quadp (p42, positive)

Average Error: 33.4 → 14.6

Time: 28.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -5.229623884976429 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.3483090680597456 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, 0\right)}{a}}{b + b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1831546 = b;
        double r1831547 = -r1831546;
        double r1831548 = r1831546 * r1831546;
        double r1831549 = 4.0;
        double r1831550 = a;
        double r1831551 = c;
        double r1831552 = r1831550 * r1831551;
        double r1831553 = r1831549 * r1831552;
        double r1831554 = r1831548 - r1831553;
        double r1831555 = sqrt(r1831554);
        double r1831556 = r1831547 + r1831555;
        double r1831557 = 2.0;
        double r1831558 = r1831557 * r1831550;
        double r1831559 = r1831556 / r1831558;
        return r1831559;
}

↓

double f(double a, double b, double c) {
        double r1831560 = b;
        double r1831561 = -5.229623884976429e-265;
        bool r1831562 = r1831560 <= r1831561;
        double r1831563 = -4.0;
        double r1831564 = a;
        double r1831565 = r1831563 * r1831564;
        double r1831566 = c;
        double r1831567 = r1831560 * r1831560;
        double r1831568 = fma(r1831565, r1831566, r1831567);
        double r1831569 = sqrt(r1831568);
        double r1831570 = r1831569 - r1831560;
        double r1831571 = r1831570 / r1831564;
        double r1831572 = 2.0;
        double r1831573 = r1831571 / r1831572;
        double r1831574 = 1.3483090680597456e+92;
        bool r1831575 = r1831560 <= r1831574;
        double r1831576 = r1831566 * r1831563;
        double r1831577 = r1831566 * r1831564;
        double r1831578 = fma(r1831577, r1831563, r1831567);
        double r1831579 = sqrt(r1831578);
        double r1831580 = r1831560 + r1831579;
        double r1831581 = r1831576 / r1831580;
        double r1831582 = r1831581 / r1831572;
        double r1831583 = 0.0;
        double r1831584 = fma(r1831564, r1831576, r1831583);
        double r1831585 = r1831584 / r1831564;
        double r1831586 = r1831560 + r1831560;
        double r1831587 = r1831585 / r1831586;
        double r1831588 = r1831587 / r1831572;
        double r1831589 = r1831575 ? r1831582 : r1831588;
        double r1831590 = r1831562 ? r1831573 : r1831589;
        return r1831590;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.4
Target	20.6
Herbie	14.6

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

Derivation

1. Split input into 3 regimes

2. if b < -5.229623884976429e-265

   1. Initial program 22.2

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

   2. Simplified 22.2

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

   4. Applied clear-num 22.3

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

   6. Applied *-un-lft-identity 22.3

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

   7. Applied *-un-lft-identity 22.3

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

   8. Applied times-frac 22.3

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

   9. Applied add-cube-cbrt 22.3

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}}{2}\]

   10. Applied times-frac 22.3

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}}}{2}\]

   11. Simplified 22.3

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}}}{2}\]

   12. Simplified 22.2

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

   if -5.229623884976429e-265 < b < 1.3483090680597456e+92

   1. Initial program 30.2

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

   2. Simplified 30.2

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

   4. Applied clear-num 30.2

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

   6. Applied flip-- 30.3

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

   7. Applied associate-/r/ 30.4

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

   8. Applied associate-/r* 30.5

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

   9. Simplified 15.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(a, -4 \cdot c, 0\right)}{a}}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{2}\]

   10. Taylor expanded around 0 9.0

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

   if 1.3483090680597456e+92 < b

   1. Initial program 57.6

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

   2. Simplified 57.6

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

   4. Applied clear-num 57.6

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

   6. Applied flip-- 57.7

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

   7. Applied associate-/r/ 57.7

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

   8. Applied associate-/r* 57.7

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

   9. Simplified 29.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(a, -4 \cdot c, 0\right)}{a}}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{2}\]

   10. Taylor expanded around 0 8.7

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a, -4 \cdot c, 0\right)}{a}}{\color{blue}{b} + b}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.229623884976429 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.3483090680597456 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a, c \cdot -4, 0\right)}{a}}{b + b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

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