Average Error: 13.7 → 0.2
Time: 33.6s
Precision: 64
Internal Precision: 128
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -4.830867026911197 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 10594.61697550749:\\ \;\;\;\;\left({\left(\sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}}\right) \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x}{\tan B}\\ \end{array}\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if F < -4.830867026911197e+28

    1. Initial program 25.9

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

    2. Simplified25.8

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

    3. Taylor expanded around -inf 0.1

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

    4. Simplified0.1

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

    if -4.830867026911197e+28 < F < 10594.61697550749

    1. Initial program 0.4

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

    2. Simplified0.3

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

    4. Applied add-sqr-sqrt0.3

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

    5. Applied unpow-prod-down0.3

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

    if 10594.61697550749 < F

    1. Initial program 25.3

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

    2. Simplified25.2

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4.830867026911197 \cdot 10^{+28}:\\ \;\;\;\;\left(\frac{1}{\sin B \cdot \left(F \cdot F\right)} - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \le 10594.61697550749:\\ \;\;\;\;\left({\left(\sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}} \cdot {\left(\sqrt{x \cdot 2 + \left(2 + F \cdot F\right)}\right)}^{\frac{-1}{2}}\right) \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \frac{\frac{1}{F \cdot F}}{\sin B}\right) - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(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))))))