Average Error: 14.0 → 0.5
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -3.046573298696304 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{F}}{\sin B} - \left(\frac{1}{\sin B} + \frac{x}{\tan B}\right)\\ \mathbf{if}\;F \le 1.644625965014904:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{x}{\tan B}\right) - \frac{\frac{1}{\sin B}}{F \cdot F}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -3.046573298696304e+83

    1. Initial program 31.5

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

    2. Applied simplify31.5

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

    3. Taylor expanded around -inf 14.9

      \[\leadsto \color{blue}{\left(\frac{1}{{F}^{3}} - \frac{1}{F}\right)} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\]

    4. Applied simplify0.2

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

    if -3.046573298696304e+83 < F < 1.644625965014904

    1. Initial program 0.9

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

    2. Applied simplify0.7

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

    4. Applied tan-quot0.8

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

    5. Applied associate-/r/0.8

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

    if 1.644625965014904 < 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. Applied simplify24.6

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

    3. Taylor expanded around inf 12.1

      \[\leadsto \color{blue}{\left(\frac{1}{F} - \frac{1}{{F}^{3}}\right)} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\]

    4. Applied simplify0.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2019053 
(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))))))