Cubic critical

Average Error: 34.3 → 13.7

Time: 8.8s

Precision: binary64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\begin{array}{l} t_0 := a \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -1.887372346820939 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(1.5, t_0, b \cdot -2\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 1.786960702039828 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, 3 \cdot \left(a \cdot c\right)\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(t_0 \cdot -1.5\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (/ c b))))
   (if (<= b -1.887372346820939e+150)
     (* (fma 1.5 t_0 (* b -2.0)) (/ 0.3333333333333333 a))
     (if (<= b 1.786960702039828e-29)
       (/
        (-
         (sqrt (fma b b (fma c (* a -3.0) (fma c (* a -3.0) (* 3.0 (* a c))))))
         b)
        (* a 3.0))
       (* (/ 0.3333333333333333 a) (* t_0 -1.5))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = a * (c / b);
	double tmp;
	if (b <= -1.887372346820939e+150) {
		tmp = fma(1.5, t_0, (b * -2.0)) * (0.3333333333333333 / a);
	} else if (b <= 1.786960702039828e-29) {
		tmp = (sqrt(fma(b, b, fma(c, (a * -3.0), fma(c, (a * -3.0), (3.0 * (a * c)))))) - b) / (a * 3.0);
	} else {
		tmp = (0.3333333333333333 / a) * (t_0 * -1.5);
	}
	return tmp;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	t_0 = Float64(a * Float64(c / b))
	tmp = 0.0
	if (b <= -1.887372346820939e+150)
		tmp = Float64(fma(1.5, t_0, Float64(b * -2.0)) * Float64(0.3333333333333333 / a));
	elseif (b <= 1.786960702039828e-29)
		tmp = Float64(Float64(sqrt(fma(b, b, fma(c, Float64(a * -3.0), fma(c, Float64(a * -3.0), Float64(3.0 * Float64(a * c)))))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(t_0 * -1.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.887372346820939e+150], N[(N[(1.5 * t$95$0 + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.786960702039828e-29], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision] + N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(t$95$0 * -1.5), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.887372346820939e150

   1. Initial program 63.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 63.2

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   3. Taylor expanded in b around -inf 10.4

      \[\leadsto \color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)} \cdot \frac{0.3333333333333333}{a} \]

   4. Simplified 2.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, a \cdot \frac{c}{b}, b \cdot -2\right)} \cdot \frac{0.3333333333333333}{a} \]

   if -1.887372346820939e150 < b < 1.78696070203982802e-29

   1. Initial program 14.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Applied egg-rr 14.7

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, 3 \cdot \left(a \cdot c\right)\right)\right)\right)}}}{3 \cdot a} \]

   if 1.78696070203982802e-29 < b

   1. Initial program 55.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 55.1

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   3. Taylor expanded in b around inf 18.6

      \[\leadsto \color{blue}{\left(-1.5 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{0.3333333333333333}{a} \]

   4. Simplified 15.8

      \[\leadsto \color{blue}{\left(\left(a \cdot \frac{c}{b}\right) \cdot -1.5\right)} \cdot \frac{0.3333333333333333}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.887372346820939 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(1.5, a \cdot \frac{c}{b}, b \cdot -2\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 1.786960702039828 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, 3 \cdot \left(a \cdot c\right)\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\left(a \cdot \frac{c}{b}\right) \cdot -1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))