quadp (p42, positive)

Average Error: 32.8 → 29.4

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1.0088751240251126 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2004564 = b;
        double r2004565 = -r2004564;
        double r2004566 = r2004564 * r2004564;
        double r2004567 = 4.0;
        double r2004568 = a;
        double r2004569 = c;
        double r2004570 = r2004568 * r2004569;
        double r2004571 = r2004567 * r2004570;
        double r2004572 = r2004566 - r2004571;
        double r2004573 = sqrt(r2004572);
        double r2004574 = r2004565 + r2004573;
        double r2004575 = 2.0;
        double r2004576 = r2004575 * r2004568;
        double r2004577 = r2004574 / r2004576;
        return r2004577;
}

↓

double f(double a, double b, double c) {
        double r2004578 = b;
        double r2004579 = 1.0088751240251126e+68;
        bool r2004580 = r2004578 <= r2004579;
        double r2004581 = a;
        double r2004582 = c;
        double r2004583 = r2004581 * r2004582;
        double r2004584 = -4.0;
        double r2004585 = r2004578 * r2004578;
        double r2004586 = fma(r2004583, r2004584, r2004585);
        double r2004587 = cbrt(r2004586);
        double r2004588 = r2004587 * r2004587;
        double r2004589 = sqrt(r2004588);
        double r2004590 = sqrt(r2004587);
        double r2004591 = r2004589 * r2004590;
        double r2004592 = sqrt(r2004591);
        double r2004593 = sqrt(r2004586);
        double r2004594 = sqrt(r2004593);
        double r2004595 = -r2004578;
        double r2004596 = fma(r2004592, r2004594, r2004595);
        double r2004597 = r2004596 / r2004581;
        double r2004598 = 2.0;
        double r2004599 = r2004597 / r2004598;
        double r2004600 = 0.0;
        double r2004601 = r2004580 ? r2004599 : r2004600;
        return r2004601;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.8
Target	20.1
Herbie	29.4

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

Derivation

1. Split input into 2 regimes

2. if b < 1.0088751240251126e+68

   1. Initial program 24.4

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

   2. Simplified 24.4

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

   4. Applied add-sqr-sqrt 24.4

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

   5. Applied sqrt-prod 24.8

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

   6. Applied fma-neg 25.0

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

   8. Applied add-cube-cbrt 25.0

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

   9. Applied sqrt-prod 25.0

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

   if 1.0088751240251126e+68 < b

   1. Initial program 57.1

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

   2. Simplified 57.1

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

   3. Taylor expanded around 0 41.9

      \[\leadsto \frac{\color{blue}{0}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.0088751240251126 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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