quadm (p42, negative)

Average Error: 33.6 → 9.9

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2616128 = b;
        double r2616129 = -r2616128;
        double r2616130 = r2616128 * r2616128;
        double r2616131 = 4.0;
        double r2616132 = a;
        double r2616133 = c;
        double r2616134 = r2616132 * r2616133;
        double r2616135 = r2616131 * r2616134;
        double r2616136 = r2616130 - r2616135;
        double r2616137 = sqrt(r2616136);
        double r2616138 = r2616129 - r2616137;
        double r2616139 = 2.0;
        double r2616140 = r2616139 * r2616132;
        double r2616141 = r2616138 / r2616140;
        return r2616141;
}

↓

double f(double a, double b, double c) {
        double r2616142 = b;
        double r2616143 = -8.1855168042470635e-53;
        bool r2616144 = r2616142 <= r2616143;
        double r2616145 = c;
        double r2616146 = r2616145 / r2616142;
        double r2616147 = -r2616146;
        double r2616148 = 3.634898599408338e+146;
        bool r2616149 = r2616142 <= r2616148;
        double r2616150 = -0.5;
        double r2616151 = -4.0;
        double r2616152 = r2616151 * r2616145;
        double r2616153 = a;
        double r2616154 = r2616142 * r2616142;
        double r2616155 = fma(r2616152, r2616153, r2616154);
        double r2616156 = sqrt(r2616155);
        double r2616157 = r2616156 + r2616142;
        double r2616158 = r2616150 * r2616157;
        double r2616159 = r2616158 / r2616153;
        double r2616160 = 2.0;
        double r2616161 = r2616142 / r2616153;
        double r2616162 = r2616145 / r2616161;
        double r2616163 = r2616162 - r2616142;
        double r2616164 = r2616160 * r2616163;
        double r2616165 = r2616164 / r2616160;
        double r2616166 = r2616165 / r2616153;
        double r2616167 = r2616149 ? r2616159 : r2616166;
        double r2616168 = r2616144 ? r2616147 : r2616167;
        return r2616168;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.6
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if b < -8.1855168042470635e-53

   1. Initial program 54.3

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

   2. Simplified 54.3

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

   4. Applied *-un-lft-identity 54.3

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

   5. Applied div-inv 54.3

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

   6. Applied times-frac 54.3

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

   7. Simplified 54.3

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

   8. Simplified 54.3

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

   9. Taylor expanded around -inf 7.5

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

   10. Simplified 7.5

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

   if -8.1855168042470635e-53 < b < 3.634898599408338e+146

   1. Initial program 13.3

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

   2. Simplified 13.3

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

   4. Applied *-un-lft-identity 13.3

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

   5. Applied div-inv 13.3

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

   6. Applied times-frac 13.4

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

   7. Simplified 13.4

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

   8. Simplified 13.4

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

   10. Applied associate-*r/ 13.3

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

   11. Simplified 13.3

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

   12. Taylor expanded around -inf 13.3

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

   13. Simplified 13.3

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

   if 3.634898599408338e+146 < b

   1. Initial program 58.5

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

   2. Simplified 58.5

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

   3. Taylor expanded around inf 10.9

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

   4. Simplified 2.1

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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