The quadratic formula (r1)

Average Error: 34.9 → 10.1

Time: 20.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r5011587 = b;
        double r5011588 = -r5011587;
        double r5011589 = r5011587 * r5011587;
        double r5011590 = 4.0;
        double r5011591 = a;
        double r5011592 = r5011590 * r5011591;
        double r5011593 = c;
        double r5011594 = r5011592 * r5011593;
        double r5011595 = r5011589 - r5011594;
        double r5011596 = sqrt(r5011595);
        double r5011597 = r5011588 + r5011596;
        double r5011598 = 2.0;
        double r5011599 = r5011598 * r5011591;
        double r5011600 = r5011597 / r5011599;
        return r5011600;
}

↓

double f(double a, double b, double c) {
        double r5011601 = b;
        double r5011602 = -3.6803290429888884e+148;
        bool r5011603 = r5011601 <= r5011602;
        double r5011604 = c;
        double r5011605 = r5011604 / r5011601;
        double r5011606 = a;
        double r5011607 = r5011601 / r5011606;
        double r5011608 = r5011605 - r5011607;
        double r5011609 = 1.0;
        double r5011610 = r5011608 * r5011609;
        double r5011611 = 4.6129908231112306e-104;
        bool r5011612 = r5011601 <= r5011611;
        double r5011613 = 1.0;
        double r5011614 = 2.0;
        double r5011615 = r5011606 * r5011614;
        double r5011616 = r5011613 / r5011615;
        double r5011617 = r5011601 * r5011601;
        double r5011618 = r5011606 * r5011604;
        double r5011619 = 4.0;
        double r5011620 = r5011618 * r5011619;
        double r5011621 = r5011617 - r5011620;
        double r5011622 = sqrt(r5011621);
        double r5011623 = r5011622 - r5011601;
        double r5011624 = r5011616 * r5011623;
        double r5011625 = -1.0;
        double r5011626 = r5011625 * r5011605;
        double r5011627 = r5011612 ? r5011624 : r5011626;
        double r5011628 = r5011603 ? r5011610 : r5011627;
        return r5011628;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.9
Target	21.3
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if b < -3.6803290429888884e+148

   1. Initial program 62.1

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

   3. Applied clear-num 62.2

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

   4. Simplified 62.2

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

   6. Applied div-inv 62.2

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

   7. Applied *-un-lft-identity 62.2

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

   8. Applied times-frac 62.2

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

   9. Applied add-cube-cbrt 62.2

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

   10. Applied times-frac 62.2

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

   11. Simplified 62.2

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

   12. Simplified 62.2

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

   13. Taylor expanded around -inf 2.3

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

   14. Simplified 2.3

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

   if -3.6803290429888884e+148 < b < 4.6129908231112306e-104

   1. Initial program 12.2

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

   3. Applied clear-num 12.3

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

   4. Simplified 12.3

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

   6. Applied div-inv 12.3

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

   7. Applied *-un-lft-identity 12.3

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

   8. Applied times-frac 12.4

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

   9. Applied add-cube-cbrt 12.4

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

   10. Applied times-frac 12.3

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

   11. Simplified 12.3

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

   12. Simplified 12.3

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

   if 4.6129908231112306e-104 < b

   1. Initial program 52.7

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.680329042988888396603264581948851078331 \cdot 10^{148}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))