Average Error: 0.4 → 0.4
Time: 11.0s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{t \cdot \left(\pi - \pi \cdot \left(v \cdot v\right)\right)} \]
\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{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{t \cdot \left(\pi - \pi \cdot \left(v \cdot v\right)\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) (sqrt (fma v (* v -6.0) 2.0)))
  (* t (- PI (* PI (* v v))))))
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) / sqrt(fma(v, (v * -6.0), 2.0))) / (t * (((double) M_PI) - (((double) M_PI) * (v * v))));
}

Error

Bits error versus v

Bits error versus t

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. Simplified0.4

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)} \cdot \left(\left(\pi \cdot t\right) \cdot \left(1 - v \cdot v\right)\right)}} \]

  3. Applied associate-/r*_binary640.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\left(\pi \cdot t\right) \cdot \left(1 - v \cdot v\right)}} \]

  4. Applied pow1_binary640.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\left(\pi \cdot t\right) \cdot \color{blue}{{\left(1 - v \cdot v\right)}^{1}}} \]

  5. Applied pow1_binary640.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\left(\pi \cdot \color{blue}{{t}^{1}}\right) \cdot {\left(1 - v \cdot v\right)}^{1}} \]

  6. Applied pow1_binary640.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\left(\color{blue}{{\pi}^{1}} \cdot {t}^{1}\right) \cdot {\left(1 - v \cdot v\right)}^{1}} \]

  7. Applied pow-prod-down_binary640.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\color{blue}{{\left(\pi \cdot t\right)}^{1}} \cdot {\left(1 - v \cdot v\right)}^{1}} \]

  8. Applied pow-prod-down_binary640.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\color{blue}{{\left(\left(\pi \cdot t\right) \cdot \left(1 - v \cdot v\right)\right)}^{1}}} \]

  9. Simplified0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{{\color{blue}{\left(t \cdot \left(\pi - \pi \cdot \left(v \cdot v\right)\right)\right)}}^{1}} \]

  10. Final simplification0.4

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}}{t \cdot \left(\pi - \pi \cdot \left(v \cdot v\right)\right)} \]

Reproduce

herbie shell --seed 2022103 
(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)))))