Average Error: 13.5 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -3.744518881504563 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{\left(F \cdot F\right) \cdot \sin B} - (\left(\frac{\cos B}{\sin B}\right) \cdot x + \left(\frac{1}{\sin B}\right))_*\\ \mathbf{if}\;F \le 14452.705243293623:\\ \;\;\;\;F \cdot \frac{\frac{1}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{1}{{F}^{2} \cdot \sin B}\right) - \frac{\cos B}{\frac{\sin B}{x}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -3.744518881504563e+43

    1. Initial program 28.0

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Applied simplify28.0

      \[\leadsto \color{blue}{(\left({\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\]
    3. Using strategy rm

    4. Applied pow-neg28.0

      \[\leadsto (\color{blue}{\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right)} \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

    5. Taylor expanded around inf 28.0

      \[\leadsto (\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \color{blue}{\left(-1 \cdot \frac{\cos B \cdot x}{\sin B}\right)})_*\]

    6. Applied simplify28.0

      \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}}\]

    7. Taylor expanded around -inf 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{{F}^{2} \cdot \sin B} - \frac{1}{\sin B}\right)} - \frac{\cos B}{\frac{\sin B}{x}}\]

    8. Applied simplify0.2

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

    if -3.744518881504563e+43 < F < 14452.705243293623

    1. Initial program 0.4

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Applied simplify0.3

      \[\leadsto \color{blue}{(\left({\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\]
    3. Using strategy rm

    4. Applied pow-neg0.3

      \[\leadsto (\color{blue}{\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right)} \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

    5. Taylor expanded around inf 0.4

      \[\leadsto (\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \color{blue}{\left(-1 \cdot \frac{\cos B \cdot x}{\sin B}\right)})_*\]

    6. Applied simplify0.4

      \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity0.4

      \[\leadsto \frac{\frac{F}{\sin B}}{{\color{blue}{\left(1 \cdot (x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}\]

    9. Applied unpow-prod-down0.4

      \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{{1}^{\left(\frac{1}{2}\right)} \cdot {\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}} - \frac{\cos B}{\frac{\sin B}{x}}\]

    10. Applied div-inv0.4

      \[\leadsto \frac{\color{blue}{F \cdot \frac{1}{\sin B}}}{{1}^{\left(\frac{1}{2}\right)} \cdot {\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}\]

    11. Applied times-frac0.4

      \[\leadsto \color{blue}{\frac{F}{{1}^{\left(\frac{1}{2}\right)}} \cdot \frac{\frac{1}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}} - \frac{\cos B}{\frac{\sin B}{x}}\]

    12. Applied simplify0.4

      \[\leadsto \color{blue}{F} \cdot \frac{\frac{1}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}\]

    if 14452.705243293623 < F

    1. Initial program 24.5

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

    2. Applied simplify24.4

      \[\leadsto \color{blue}{(\left({\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*}\]
    3. Using strategy rm

    4. Applied pow-neg24.4

      \[\leadsto (\color{blue}{\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right)} \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{-x}{\tan B}\right))_*\]

    5. Taylor expanded around inf 24.4

      \[\leadsto (\left(\frac{1}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(\frac{F}{\sin B}\right) + \color{blue}{\left(-1 \cdot \frac{\cos B \cdot x}{\sin B}\right)})_*\]

    6. Applied simplify24.4

      \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} - \frac{\cos B}{\frac{\sin B}{x}}}\]

    7. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - \frac{1}{{F}^{2} \cdot \sin B}\right)} - \frac{\cos B}{\frac{\sin B}{x}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))