Average Error: 13.8 → 0.3
Time: 53.0s
Precision: 64
Internal Precision: 384
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -9.76446544682648 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{\sin B}}{F} - (\left(\frac{x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_*\\ \mathbf{if}\;F \le 1778.2395190060415:\\ \;\;\;\;(\left({\left((F \cdot F + \left((2 \cdot x + 2)_*\right))_*\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \left(\frac{F}{\sin B}\right) + \left(\frac{\cos B \cdot \left(-x\right)}{\sin B}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-x}{\sin B}\right) \cdot \left(\cos B\right) + \left(\frac{1}{\sin B}\right))_* - \frac{\frac{1}{F \cdot F}}{\sin B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -9.76446544682648e+62

    1. Initial program 29.9

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

    2. Applied simplify29.8

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

    4. Applied div-inv29.8

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

    5. Taylor expanded around inf 29.8

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

    6. Applied simplify29.8

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

    7. Taylor expanded around -inf 15.0

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

    8. Applied simplify0.2

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

    if -9.76446544682648e+62 < F < 1778.2395190060415

    1. Initial program 0.5

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

    2. Applied simplify0.4

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

    4. Applied div-inv0.4

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

    5. Taylor expanded around inf 0.5

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

    6. Applied simplify0.5

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

    if 1778.2395190060415 < F

    1. Initial program 24.8

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

    2. Applied simplify24.7

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

    4. Applied div-inv24.7

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

    5. Taylor expanded around inf 24.8

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

    6. Applied simplify24.8

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

    7. Taylor expanded around inf 11.7

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

    8. Applied simplify0.2

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

Runtime

Time bar (total: 53.0s)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +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))))))