Average Error: 13.5 → 10.6
Time: 13.7s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\]
\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)}
↓
F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}
(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
↓
(FPCore (F B x)
 :precision binary64
 (- (* F (/ (pow (+ (+ (* F F) 2.0) (* 2.0 x)) -0.5) (sin B))) (/ x (tan B))))
double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}
↓
double code(double F, double B, double x) {
	return (F * (pow((((F * F) + 2.0) + (2.0 * x)), -0.5) / sin(B))) - (x / tan(B));
}

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. Initial program 13.5

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

  2. Simplified13.5

    \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
  3. Using strategy rm

  4. Applied div-inv_binary6413.5

    \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\]

  5. Applied associate-*l*_binary6410.6

    \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\right)} - \frac{x}{\tan B}\]

  6. Simplified10.6

    \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B}\]

  7. Final simplification10.6

    \[\leadsto F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\]

Reproduce

herbie shell --seed 2020277 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))