Average Error: 5.5 → 1.7
Time: 9.8s
Precision: binary64
\[[y, z] = \mathsf{sort}([y, z]) \[j, k] = \mathsf{sort}([j, k]) \\]
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
↓
\[\begin{array}{l} t_1 := \left(\left(\left(t \cdot \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{if}\;t \leq -7.432234198452891 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.0923338603745131 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x y z t a b c i j k)
 :precision binary64
 (-
  (-
   (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c))
   (* (* x 4.0) i))
  (* (* j 27.0) k)))
↓
(FPCore (x y z t a b c i j k)
 :precision binary64
 (let* ((t_1
         (-
          (-
           (+ (- (* t (* 18.0 (* y (* z x)))) (* t (* a 4.0))) (* b c))
           (* (* x 4.0) i))
          (* (* j 27.0) k))))
   (if (<= t -7.432234198452891e+53)
     t_1
     (if (<= t 1.0923338603745131e-8)
       (fma
        x
        (fma 18.0 (* y (* t z)) (* i -4.0))
        (fma a (* t -4.0) (fma -27.0 (* j k) (* b c))))
       t_1))))
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return (((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double t_1 = ((((t * (18.0 * (y * (z * x)))) - (t * (a * 4.0))) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k);
	double tmp;
	if (t <= -7.432234198452891e+53) {
		tmp = t_1;
	} else if (t <= 1.0923338603745131e-8) {
		tmp = fma(x, fma(18.0, (y * (t * z)), (i * -4.0)), fma(a, (t * -4.0), fma(-27.0, (j * k), (b * c))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x, y, z, t, a, b, c, i, j, k)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 18.0) * y) * z) * t) - Float64(Float64(a * 4.0) * t)) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
end
↓
function code(x, y, z, t, a, b, c, i, j, k)
	t_1 = Float64(Float64(Float64(Float64(Float64(t * Float64(18.0 * Float64(y * Float64(z * x)))) - Float64(t * Float64(a * 4.0))) + Float64(b * c)) - Float64(Float64(x * 4.0) * i)) - Float64(Float64(j * 27.0) * k))
	tmp = 0.0
	if (t <= -7.432234198452891e+53)
		tmp = t_1;
	elseif (t <= 1.0923338603745131e-8)
		tmp = fma(x, fma(18.0, Float64(y * Float64(t * z)), Float64(i * -4.0)), fma(a, Float64(t * -4.0), fma(-27.0, Float64(j * k), Float64(b * c))));
	else
		tmp = t_1;
	end
	return tmp
end

code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := N[(N[(N[(N[(N[(N[(N[(N[(x * 18.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_, c_, i_, j_, k_] := Block[{t$95$1 = N[(N[(N[(N[(N[(t * N[(18.0 * N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision] - N[(N[(x * 4.0), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision] - N[(N[(j * 27.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.432234198452891e+53], t$95$1, If[LessEqual[t, 1.0923338603745131e-8], N[(x * N[(18.0 * N[(y * N[(t * z), $MachinePrecision]), $MachinePrecision] + N[(i * -4.0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t * -4.0), $MachinePrecision] + N[(-27.0 * N[(j * k), $MachinePrecision] + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
t_1 := \left(\left(\left(t \cdot \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\
\mathbf{if}\;t \leq -7.432234198452891 \cdot 10^{+53}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 1.0923338603745131 \cdot 10^{-8}:\\
\;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original5.5
Target1.6
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;t < -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t < 165.68027943805222:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.43223419845289062e53 or 1.09233386037451314e-8 < t

    1. Initial program 1.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

    2. Taylor expanded in x around 0 2.0

      \[\leadsto \left(\left(\left(\color{blue}{\left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right)\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

    if -7.43223419845289062e53 < t < 1.09233386037451314e-8

    1. Initial program 7.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]

    2. Simplified1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(z \cdot t\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.432234198452891 \cdot 10^{+53}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;t \leq 1.0923338603745131 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(18, y \cdot \left(t \cdot z\right), i \cdot -4\right), \mathsf{fma}\left(a, t \cdot -4, \mathsf{fma}\left(-27, j \cdot k, b \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(18 \cdot \left(y \cdot \left(z \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array} \]

Reproduce

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

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))