Average Error: 5.5 → 1.6
Time: 10.3s
Precision: binary64
\[[y, z] = \mathsf{sort}([y, z]) \\]
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -2.5334842913125285 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t, \mathsf{fma}\left(y, 18 \cdot \left(z \cdot x\right), a \cdot -4\right), \mathsf{fma}\left(k \cdot j, -27, -4 \cdot \left(x \cdot i\right)\right)\right)\right)\\ \mathbf{elif}\;t \leq 4.479382283497473 \cdot 10^{-94}:\\ \;\;\;\;\left(c \cdot b + 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right)\right) - \left(\left(x \cdot i\right) \cdot 4 + \left(4 \cdot \left(t \cdot a\right) + \left(k \cdot j\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(y \cdot \left(18 \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right)\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \]
(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
 (if (<= t -2.5334842913125285e-17)
   (fma
    c
    b
    (fma
     t
     (fma y (* 18.0 (* z x)) (* a -4.0))
     (fma (* k j) -27.0 (* -4.0 (* x i)))))
   (if (<= t 4.479382283497473e-94)
     (-
      (+ (* c b) (* 18.0 (* y (* x (* t z)))))
      (+ (* (* x i) 4.0) (+ (* 4.0 (* t a)) (* (* k j) 27.0))))
     (-
      (-
       (+ (* c b) (- (* t (* z (* y (* 18.0 x)))) (* t (* a 4.0))))
       (* i (* x 4.0)))
      (* k (* j 27.0))))))
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 tmp;
	if (t <= -2.5334842913125285e-17) {
		tmp = fma(c, b, fma(t, fma(y, (18.0 * (z * x)), (a * -4.0)), fma((k * j), -27.0, (-4.0 * (x * i)))));
	} else if (t <= 4.479382283497473e-94) {
		tmp = ((c * b) + (18.0 * (y * (x * (t * z))))) - (((x * i) * 4.0) + ((4.0 * (t * a)) + ((k * j) * 27.0)));
	} else {
		tmp = (((c * b) + ((t * (z * (y * (18.0 * x)))) - (t * (a * 4.0)))) - (i * (x * 4.0))) - (k * (j * 27.0));
	}
	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)
	tmp = 0.0
	if (t <= -2.5334842913125285e-17)
		tmp = fma(c, b, fma(t, fma(y, Float64(18.0 * Float64(z * x)), Float64(a * -4.0)), fma(Float64(k * j), -27.0, Float64(-4.0 * Float64(x * i)))));
	elseif (t <= 4.479382283497473e-94)
		tmp = Float64(Float64(Float64(c * b) + Float64(18.0 * Float64(y * Float64(x * Float64(t * z))))) - Float64(Float64(Float64(x * i) * 4.0) + Float64(Float64(4.0 * Float64(t * a)) + Float64(Float64(k * j) * 27.0))));
	else
		tmp = Float64(Float64(Float64(Float64(c * b) + Float64(Float64(t * Float64(z * Float64(y * Float64(18.0 * x)))) - Float64(t * Float64(a * 4.0)))) - Float64(i * Float64(x * 4.0))) - Float64(k * Float64(j * 27.0)));
	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_] := If[LessEqual[t, -2.5334842913125285e-17], N[(c * b + N[(t * N[(y * N[(18.0 * N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(k * j), $MachinePrecision] * -27.0 + N[(-4.0 * N[(x * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.479382283497473e-94], N[(N[(N[(c * b), $MachinePrecision] + N[(18.0 * N[(y * N[(x * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x * i), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision] + N[(N[(k * j), $MachinePrecision] * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] + N[(N[(t * N[(z * N[(y * N[(18.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(k * N[(j * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\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}
\mathbf{if}\;t \leq -2.5334842913125285 \cdot 10^{-17}:\\
\;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t, \mathsf{fma}\left(y, 18 \cdot \left(z \cdot x\right), a \cdot -4\right), \mathsf{fma}\left(k \cdot j, -27, -4 \cdot \left(x \cdot i\right)\right)\right)\right)\\

\mathbf{elif}\;t \leq 4.479382283497473 \cdot 10^{-94}:\\
\;\;\;\;\left(c \cdot b + 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right)\right) - \left(\left(x \cdot i\right) \cdot 4 + \left(4 \cdot \left(t \cdot a\right) + \left(k \cdot j\right) \cdot 27\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(y \cdot \left(18 \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right)\right) - k \cdot \left(j \cdot 27\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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

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

Derivation

  1. Split input into 3 regimes

  2. if t < -2.5334842913125285e-17

    1. Initial program 1.8

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

    2. Simplified12.0

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

    3. Taylor expanded in x around 0 5.6

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

    4. Simplified2.0

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

    if -2.5334842913125285e-17 < t < 4.47938228349747307e-94

    1. Initial program 8.6

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

    2. Simplified1.3

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

    3. Taylor expanded in x around 0 4.3

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

    4. Applied associate-*r*_binary640.9

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

    if 4.47938228349747307e-94 < t

    1. Initial program 2.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5334842913125285 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \mathsf{fma}\left(t, \mathsf{fma}\left(y, 18 \cdot \left(z \cdot x\right), a \cdot -4\right), \mathsf{fma}\left(k \cdot j, -27, -4 \cdot \left(x \cdot i\right)\right)\right)\right)\\ \mathbf{elif}\;t \leq 4.479382283497473 \cdot 10^{-94}:\\ \;\;\;\;\left(c \cdot b + 18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right)\right) - \left(\left(x \cdot i\right) \cdot 4 + \left(4 \cdot \left(t \cdot a\right) + \left(k \cdot j\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(t \cdot \left(z \cdot \left(y \cdot \left(18 \cdot x\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - i \cdot \left(x \cdot 4\right)\right) - k \cdot \left(j \cdot 27\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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)))