quadm (p42, negative)

Average Error: 34.1 → 9.5

Time: 16.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b} \cdot 1\\ \mathbf{elif}\;b \le 3.389622183547600500259532466700370210031 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{1}{-2}}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, {b}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{\left(-a\right) \cdot 2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r61909 = b;
        double r61910 = -r61909;
        double r61911 = r61909 * r61909;
        double r61912 = 4.0;
        double r61913 = a;
        double r61914 = c;
        double r61915 = r61913 * r61914;
        double r61916 = r61912 * r61915;
        double r61917 = r61911 - r61916;
        double r61918 = sqrt(r61917);
        double r61919 = r61910 - r61918;
        double r61920 = 2.0;
        double r61921 = r61920 * r61913;
        double r61922 = r61919 / r61921;
        return r61922;
}

↓

double f(double a, double b, double c) {
        double r61923 = b;
        double r61924 = -4.356959927988237e-56;
        bool r61925 = r61923 <= r61924;
        double r61926 = c;
        double r61927 = r61926 / r61923;
        double r61928 = 1.0;
        double r61929 = r61927 * r61928;
        double r61930 = -r61929;
        double r61931 = 3.3896221835476005e+130;
        bool r61932 = r61923 <= r61931;
        double r61933 = 1.0;
        double r61934 = 2.0;
        double r61935 = -r61934;
        double r61936 = r61933 / r61935;
        double r61937 = a;
        double r61938 = r61936 / r61937;
        double r61939 = -r61937;
        double r61940 = 4.0;
        double r61941 = r61939 * r61940;
        double r61942 = 2.0;
        double r61943 = pow(r61923, r61942);
        double r61944 = fma(r61941, r61926, r61943);
        double r61945 = sqrt(r61944);
        double r61946 = r61923 + r61945;
        double r61947 = r61938 * r61946;
        double r61948 = r61923 + r61923;
        double r61949 = r61939 * r61934;
        double r61950 = r61948 / r61949;
        double r61951 = r61932 ? r61947 : r61950;
        double r61952 = r61925 ? r61930 : r61951;
        return r61952;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.1
Target	21.1
Herbie	9.5

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

Derivation

1. Split input into 3 regimes

2. if b < -4.356959927988237e-56

   1. Initial program 54.0

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

   2. Simplified 54.0

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

   4. Applied fma-udef 54.0

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

   5. Simplified 54.0

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

   7. Applied frac-2neg 54.0

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

   8. Simplified 54.0

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

   9. Simplified 54.0

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

   11. Applied div-inv 54.0

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

   12. Simplified 54.0

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

   13. Taylor expanded around -inf 7.7

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

   14. Simplified 7.7

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

   if -4.356959927988237e-56 < b < 3.3896221835476005e+130

   1. Initial program 12.6

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

   2. Simplified 12.6

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

   4. Applied fma-udef 12.6

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

   5. Simplified 12.6

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

   7. Applied frac-2neg 12.6

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

   8. Simplified 12.6

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

   9. Simplified 12.6

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

   11. Applied div-inv 12.8

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

   12. Simplified 12.8

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

   if 3.3896221835476005e+130 < b

   1. Initial program 56.2

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

   2. Simplified 56.2

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

   4. Applied fma-udef 56.2

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

   5. Simplified 56.2

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

   7. Applied frac-2neg 56.2

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

   8. Simplified 56.2

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

   9. Simplified 56.2

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

   10. Taylor expanded around 0 2.6

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

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b} \cdot 1\\ \mathbf{elif}\;b \le 3.389622183547600500259532466700370210031 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{1}{-2}}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, {b}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{\left(-a\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))