Average Error: 0.5 → 0.4
Time: 10.5s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]
\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{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\right)\right)} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}

(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
 (*
  (/ (fma (* v v) -5.0 1.0) (* t (* (- 1.0 (* v v)) (* PI (sqrt 2.0)))))
  (sqrt (/ 1.0 (fma (* v v) -3.0 1.0)))))
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt(2.0 * (1.0 - (3.0 * (v * v))))) * (1.0 - (v * v)));
}
↓
double code(double v, double t) {
	return (fma((v * v), -5.0, 1.0) / (t * ((1.0 - (v * v)) * (((double) M_PI) * sqrt(2.0))))) * sqrt(1.0 / fma((v * v), -3.0, 1.0));
}

Error

Bits error versus v

Bits error versus t

Derivation

  1. Initial program 0.5

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

  2. Applied add-sqr-sqrt_binary640.9

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

  3. Applied associate-*r*_binary640.8

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

  4. Taylor expanded in t around 0 0.5

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

  5. Simplified0.5

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

  6. Taylor expanded in t around 0 0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2021305 
(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)))))