The quadratic formula (r1)

Average Error: 34.9 → 10.1

Time: 18.3s

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 r7190785 = b;
        double r7190786 = -r7190785;
        double r7190787 = r7190785 * r7190785;
        double r7190788 = 4.0;
        double r7190789 = a;
        double r7190790 = r7190788 * r7190789;
        double r7190791 = c;
        double r7190792 = r7190790 * r7190791;
        double r7190793 = r7190787 - r7190792;
        double r7190794 = sqrt(r7190793);
        double r7190795 = r7190786 + r7190794;
        double r7190796 = 2.0;
        double r7190797 = r7190796 * r7190789;
        double r7190798 = r7190795 / r7190797;
        return r7190798;
}

↓

double f(double a, double b, double c) {
        double r7190799 = b;
        double r7190800 = -3.6803290429888884e+148;
        bool r7190801 = r7190799 <= r7190800;
        double r7190802 = c;
        double r7190803 = r7190802 / r7190799;
        double r7190804 = a;
        double r7190805 = r7190799 / r7190804;
        double r7190806 = r7190803 - r7190805;
        double r7190807 = 1.0;
        double r7190808 = r7190806 * r7190807;
        double r7190809 = 4.6129908231112306e-104;
        bool r7190810 = r7190799 <= r7190809;
        double r7190811 = 1.0;
        double r7190812 = 2.0;
        double r7190813 = r7190804 * r7190812;
        double r7190814 = r7190811 / r7190813;
        double r7190815 = r7190799 * r7190799;
        double r7190816 = r7190804 * r7190802;
        double r7190817 = 4.0;
        double r7190818 = r7190816 * r7190817;
        double r7190819 = r7190815 - r7190818;
        double r7190820 = sqrt(r7190819);
        double r7190821 = r7190820 - r7190799;
        double r7190822 = r7190814 * r7190821;
        double r7190823 = -1.0;
        double r7190824 = r7190823 * r7190803;
        double r7190825 = r7190810 ? r7190822 : r7190824;
        double r7190826 = r7190801 ? r7190808 : r7190825;
        return r7190826;
}

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)))