{\pi}{2} - \tan^{-1} \left(\frac{x1}{y1}\right)\right)\right)}\]
    2. Applied simplify58.9

      \[\leadsto \color{blue}{\frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \sqrt{y1 \cdot y1 + x1 \cdot x1}}{\sin \left(\pi - 2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}}\]
    3. Taylor expanded around 0 55.7

      \[\leadsto \frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \color{blue}{x1}}{\sin \left(\pi - 2 \cdot \tan^{-1} \left(\frac{x1}{y1}\right)\right)}\]
    4. Applied simplify4.6

      \[\leadsto \color{blue}{\frac{x1 \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right) \cdot 2\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Applied simplify7.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x1 \le -7.560066695063207 \cdot 10^{+105}:\\ \;\;\;\;\frac{\left(-x1\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right) \cdot 2\right)}\\ \mathbf{if}\;x1 \le 1.4695493763731748 \cdot 10^{+127}:\\ \;\;\;\;\frac{\sqrt{x1 \cdot x1 + y1 \cdot y1} \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)}{\left(\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot \cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\tan^{-1} \left(\frac{x1}{y1}\right)\right) \cdot x1}{\sin \left(\tan^{-1} \left(\frac{x1}{y1}\right) \cdot 2\right)}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (y1 x1)
  :name "(sqrt(y1 * y1 + x1 * x1) * sin((PI/2) - atan(x1 / y1))) / (sin(PI - 2 * ((PI/2) - atan(x1 / y1))))"
  (/ (* (sqrt (+ (* y1 y1) (* x1 x1))) (sin (- (/ PI 2) (atan (/ x1 y1))))) (sin (- PI (* 2 (- (/ PI 2) (atan (/ x1 y1))))))))