quadp (p42, positive)

Average Error: 34.1 → 6.5

Time: 2.4min

Precision: binary64

Cost: 8387

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -4.89437030841433 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.613044434627782 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{a \cdot -4}{\frac{-1}{c}}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.816931246986667 \cdot 10^{+76}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

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 -4.89437030841433e+152)
   (- (/ c b) (/ b a))
   (if (<= b -6.613044434627782e-283)
     (/ (- (sqrt (- (* b b) (/ (* a -4.0) (/ -1.0 c)))) b) (* a 2.0))
     (if (<= b 4.816931246986667e+76)
       (/ (* c 2.0) (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))))
       (/ (- c) b)))))

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 <= -4.89437030841433e+152) {
		tmp = (c / b) - (b / a);
	} else if (b <= -6.613044434627782e-283) {
		tmp = (sqrt((b * b) - ((a * -4.0) / (-1.0 / c))) - b) / (a * 2.0);
	} else if (b <= 4.816931246986667e+76) {
		tmp = (c * 2.0) / (-b - sqrt((b * b) - (4.0 * (c * a))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	20.7
Herbie	6.5

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

Alternatives

Alternative 1
Error	6.5
Cost	8387

\[\begin{array}{l} \mathbf{if}\;b \leq -4.0737268480388477 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -2.8094746472583994 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.816931246986667 \cdot 10^{+76}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 2
Error	6.5
Cost	8387

\[\begin{array}{l} \mathbf{if}\;b \leq -4.191727865581558 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -9.078145301820259 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 9.7059850177998 \cdot 10^{+76}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 3
Error	10.1
Cost	8002

\[\begin{array}{l} \mathbf{if}\;b \leq -6.065441046468951 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.0684527239595751 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 4
Error	13.7
Cost	7874

\[\begin{array}{l} \mathbf{if}\;b \leq -6.666572808649524 \cdot 10^{-25}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.595244068316303 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 5
Error	13.4
Cost	8260

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0036134703492082 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.624296268777243 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.1306939922765858 \cdot 10^{-52}:\\ \;\;\;\;2 \cdot \frac{c}{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}\\ \mathbf{elif}\;b \leq 6.322625190231006 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 6
Error	13.9
Cost	7618

\[\begin{array}{l} \mathbf{if}\;b \leq -5.614797585418129 \cdot 10^{-41}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.322625190231006 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 7
Error	13.8
Cost	7618

\[\begin{array}{l} \mathbf{if}\;b \leq -1.036257904214663 \cdot 10^{-42}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.322625190231006 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 8
Error	22.9
Cost	7169

\[\begin{array}{l} \mathbf{if}\;b \leq 4.2341774179204834 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left|b\right| - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 9
Error	22.6
Cost	577

\[\begin{array}{l} \mathbf{if}\;b \leq 8.529742819354031 \cdot 10^{-226}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Alternative 10
Error	39.7
Cost	256

\[\frac{-c}{b}\]

Alternative 11
Error	56.2
Cost	64

\[0\]

Alternative 12
Error	61.6
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if b < -4.89437030841433042e152

   1. Initial program 63.4

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

   2. Simplified 63.4

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

   3. Taylor expanded around -inf 2.2

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

   4. Simplified 2.2

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

   if -4.89437030841433042e152 < b < -6.61304443462778203e-283

   1. Initial program 8.5

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

   3. Applied add-exp-log_binary64_457 37.6

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

   4. Applied add-exp-log_binary64_457 56.6

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

   5. Applied prod-exp_binary64_468 56.6

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

   6. Applied add-exp-log_binary64_457 56.6

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

   7. Applied prod-exp_binary64_468 56.6

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

   8. Simplified 44.9

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

   9. Taylor expanded around -inf 56.1

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

   10. Simplified 8.5

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

   11. Simplified 8.5

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

   if -6.61304443462778203e-283 < b < 4.81693124698666678e76

   1. Initial program 29.6

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

   3. Applied flip-+_binary64_393 29.7

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

   4. Simplified 16.0

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

   6. Applied *-un-lft-identity_binary64_419 16.0

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

   7. Applied times-frac_binary64_425 16.0

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

   8. Applied times-frac_binary64_425 16.0

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

   9. Simplified 16.0

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

   10. Simplified 8.8

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

   12. Applied associate-*r/_binary64_361 8.8

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

   13. Applied associate-*r/_binary64_361 8.8

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

   14. Simplified 8.8

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

   15. Simplified 8.8

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

   if 4.81693124698666678e76 < b

   1. Initial program 58.7

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

   3. Applied flip-+_binary64_393 58.7

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

   4. Simplified 29.6

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

   6. Applied *-un-lft-identity_binary64_419 29.6

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

   7. Applied times-frac_binary64_425 29.6

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

   8. Applied times-frac_binary64_425 29.6

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

   9. Simplified 29.6

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

   10. Simplified 26.9

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

   11. Taylor expanded around 0 3.2

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

   12. Simplified 3.2

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

   13. Simplified 3.2

       \[\leadsto \color{blue}{\frac{-c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.89437030841433 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -6.613044434627782 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{a \cdot -4}{\frac{-1}{c}}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.816931246986667 \cdot 10^{+76}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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)))