Average Error: 2.9 → 0.6
Time: 7.8s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \[a, b] = \mathsf{sort}([a, b]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 3.322523053190084 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, 2 \cdot x\right) - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(x, 2, -9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\\ \end{array} \]
(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 3.322523053190084e-178)
   (- (fma (* 27.0 a) b (* 2.0 x)) (* 9.0 (* y (* z t))))
   (fma a (* 27.0 b) (fma x 2.0 (* -9.0 (* t (* z y)))))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 3.322523053190084e-178) {
		tmp = fma((27.0 * a), b, (2.0 * x)) - (9.0 * (y * (z * t)));
	} else {
		tmp = fma(a, (27.0 * b), fma(x, 2.0, (-9.0 * (t * (z * y)))));
	}
	return tmp;
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * 2.0) - Float64(Float64(Float64(y * 9.0) * z) * t)) + Float64(Float64(a * 27.0) * b))
end
↓
function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 3.322523053190084e-178)
		tmp = Float64(fma(Float64(27.0 * a), b, Float64(2.0 * x)) - Float64(9.0 * Float64(y * Float64(z * t))));
	else
		tmp = fma(a, Float64(27.0 * b), fma(x, 2.0, Float64(-9.0 * Float64(t * Float64(z * y)))));
	end
	return tmp
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * 2.0), $MachinePrecision] - N[(N[(N[(y * 9.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 27.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 3.322523053190084e-178], N[(N[(N[(27.0 * a), $MachinePrecision] * b + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] - N[(9.0 * N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(27.0 * b), $MachinePrecision] + N[(x * 2.0 + N[(-9.0 * N[(t * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \leq 3.322523053190084 \cdot 10^{-178}:\\
\;\;\;\;\mathsf{fma}\left(27 \cdot a, b, 2 \cdot x\right) - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(x, 2, -9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original2.9
Target3.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < 3.32252305319008375e-178

    1. Initial program 4.0

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    2. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(x, 2, \left(a \cdot 27\right) \cdot b\right)\right)} \]

    3. Taylor expanded in y around 0 0.6

      \[\leadsto \color{blue}{\left(27 \cdot \left(a \cdot b\right) + 2 \cdot x\right) - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)} \]

    4. Applied egg-rr0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(27 \cdot a, b, 2 \cdot x\right)} - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right) \]

    if 3.32252305319008375e-178 < z

    1. Initial program 0.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    2. Taylor expanded in y around 0 0.5

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b \]

    3. Applied egg-rr0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(x, 2, -9 \cdot \left(\left(y \cdot z\right) \cdot t\right)\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.322523053190084 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(27 \cdot a, b, 2 \cdot x\right) - 9 \cdot \left(y \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, \mathsf{fma}\left(x, 2, -9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))