Average Error: 13.7 → 2.2
Time: 12.5s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -6.712874546376408 \cdot 10^{187}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)} - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 1.3264063080845111 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} + 1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -6.712874546376408e187

    1. Initial program 43.2

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

    2. Simplified43.2

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

    4. Applied associate-*l/37.0

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

    6. Applied associate-*r/37.0

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

    8. Applied pow-neg37.0

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

    9. Applied un-div-inv37.0

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

    10. Applied associate-/l/37.0

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

    11. Taylor expanded around -inf 4.0

      \[\leadsto \frac{F}{\sin B \cdot \color{blue}{\left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)}} - \frac{x \cdot 1}{\tan B}\]

    if -6.712874546376408e187 < F < 1.3264063080845111e154

    1. Initial program 3.8

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

    2. Simplified3.8

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

    4. Applied associate-*l/1.6

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

    6. Applied pow-neg1.6

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

    7. Applied un-div-inv1.6

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

    if 1.3264063080845111e154 < F

    1. Initial program 43.2

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

    2. Simplified43.2

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

    4. Applied associate-*l/37.5

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

    6. Applied associate-*r/37.5

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

    8. Applied pow-neg37.5

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

    9. Applied un-div-inv37.5

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

    10. Applied associate-/l/37.5

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

    11. Taylor expanded around inf 4.0

      \[\leadsto \frac{F}{\sin B \cdot \color{blue}{\left(e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} + 1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)}} - \frac{x \cdot 1}{\tan B}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -6.712874546376408 \cdot 10^{187}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)} - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 1.3264063080845111 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{F}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} + 1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (neg (/ 1.0 2.0))))))