Average Error: 3.2 → 0.5
Time: 6.5s
Precision: binary64
\[ \begin{array}{c}[y, z, t] = \mathsf{sort}([y, z, t])\\ [a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq 4.7492635233917384 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x \cdot 2\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 + t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]
(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 (<= t 4.7492635233917384e+27)
   (+ (fma (* -9.0 (* t y)) z (* x 2.0)) (* 27.0 (* a b)))
   (+ (+ (* x 2.0) (* t (* -9.0 (* y z)))) (* b (* 27.0 a)))))
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 (t <= 4.7492635233917384e+27) {
		tmp = fma((-9.0 * (t * y)), z, (x * 2.0)) + (27.0 * (a * b));
	} else {
		tmp = ((x * 2.0) + (t * (-9.0 * (y * z)))) + (b * (27.0 * a));
	}
	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 (t <= 4.7492635233917384e+27)
		tmp = Float64(fma(Float64(-9.0 * Float64(t * y)), z, Float64(x * 2.0)) + Float64(27.0 * Float64(a * b)));
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(t * Float64(-9.0 * Float64(y * z)))) + Float64(b * Float64(27.0 * a)));
	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[t, 4.7492635233917384e+27], N[(N[(N[(-9.0 * N[(t * y), $MachinePrecision]), $MachinePrecision] * z + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + N[(27.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(t * N[(-9.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(27.0 * a), $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}\;t \leq 4.7492635233917384 \cdot 10^{+27}:\\
\;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x \cdot 2\right) + 27 \cdot \left(a \cdot b\right)\\

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

Original3.2
Target3.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < 4.74926352339173839e27

    1. Initial program 5.4

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

    2. Taylor expanded in y around 0 0.7

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

    3. Taylor expanded in a around 0 0.6

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

    4. Applied egg-rr0.2

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

    if 4.74926352339173839e27 < t

    1. Initial program 0.9

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

    2. Taylor expanded in y around 0 0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7492635233917384 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot \left(t \cdot y\right), z, x \cdot 2\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 + t \cdot \left(-9 \cdot \left(y \cdot z\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(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)))