Cubic critical

Average Error: 34.1 → 13.7

Time: 7.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}{3 \cdot a}\\ \end{array} \]

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 -5.2e+42)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 6e-23)
     (/ (fma -1.0 b (sqrt (fma c (* a -3.0) (* b b)))) (* 3.0 a))
     (/ (* (* a (/ c b)) -1.5) (* 3.0 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 <= -5.2e+42) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 6e-23) {
		tmp = fma(-1.0, b, sqrt(fma(c, (a * -3.0), (b * b)))) / (3.0 * a);
	} else {
		tmp = ((a * (c / b)) * -1.5) / (3.0 * 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 <= -5.2e+42)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 6e-23)
		tmp = Float64(fma(-1.0, b, sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(c / b)) * -1.5) / Float64(3.0 * 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, -5.2e+42], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-23], N[(N[(-1.0 * b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] / N[(3.0 * 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 < -5.1999999999999998e42

   1. Initial program 36.8

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

   2. Taylor expanded in b around -inf 6.7

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

   if -5.1999999999999998e42 < b < 6.00000000000000006e-23

   1. Initial program 15.4

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

   2. Applied egg-rr 15.4

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

   3. Applied egg-rr 15.4

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

   4. Taylor expanded in b around 0 15.5

      \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(c \cdot a\right)}}\right)}{3 \cdot a} \]

   5. Simplified 15.4

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

   if 6.00000000000000006e-23 < b

   1. Initial program 54.9

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

   2. Taylor expanded in b around inf 18.9

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{c \cdot a}{b}}}{3 \cdot a} \]

   3. Simplified 15.7

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

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}{3 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))