Average Error: 13.4 → 0.3
Time: 54.2s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -7.174018771573197 \cdot 10^{+61}:\\ \;\;\;\;\left(\frac{1}{{F}^{2}} - 1\right) \cdot \frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{elif}\;F \le 40015693.24407982:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - \frac{1}{{F}^{2}}\right) - \cos B \cdot \frac{x}{\sin B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -7.174018771573197e+61

    1. Initial program 28.7

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

    2. Initial simplification28.7

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

    4. Applied div-inv28.7

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

    5. Applied associate-*r*22.9

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

    7. Applied tan-quot23.0

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

    8. Applied associate-/r/23.0

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

    9. Taylor expanded around -inf 0.2

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

    if -7.174018771573197e+61 < F < 40015693.24407982

    1. Initial program 0.6

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

    2. Initial simplification0.5

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

    4. Applied div-inv0.5

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

    5. Applied associate-*r*0.3

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

    7. Applied un-div-inv0.3

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

    9. Applied div-inv0.4

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

    if 40015693.24407982 < F

    1. Initial program 24.6

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

    2. Initial simplification24.6

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

    4. Applied div-inv24.6

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

    5. Applied associate-*r*19.7

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

    7. Applied tan-quot19.7

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

    8. Applied associate-/r/19.7

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

    9. Taylor expanded around inf 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -7.174018771573197 \cdot 10^{+61}:\\ \;\;\;\;\left(\frac{1}{{F}^{2}} - 1\right) \cdot \frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{elif}\;F \le 40015693.24407982:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - \frac{1}{{F}^{2}}\right) - \cos B \cdot \frac{x}{\sin B}\\ \end{array}\]

Runtime

Time bar (total: 54.2s)Debug logProfile

herbie shell --seed 2018348 
(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))))))