Cubic critical

Average Error: 34.0 → 14.0

Time: 8.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \frac{c}{b}\right) \cdot -1.5\right) \cdot \frac{0.3333333333333333}{a}\\ \end{array} \]

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 -1e+144)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 9.5e-99)
     (/ (fma -1.0 b (sqrt (fma b b (* c (* a -3.0))))) (* 3.0 a))
     (* (* (* a (/ c b)) -1.5) (/ 0.3333333333333333 a)))))

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 <= -1e+144) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 9.5e-99) {
		tmp = fma(-1.0, b, sqrt(fma(b, b, (c * (a * -3.0))))) / (3.0 * a);
	} else {
		tmp = ((a * (c / b)) * -1.5) * (0.3333333333333333 / a);
	}
	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 <= -1e+144)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 9.5e-99)
		tmp = Float64(fma(-1.0, b, sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(c / b)) * -1.5) * Float64(0.3333333333333333 / a));
	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, -1e+144], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.5e-99], N[(N[(-1.0 * b + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] * N[(0.3333333333333333 / a), $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 < -1.00000000000000002e144

   1. Initial program 60.4

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

   2. Taylor expanded in b around -inf 2.4

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

   if -1.00000000000000002e144 < b < 9.5000000000000008e-99

   1. Initial program 12.4

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

   2. Applied egg-rr 12.4

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

   if 9.5000000000000008e-99 < b

   1. Initial program 51.5

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

   2. Simplified 51.5

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

   3. Taylor expanded in b around inf 22.2

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

   4. Simplified 19.1

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

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot \frac{c}{b}\right) \cdot -1.5\right) \cdot \frac{0.3333333333333333}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))