Average Error: 5.6 → 0.2
Time: 2.4s
Precision: binary64
\[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]
↓
\[\frac{1 - x}{\frac{y}{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}} \]
(FPCore (x y) :precision binary64 (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))
↓
(FPCore (x y)
 :precision binary64
 (/ (- 1.0 x) (/ y (fma x -0.3333333333333333 1.0))))
double code(double x, double y) {
	return ((1.0 - x) * (3.0 - x)) / (y * 3.0);
}
↓
double code(double x, double y) {
	return (1.0 - x) / (y / fma(x, -0.3333333333333333, 1.0));
}

function code(x, y)
	return Float64(Float64(Float64(1.0 - x) * Float64(3.0 - x)) / Float64(y * 3.0))
end
↓
function code(x, y)
	return Float64(Float64(1.0 - x) / Float64(y / fma(x, -0.3333333333333333, 1.0)))
end

code[x_, y_] := N[(N[(N[(1.0 - x), $MachinePrecision] * N[(3.0 - x), $MachinePrecision]), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] / N[(y / N[(x * -0.3333333333333333 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3}
↓
\frac{1 - x}{\frac{y}{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}}

Error

Bits error versus x

Bits error versus y

Target

Original5.6
Target0.1
Herbie0.2
\[\frac{1 - x}{y} \cdot \frac{3 - x}{3} \]

Derivation

  1. Initial program 5.6

    \[\frac{\left(1 - x\right) \cdot \left(3 - x\right)}{y \cdot 3} \]

  2. Simplified0.4

    \[\leadsto \color{blue}{\left(1 - x\right) \cdot \left(\left(3 - x\right) \cdot \frac{0.3333333333333333}{y}\right)} \]

  3. Applied egg-rr0.2

    \[\leadsto \left(1 - x\right) \cdot \color{blue}{\frac{1}{\frac{y}{0.3333333333333333 \cdot \left(3 - x\right)}}} \]

  4. Taylor expanded in y around 0 5.7

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\left(3 - x\right) \cdot \left(1 - x\right)}{y}} \]

  5. Simplified0.2

    \[\leadsto \color{blue}{\frac{1 - x}{\frac{y}{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}}} \]

  6. Final simplification0.2

    \[\leadsto \frac{1 - x}{\frac{y}{\mathsf{fma}\left(x, -0.3333333333333333, 1\right)}} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:bezierFromSweepQ1 from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (* (/ (- 1.0 x) y) (/ (- 3.0 x) 3.0))

  (/ (* (- 1.0 x) (- 3.0 x)) (* y 3.0)))