Average Error: 33.9 → 0.4
Time: 7.5s
Precision: binary64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right) \]
(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))
↓
(FPCore (x y z t) :precision binary64 (fma (/ z t) (/ z t) (pow (/ x y) 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((z / t), (z / t), pow((x / y), 2.0));
}

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end
↓
function code(x, y, z, t)
	return fma(Float64(z / t), Float64(z / t), (Float64(x / y) ^ 2.0))
end

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.9
Target0.4
Herbie0.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \]

Derivation

  1. Initial program 33.9

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

  2. Simplified29.9

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

  3. Applied egg-rr13.4

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

  4. Applied egg-rr4.2

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

  5. Applied egg-rr0.4

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

  6. Applied egg-rr0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2022133 
(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))))