quadm (p42, negative)

Average Error: 33.3 → 6.6

Time: 3.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)}}\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)} + b\right) \cdot \frac{\frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r13170101 = b;
        double r13170102 = -r13170101;
        double r13170103 = r13170101 * r13170101;
        double r13170104 = 4.0;
        double r13170105 = a;
        double r13170106 = c;
        double r13170107 = r13170105 * r13170106;
        double r13170108 = r13170104 * r13170107;
        double r13170109 = r13170103 - r13170108;
        double r13170110 = sqrt(r13170109);
        double r13170111 = r13170102 - r13170110;
        double r13170112 = 2.0;
        double r13170113 = r13170112 * r13170105;
        double r13170114 = r13170111 / r13170113;
        return r13170114;
}

↓

double f(double a, double b, double c) {
        double r13170115 = b;
        double r13170116 = -3.411206454162785e+120;
        bool r13170117 = r13170115 <= r13170116;
        double r13170118 = c;
        double r13170119 = r13170118 / r13170115;
        double r13170120 = -r13170119;
        double r13170121 = 8.142093116881289e-248;
        bool r13170122 = r13170115 <= r13170121;
        double r13170123 = -2.0;
        double r13170124 = a;
        double r13170125 = r13170124 * r13170118;
        double r13170126 = -4.0;
        double r13170127 = r13170125 * r13170126;
        double r13170128 = fma(r13170115, r13170115, r13170127);
        double r13170129 = sqrt(r13170128);
        double r13170130 = r13170115 - r13170129;
        double r13170131 = r13170123 / r13170130;
        double r13170132 = r13170118 * r13170131;
        double r13170133 = 5.419916601733116e+77;
        bool r13170134 = r13170115 <= r13170133;
        double r13170135 = r13170115 * r13170115;
        double r13170136 = fma(r13170125, r13170126, r13170135);
        double r13170137 = sqrt(r13170136);
        double r13170138 = r13170137 + r13170115;
        double r13170139 = -0.5;
        double r13170140 = r13170139 / r13170124;
        double r13170141 = r13170138 * r13170140;
        double r13170142 = r13170115 / r13170124;
        double r13170143 = r13170119 - r13170142;
        double r13170144 = r13170134 ? r13170141 : r13170143;
        double r13170145 = r13170122 ? r13170132 : r13170144;
        double r13170146 = r13170117 ? r13170120 : r13170145;
        return r13170146;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.3
Target	20.6
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if b < -3.411206454162785e+120

   1. Initial program 59.4

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

   2. Simplified 59.4

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

   4. Applied *-un-lft-identity 59.4

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

   5. Applied div-inv 59.4

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

   6. Applied times-frac 59.4

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

   7. Simplified 59.4

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

   8. Simplified 59.4

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

   9. Taylor expanded around -inf 1.9

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

   10. Simplified 1.9

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

   if -3.411206454162785e+120 < b < 8.142093116881289e-248

   1. Initial program 30.9

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

   2. Simplified 30.9

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

   4. Applied *-un-lft-identity 30.9

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

   5. Applied div-inv 30.9

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

   6. Applied times-frac 31.0

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

   7. Simplified 31.0

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

   8. Simplified 31.0

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

   10. Applied flip-+ 31.1

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

   11. Applied distribute-neg-frac 31.1

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

   12. Applied frac-times 36.0

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

   13. Simplified 20.1

       \[\leadsto \frac{\color{blue}{\left(0 - a \cdot \left(c \cdot -4\right)\right) \cdot \frac{-1}{2}}}{\left(b - \sqrt{\mathsf{fma}\left(a, \left(c \cdot -4\right), \left(b \cdot b\right)\right)}\right) \cdot a}\]
   14. Using strategy rm

   15. Applied sub0-neg 20.1

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

   16. Applied distribute-lft-neg-out 20.1

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

   17. Applied distribute-frac-neg 20.1

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

   18. Simplified 8.9

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

   if 8.142093116881289e-248 < b < 5.419916601733116e+77

   1. Initial program 8.5

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

   2. Simplified 8.5

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

   4. Applied *-un-lft-identity 8.5

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

   5. Applied div-inv 8.5

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

   6. Applied times-frac 8.7

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

   7. Simplified 8.7

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

   8. Simplified 8.7

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

   9. Taylor expanded around -inf 8.7

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

   10. Simplified 8.7

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

   if 5.419916601733116e+77 < b

   1. Initial program 40.5

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

   2. Simplified 40.5

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

   4. Applied *-un-lft-identity 40.5

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

   5. Applied div-inv 40.5

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

   6. Applied times-frac 40.6

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

   7. Simplified 40.6

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

   8. Simplified 40.6

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

   9. Taylor expanded around inf 4.7

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)}}\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\left(a \cdot c\right), -4, \left(b \cdot b\right)\right)} + b\right) \cdot \frac{\frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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)))