Average Error: 13.2 → 0.2
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -6.91821836359706 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{1}{F \cdot F}}{\sin B} - \left(\frac{x}{\tan B} + \frac{1}{\sin B}\right)\\ \mathbf{if}\;F \le 3.086982648980967 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot {\left((x \cdot 2 + \left((F \cdot F + 2)_*\right))_*\right)}^{\left(\frac{-1}{2}\right)}\right) + \frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x}{\tan B}\right) - \frac{\frac{\frac{1}{F}}{F}}{\sin B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -6.91821836359706e+45

    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 simplify26.8

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

    4. Applied clear-num26.8

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

    6. Applied fma-udef26.8

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

    7. Applied simplify26.2

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

    8. Taylor expanded around -inf 0.2

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

    9. Applied simplify0.2

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

    if -6.91821836359706e+45 < F < 3.086982648980967e+29

    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 clear-num0.4

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

    6. Applied fma-udef0.4

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

    7. Applied simplify0.3

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

    9. Applied div-inv0.4

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

    10. Applied *-un-lft-identity0.4

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

    11. Applied unpow-prod-down0.4

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

    12. Applied times-frac0.3

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

    13. Applied simplify0.3

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

    14. Applied simplify0.3

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

    if 3.086982648980967e+29 < F

    1. Initial program 26.7

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

    2. 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 clear-num26.7

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

    6. Applied fma-udef26.7

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

    7. Applied simplify25.9

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

    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: 1.3m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +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))))))