Average Error: 13.7 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -8.79840120370125 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F}}{\sin B} - \left(\frac{x}{\tan B} + \frac{1}{\sin B}\right)\\ \mathbf{if}\;F \le 3.260214795753425 \cdot 10^{+28}:\\ \;\;\;\;\frac{F}{{\left((F \cdot F + \left((2 \cdot x + 2)_*\right))_*\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B} + \frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x}{\tan B}\right) - \frac{\frac{\frac{1}{F}}{F}}{\sin B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -8.79840120370125e+38

    1. Initial program 27.6

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

    2. Applied simplify27.6

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

    4. Applied pow-neg27.6

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

    6. Applied fma-udef27.6

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

    7. Applied simplify27.6

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

    8. Taylor expanded around -inf 0.1

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

    9. Applied simplify0.1

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

    if -8.79840120370125e+38 < F < 3.260214795753425e+28

    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 pow-neg0.4

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

    6. Applied fma-udef0.4

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

    7. Applied simplify0.4

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

    9. Applied div-inv0.4

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

    10. Applied associate-/l*0.3

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

    11. Applied simplify0.3

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

    if 3.260214795753425e+28 < F

    1. Initial program 26.8

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

    2. Applied simplify26.7

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

    4. Applied pow-neg26.7

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

    6. Applied fma-udef26.7

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

    7. Applied simplify26.7

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

    8. Taylor expanded around inf 0.2

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

    9. Applied simplify0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018166 +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))))))