Average Error: 0.4 → 0.2
Time: 3.5s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{\pi}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{\pi}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))
↓
(FPCore (v t)
 :precision binary64
 (*
  (/
   (/
    (/ (sqrt (- 1.0 (* 5.0 (* v v)))) (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0)))))
    PI)
   t)
  (/ (sqrt (- 1.0 (* 5.0 (* v v)))) (- 1.0 (* v v)))))
double code(double v, double t) {
	return (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (((double) (((double) M_PI) * t)) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))))) * ((double) (1.0 - ((double) (v * v)))))));
}
↓
double code(double v, double t) {
	return ((double) ((((((double) sqrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))) / ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (((double) (v * v)) * 3.0))))))))) / ((double) M_PI)) / t) * (((double) sqrt(((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))))) / ((double) (1.0 - ((double) (v * v)))))));
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.5

    \[\leadsto \frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  4. Applied times-frac_binary640.5

    \[\leadsto \color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}}\]

  5. Simplified0.5

    \[\leadsto \color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)} \cdot \left(\pi \cdot t\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
  6. Using strategy rm

  7. Applied associate-/r*_binary640.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{\pi \cdot t}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
  8. Using strategy rm

  9. Applied associate-/r*_binary640.2

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{\pi}}{t}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

  10. Final simplification0.2

    \[\leadsto \frac{\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{\pi}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))