The quadratic formula (r2)

Average Error: 33.7 → 11.2

Time: 4.8s

Precision: 64

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.77413468628334187 \cdot 10^{-17}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b} \cdot \sqrt[3]{b}, \sqrt[3]{b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

C

double code(double a, double b, double c) {
	return ((-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a));
}

↓

double code(double a, double b, double c) {
	double VAR;
	if ((b <= -3.774134686283342e-17)) {
		VAR = (-1.0 * (c / b));
	} else {
		double VAR_1;
		if ((b <= 1.2919983862558445e+30)) {
			VAR_1 = (fma(-(cbrt(b) * cbrt(b)), cbrt(b), -sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a));
		} else {
			VAR_1 = (1.0 * ((c / b) - (b / a)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.7
Target	20.5
Herbie	11.2

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -3.774134686283342e-17

   1. Initial program 54.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Taylor expanded around -inf 6.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   if -3.774134686283342e-17 < b < 1.2919983862558445e+30

   1. Initial program 16.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 17.0

      \[\leadsto \frac{\left(-\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   4. Applied distribute-lft-neg-in 17.0

      \[\leadsto \frac{\color{blue}{\left(-\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}} - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   5. Applied fma-neg 17.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-\sqrt[3]{b} \cdot \sqrt[3]{b}, \sqrt[3]{b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]

   if 1.2919983862558445e+30 < b

   1. Initial program 34.2

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Taylor expanded around inf 6.6

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

   3. Simplified 6.6

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.77413468628334187 \cdot 10^{-17}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b} \cdot \sqrt[3]{b}, \sqrt[3]{b}, -\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))