Average Error: 13.7 → 0.5
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -7.393823576386876 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\tan B} \cdot \left(-x\right) + \frac{\frac{1}{F \cdot F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 63313.348454549254:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} + \frac{1}{\tan B} \cdot \left(-x\right)\\ \end{array}\]

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 < -7.393823576386876e+159

    1. Initial program 42.4

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -7.393823576386876e+159 < F < 63313.348454549254

    1. Initial program 2.2

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

    3. Applied pow-neg2.2

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

    4. Applied frac-times0.8

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

    5. Simplified0.8

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

    7. Applied un-div-inv0.7

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

    9. Applied clear-num0.8

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

    if 63313.348454549254 < F

    1. Initial program 24.1

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -7.393823576386876 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\tan B} \cdot \left(-x\right) + \frac{\frac{1}{F \cdot F} - 1}{\sin B}\\ \mathbf{elif}\;F \le 63313.348454549254:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{1}{2}\right)}} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{F \cdot F}}{\sin B} + \frac{1}{\tan B} \cdot \left(-x\right)\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018214 
(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))))))