quadp (p42, positive)

Average Error: 34.8 → 22.1

Time: 14.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.353329681310028 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.58280276704199 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.382688758558198 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{a \cdot \left(c \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt[3]{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.353329681310028e+154)
   (/ (- b) (* a 2.0))
   (if (<= b 5.58280276704199e-129)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 9.382688758558198e+153)
       (*
        (*
         (/
          (+ (* a (* c -4.0)) (* 0.0 (* b b)))
          (+ b (sqrt (- (* b b) (* 4.0 (* a c))))))
         (* (cbrt 0.5) (cbrt 0.5)))
        (/ (cbrt 0.5) a))
       0.0))))

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 tmp;
	if (b <= -1.353329681310028e+154) {
		tmp = -b / (a * 2.0);
	} else if (b <= 5.58280276704199e-129) {
		tmp = (sqrt((b * b) - (4.0 * (a * c))) - b) / (a * 2.0);
	} else if (b <= 9.382688758558198e+153) {
		tmp = ((((a * (c * -4.0)) + (0.0 * (b * b))) / (b + sqrt((b * b) - (4.0 * (a * c))))) * (cbrt(0.5) * cbrt(0.5))) * (cbrt(0.5) / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.8
Target	21.7
Herbie	22.1

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -1.3533296813100281e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around inf 52.3

      \[\leadsto \frac{\color{blue}{0} - b}{a \cdot 2}\]

   if -1.3533296813100281e154 < b < 5.58280276704198988e-129

   1. Initial program 12.0

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

   2. Simplified 12.0

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

   if 5.58280276704198988e-129 < b < 9.38268875855819839e153

   1. Initial program 42.3

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

   2. Simplified 42.3

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

   4. Applied div-inv_binary64 42.3

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{a \cdot 2}}\]

   5. Simplified 42.3

      \[\leadsto \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \color{blue}{\frac{0.5}{a}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity_binary64 42.3

      \[\leadsto \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{\color{blue}{1 \cdot a}}\]

   8. Applied add-cube-cbrt_binary64 42.5

      \[\leadsto \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right) \cdot \sqrt[3]{0.5}}}{1 \cdot a}\]

   9. Applied times-frac_binary64 42.5

      \[\leadsto \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{1} \cdot \frac{\sqrt[3]{0.5}}{a}\right)}\]

   10. Applied associate-*r*_binary64 42.5

       \[\leadsto \color{blue}{\left(\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}}{1}\right) \cdot \frac{\sqrt[3]{0.5}}{a}}\]

   11. Simplified 42.5

       \[\leadsto \color{blue}{\left(\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right)} \cdot \frac{\sqrt[3]{0.5}}{a}\]
   12. Using strategy rm

   13. Applied flip--_binary64 42.5

       \[\leadsto \left(\color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b \cdot b}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt[3]{0.5}}{a}\]

   14. Simplified 16.0

       \[\leadsto \left(\frac{\color{blue}{a \cdot \left(c \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + b} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt[3]{0.5}}{a}\]

   15. Simplified 16.0

       \[\leadsto \left(\frac{a \cdot \left(c \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt[3]{0.5}}{a}\]

   if 9.38268875855819839e153 < b

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around 0 37.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 4 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.353329681310028 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 5.58280276704199 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.382688758558198 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{a \cdot \left(c \cdot -4\right) + 0 \cdot \left(b \cdot b\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \left(\sqrt[3]{0.5} \cdot \sqrt[3]{0.5}\right)\right) \cdot \frac{\sqrt[3]{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020285 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected #f

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))