Average Error: 13.7 → 0.7
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -9.352880665213902 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\tan B} \cdot \left(-x\right) + \frac{\frac{1}{F \cdot F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 5.770972431265831 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} + \frac{1}{\tan B} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} + \frac{-x}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -9.352880665213902e+55

    1. Initial program 29.6

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -9.352880665213902e+55 < F < 5.770972431265831e-09

    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. Using strategy rm

    3. Applied pow-neg0.6

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

    4. Applied un-div-inv0.5

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

    5. Simplified0.5

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

    if 5.770972431265831e-09 < F

    1. Initial program 23.3

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

    3. Applied pow-neg23.3

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

    4. Applied frac-times18.7

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

    5. Simplified18.7

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

    6. Simplified18.7

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

    8. Applied un-div-inv18.6

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

    9. Taylor expanded around inf 1.3

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

    10. Simplified1.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -9.352880665213902 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\tan B} \cdot \left(-x\right) + \frac{\frac{1}{F \cdot F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 5.770972431265831 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)}} + \frac{1}{\tan B} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} + \frac{-x}{\tan B}\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018214 +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))))))