Average Error: 34.7 → 10.3
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.6537017569063518 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r85066 = b;
        double r85067 = -r85066;
        double r85068 = r85066 * r85066;
        double r85069 = 4.0;
        double r85070 = a;
        double r85071 = c;
        double r85072 = r85070 * r85071;
        double r85073 = r85069 * r85072;
        double r85074 = r85068 - r85073;
        double r85075 = sqrt(r85074);
        double r85076 = r85067 - r85075;
        double r85077 = 2.0;
        double r85078 = r85077 * r85070;
        double r85079 = r85076 / r85078;
        return r85079;
}
↓
double f(double a, double b, double c) {
        double r85080 = b;
        double r85081 = -4.653701756906352e-82;
        bool r85082 = r85080 <= r85081;
        double r85083 = -1.0;
        double r85084 = c;
        double r85085 = r85084 / r85080;
        double r85086 = r85083 * r85085;
        double r85087 = 1.0479007947857462e+99;
        bool r85088 = r85080 <= r85087;
        double r85089 = -r85080;
        double r85090 = r85080 * r85080;
        double r85091 = 4.0;
        double r85092 = a;
        double r85093 = r85092 * r85084;
        double r85094 = r85091 * r85093;
        double r85095 = r85090 - r85094;
        double r85096 = sqrt(r85095);
        double r85097 = r85089 - r85096;
        double r85098 = 2.0;
        double r85099 = r85098 * r85092;
        double r85100 = r85097 / r85099;
        double r85101 = 1.0;
        double r85102 = r85080 / r85092;
        double r85103 = r85085 - r85102;
        double r85104 = r85101 * r85103;
        double r85105 = r85088 ? r85100 : r85104;
        double r85106 = r85082 ? r85086 : r85105;
        return r85106;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.7
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.653701756906352e-82

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.2

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

    if -4.653701756906352e-82 < b < 1.0479007947857462e+99

    1. Initial program 13.5

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

    if 1.0479007947857462e+99 < b

    1. Initial program 47.7

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))