Average Error: 11.3 → 1.2
Time: 20.2s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -9.6213620448187684 \cdot 10^{158}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 3.0381154039953365 \cdot 10^{151}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} + 1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -9.6213620448187684e158

    1. Initial program 37.7

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

    2. Simplified37.7

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

    4. Applied pow-neg37.7

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

    5. Applied frac-times33.7

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

    6. Simplified33.7

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

    7. Taylor expanded around -inf 3.6

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

    if -9.6213620448187684e158 < F < 3.0381154039953365e151

    1. Initial program 2.6

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

    2. Simplified2.6

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

    4. Applied pow-neg2.6

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

    5. Applied frac-times0.5

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

    6. Simplified0.5

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

    8. Applied associate-*r/0.4

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

    if 3.0381154039953365e151 < F

    1. Initial program 38.2

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

    2. Simplified38.2

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

    4. Applied pow-neg38.2

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

    5. Applied frac-times34.8

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

    6. Simplified34.8

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

    7. Taylor expanded around inf 3.8

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -9.6213620448187684 \cdot 10^{158}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}}{{F}^{2}} + e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{-1}{F}\right)\right)}\right)} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 3.0381154039953365 \cdot 10^{151}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)} + 1 \cdot \frac{e^{0.5 \cdot \left(\log 1 - 2 \cdot \log \left(\frac{1}{F}\right)\right)}}{{F}^{2}}\right)} - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))))