Cubic critical

Average Error: 34.0 → 10.6

Time: 8.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{-0.6666666666666666}{\frac{a}{b}}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e+88)
   (fma 0.5 (/ c b) (/ -0.6666666666666666 (/ a b)))
   (if (<= b 2.5e-140)
     (/ (- (sqrt (fma 1.0 (* b b) (* c (* a -3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

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 tmp;
	if (b <= -5.5e+88) {
		tmp = fma(0.5, (c / b), (-0.6666666666666666 / (a / b)));
	} else if (b <= 2.5e-140) {
		tmp = (sqrt(fma(1.0, (b * b), (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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)
	tmp = 0.0
	if (b <= -5.5e+88)
		tmp = fma(0.5, Float64(c / b), Float64(-0.6666666666666666 / Float64(a / b)));
	elseif (b <= 2.5e-140)
		tmp = Float64(Float64(sqrt(fma(1.0, Float64(b * b), Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	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_] := If[LessEqual[b, -5.5e+88], N[(0.5 * N[(c / b), $MachinePrecision] + N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-140], N[(N[(N[Sqrt[N[(1.0 * N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -5.5e88

   1. Initial program 43.7

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

   2. Taylor expanded in b around -inf 4.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}} \]

   3. Simplified 4.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{-0.6666666666666666}{\frac{a}{b}}\right)} \]

   if -5.5e88 < b < 2.50000000000000007e-140

   1. Initial program 11.2

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

   2. Applied egg-rr 11.2

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(1, b \cdot b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a} \]

   if 2.50000000000000007e-140 < b

   1. Initial program 50.4

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

   2. Taylor expanded in b around inf 12.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{-0.6666666666666666}{\frac{a}{b}}\right)\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(1, b \cdot b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))