Average Error: 33.5 → 0.4
Time: 8.5s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right) \]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma (/ x y) (/ x y) (* (/ z t) (/ z t))))
double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}
↓
double code(double x, double y, double z, double t) {
	return fma((x / y), (x / y), ((z / t) * (z / t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.5
Target0.4
Herbie0.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \]

Derivation

  1. Initial program 33.5

    \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

  2. Simplified29.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{y \cdot y}, \frac{z \cdot z}{t \cdot t}\right)} \]

  3. Applied add-sqr-sqrt_binary6429.5

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, \frac{x}{y \cdot y}, \frac{z \cdot z}{t \cdot t}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{x}{y \cdot y}, \frac{z \cdot z}{t \cdot t}\right)}} \]

  4. Simplified29.5

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \sqrt{\mathsf{fma}\left(x, \frac{x}{y \cdot y}, \frac{z \cdot z}{t \cdot t}\right)} \]

  5. Simplified0.4

    \[\leadsto \mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \cdot \color{blue}{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \]

  6. Applied add-sqr-sqrt_binary640.6

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right)} \cdot \mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right) \]

  7. Applied associate-*l*_binary640.6

    \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)\right)} \]

  8. Applied associate-*r*_binary640.6

    \[\leadsto \color{blue}{\left(\sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \cdot \sqrt{\mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)}\right) \cdot \mathsf{hypot}\left(\frac{z}{t}, \frac{x}{y}\right)} \]

  9. Taylor expanded in z around 0 33.5

    \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}} + \frac{{x}^{2}}{{y}^{2}}} \]

  10. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right)} \]

  11. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{z}{t} \cdot \frac{z}{t}\right) \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :herbie-target
  (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))