Average Error: 21.3 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
↓
\[\mathsf{fma}\left(1, {\left(\frac{x}{y}\right)}^{2}, {\left(\frac{z}{t}\right)}^{2}\right) \]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(1, {\left(\frac{x}{y}\right)}^{2}, {\left(\frac{z}{t}\right)}^{2}\right)

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma 1.0 (pow (/ x y) 2.0) (pow (/ z t) 2.0)))
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(1.0, pow((x / y), 2.0), pow((z / t), 2.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original21.3
Target0.2
Herbie0.2
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \]

Derivation

  1. Initial program 21.3

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

  2. Simplified17.4

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

  3. Applied add-sqr-sqrt_binary6417.4

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

  4. Simplified17.4

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

  5. Simplified0.2

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

  6. Taylor expanded in z around 0 21.3

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

  7. Simplified0.2

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

  8. Applied *-un-lft-identity_binary640.2

    \[\leadsto \color{blue}{1 \cdot {\left(\frac{x}{y}\right)}^{2}} + {\left(\frac{z}{t}\right)}^{2} \]

  9. Applied fma-def_binary640.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, {\left(\frac{x}{y}\right)}^{2}, {\left(\frac{z}{t}\right)}^{2}\right)} \]

  10. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(1, {\left(\frac{x}{y}\right)}^{2}, {\left(\frac{z}{t}\right)}^{2}\right) \]

Reproduce

herbie shell --seed 2022067 
(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))))