Average Error: 11.7 → 0.1
Time: 4.7s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[\mathsf{fma}\left(1, x, \frac{-2}{2 \cdot \frac{z}{y} - \frac{t}{z}}\right) \]
(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma 1.0 x (/ -2.0 (- (* 2.0 (/ z y)) (/ t z)))))
double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}
↓
double code(double x, double y, double z, double t) {
	return fma(1.0, x, (-2.0 / ((2.0 * (z / y)) - (t / z))));
}

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end
↓
function code(x, y, z, t)
	return fma(1.0, x, Float64(-2.0 / Float64(Float64(2.0 * Float64(z / y)) - Float64(t / z))))
end

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(1.0 * x + N[(-2.0 / N[(N[(2.0 * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
\mathsf{fma}\left(1, x, \frac{-2}{2 \cdot \frac{z}{y} - \frac{t}{z}}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.7
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

  1. Initial program 11.7

    \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

  2. Simplified0.1

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

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))