Average Error: 11.9 → 0.1
Time: 4.0s
Precision: binary64
\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
↓
\[x - {\left(\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)\right)}^{-1} \]
(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
 (- x (pow (fma -0.5 (/ t z) (/ z y)) -1.0)))
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 x - pow(fma(-0.5, (t / z), (z / y)), -1.0);
}

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 Float64(x - (fma(-0.5, Float64(t / z), Float64(z / y)) ^ -1.0))
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[(x - N[Power[N[(-0.5 * N[(t / z), $MachinePrecision] + N[(z / y), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}
↓
x - {\left(\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)\right)}^{-1}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original11.9
Target0.1
Herbie0.1
\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

  1. Initial program 11.9

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

  2. Simplified2.9

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

  3. Applied egg-rr3.0

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

  4. Taylor expanded in z around 0 0.1

    \[\leadsto x - {\color{blue}{\left(\frac{z}{y} - 0.5 \cdot \frac{t}{z}\right)}}^{-1} \]

  5. Simplified0.1

    \[\leadsto x - {\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)\right)}}^{-1} \]

  6. Final simplification0.1

    \[\leadsto x - {\left(\mathsf{fma}\left(-0.5, \frac{t}{z}, \frac{z}{y}\right)\right)}^{-1} \]

Reproduce

herbie shell --seed 2022133 
(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)))))