Average Error: 1.5 → 1.5
Time: 26.0s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]
\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}
double f(double a, double b, double c) {
        double r1781104 = b;
        double r1781105 = -r1781104;
        double r1781106 = r1781104 * r1781104;
        double r1781107 = 4.0;
        double r1781108 = /* ERROR: no posit support in C */;
        double r1781109 = a;
        double r1781110 = c;
        double r1781111 = r1781109 * r1781110;
        double r1781112 = r1781108 * r1781111;
        double r1781113 = r1781106 - r1781112;
        double r1781114 = sqrt(r1781113);
        double r1781115 = r1781105 + r1781114;
        double r1781116 = 2.0;
        double r1781117 = /* ERROR: no posit support in C */;
        double r1781118 = r1781117 * r1781109;
        double r1781119 = r1781115 / r1781118;
        return r1781119;
}
↓
double f(double a, double b, double c) {
        double r1781120 = b;
        double r1781121 = r1781120 * r1781120;
        double r1781122 = c;
        double r1781123 = a;
        double r1781124 = r1781122 * r1781123;
        double r1781125 = 4.0;
        double r1781126 = r1781124 * r1781125;
        double r1781127 = r1781121 - r1781126;
        double r1781128 = sqrt(r1781127);
        double r1781129 = r1781128 - r1781120;
        double r1781130 = 2.0;
        double r1781131 = r1781129 / r1781130;
        double r1781132 = r1781131 / r1781123;
        return r1781132;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied associate-/r*1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))