quadm (p42, negative)

Average Error: 33.6 → 6.4

Time: 20.3s

Precision: binary64

Cost: 8387

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.675391325498039 \cdot 10^{+126}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 6.382555368342549 \cdot 10^{-262}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 1.1102892118009466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\]

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 -3.675391325498039e+126)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 6.382555368342549e-262)
     (* -0.5 (* 4.0 (/ c (- b (sqrt (- (* b b) (* 4.0 (* c a))))))))
     (if (<= b 1.1102892118009466e+81)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
       (* -0.5 (/ (+ b b) a))))))

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 <= -3.675391325498039e+126) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 6.382555368342549e-262) {
		tmp = -0.5 * (4.0 * (c / (b - sqrt((b * b) - (4.0 * (c * a))))));
	} else if (b <= 1.1102892118009466e+81) {
		tmp = (-b - sqrt((b * b) - (4.0 * (c * a)))) / (2.0 * a);
	} else {
		tmp = -0.5 * ((b + b) / a);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.2
Herbie	6.4

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

Alternatives

Alternative 1
Error	34.3
Cost	46784

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

Alternative 2
Error	29.6
Cost	41536

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

Alternative 3
Error	34.1
Cost	41152

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

Alternative 4
Error	34.1
Cost	40896

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

Alternative 5
Error	34.1
Cost	40896

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

Alternative 6
Error	35.5
Cost	40640

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

Alternative 7
Error	52.3
Cost	34176

\[-0.5 \cdot \frac{b + \frac{\sqrt{{b}^{6} - {\left(4 \cdot \left(a \cdot c\right)\right)}^{3}}}{\sqrt{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)\right)}}}{a}\]

Alternative 8
Error	42.5
Cost	28224

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

Alternative 9
Error	36.3
Cost	28096

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

Alternative 10
Error	47.9
Cost	27456

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

Alternative 11
Error	34.0
Cost	27328

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

Alternative 12
Error	34.7
Cost	27200

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

Alternative 13
Error	35.0
Cost	27200

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

Alternative 14
Error	34.2
Cost	26944

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

Alternative 15
Error	57.3
Cost	20992

\[-0.5 \cdot \frac{b + \frac{\sqrt{{b}^{4} - a \cdot \left(c \cdot \left(\left(a \cdot c\right) \cdot 16\right)\right)}}{\sqrt{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}{a}\]

Alternative 16
Error	49.0
Cost	20416

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

Alternative 17
Error	42.6
Cost	20224

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

Alternative 18
Error	37.2
Cost	20160

\[-0.5 \cdot \frac{b + e^{\log \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a}\]

Alternative 19
Error	35.7
Cost	20160

\[-0.5 \cdot \frac{e^{\log \left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{a}\]

Alternative 20
Error	60.1
Cost	20160

\[-0.5 \cdot \frac{\log \left(e^{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}{a}\]

Alternative 21
Error	61.4
Cost	20160

\[-0.5 \cdot \frac{b + \log \left(e^{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}{a}\]

Alternative 22
Error	48.9
Cost	20160

\[-0.5 \cdot e^{\log \left(\frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\]

Alternative 23
Error	32.0
Cost	7744

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

Alternative 24
Error	33.7
Cost	7488

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

Alternative 25
Error	33.7
Cost	7488

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

Alternative 26
Error	41.3
Cost	7488

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

Alternative 27
Error	28.9
Cost	7488

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

Alternative 28
Error	33.6
Cost	7424

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

Alternative 29
Error	45.3
Cost	7424

\[-0.5 \cdot \left(2 \cdot \left(\frac{c}{b} + \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right)\right)\]

Alternative 30
Error	33.6
Cost	7360

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

Alternative 31
Error	45.9
Cost	7360

\[-0.5 \cdot \frac{-4 \cdot \frac{a \cdot c}{\sqrt{a \cdot \left(c \cdot -4\right)}}}{a}\]

Alternative 32
Error	33.6
Cost	7360

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

Alternative 33
Error	44.5
Cost	7232

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

Alternative 34
Error	44.5
Cost	7232

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

Alternative 35
Error	44.0
Cost	7104

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

Alternative 36
Error	44.5
Cost	7104

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

Alternative 37
Error	45.7
Cost	7104

\[-0.5 \cdot \left(-4 \cdot \frac{c}{\sqrt{a \cdot \left(c \cdot -4\right)}}\right)\]

Alternative 38
Error	43.9
Cost	6976

\[-0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a}\]

Alternative 39
Error	39.9
Cost	1088

\[-0.5 \cdot \left(4 \cdot \frac{c}{b - \left(2 \cdot \frac{a \cdot c}{b} - b\right)}\right)\]

Alternative 40
Error	39.9
Cost	960

\[-0.5 \cdot \left(4 \cdot \frac{c}{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}\right)\]

Alternative 41
Error	44.9
Cost	704

\[-0.5 \cdot \frac{2 \cdot \frac{a \cdot c}{b}}{a}\]

Alternative 42
Error	45.6
Cost	704

\[-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\]

Alternative 43
Error	45.5
Cost	576

\[-0.5 \cdot \frac{1}{\frac{a}{b + b}}\]

Alternative 44
Error	45.4
Cost	448

\[-0.5 \cdot \frac{b + b}{a}\]

Alternative 45
Error	56.2
Cost	448

\[-0.5 \cdot \frac{b - b}{a}\]

Alternative 46
Error	39.4
Cost	448

\[-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\]

Alternative 47
Error	61.6
Cost	64

\[1\]

Alternative 48
Error	56.2
Cost	64

\[0\]

Alternative 49
Error	61.6
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if b < -3.6753913254980387e126

   1. Initial program 60.8

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

   2. Simplified 60.8

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

   3. Taylor expanded around -inf 1.7

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)}\]

   4. Simplified 1.7

      \[\leadsto \color{blue}{-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)}\]

   if -3.6753913254980387e126 < b < 6.38255536834254894e-262

   1. Initial program 32.5

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

   2. Simplified 32.5

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

   4. Applied flip-+_binary64_378 32.5

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

   5. Simplified 16.2

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

   7. Applied *-un-lft-identity_binary64_404 16.2

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

   8. Applied *-un-lft-identity_binary64_404 16.2

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

   9. Applied times-frac_binary64_410 16.2

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

   10. Applied times-frac_binary64_410 16.2

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

   11. Simplified 16.2

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

   12. Simplified 9.0

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

   13. Simplified 9.0

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

   if 6.38255536834254894e-262 < b < 1.11028921180094657e81

   1. Initial program 8.1

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

   2. Simplified 8.1

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

   if 1.11028921180094657e81 < b

   1. Initial program 43.1

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

   2. Simplified 43.1

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

   3. Taylor expanded around inf 3.8

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{b}}{a}\]

   4. Simplified 3.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + b}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.675391325498039 \cdot 10^{+126}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 6.382555368342549 \cdot 10^{-262}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 1.1102892118009466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected #f

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

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