Average Error: 0.4 → 0.1
Time: 5.0s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}}{t}}{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \]
\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{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}}{t}}{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}

(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) (* PI (* (sqrt 2.0) (- 1.0 (* v v))))) t)
  (sqrt (- 1.0 (* (* v v) 3.0)))))
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) / (((double) M_PI) * (sqrt(2.0) * (1.0 - (v * v))))) / t) / sqrt(1.0 - ((v * v) * 3.0));
}

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. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt_binary640.4

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

  6. Applied times-frac_binary640.3

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

  7. Applied associate-*l/_binary640.1

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

  8. Simplified0.1

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

  10. Applied sqrt-div_binary640.1

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

  11. Applied associate-*r/_binary640.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2021211 
(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)))))