Average Error: 34.7 → 11.0
Time: 8.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.87486430558009272 \cdot 10^{54}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le -9.8475017814633646 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{4}{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.09798512605357415 \cdot 10^{-61}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le 6.9721274759377412 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.87486430558009272 \cdot 10^{54}:\\
\;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\

\mathbf{elif}\;b \le -9.8475017814633646 \cdot 10^{24}:\\
\;\;\;\;\frac{\frac{4}{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\

\mathbf{elif}\;b \le -3.09798512605357415 \cdot 10^{-61}:\\
\;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\

\mathbf{elif}\;b \le 6.9721274759377412 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r150075 = b;
        double r150076 = -r150075;
        double r150077 = r150075 * r150075;
        double r150078 = 4.0;
        double r150079 = a;
        double r150080 = r150078 * r150079;
        double r150081 = c;
        double r150082 = r150080 * r150081;
        double r150083 = r150077 - r150082;
        double r150084 = sqrt(r150083);
        double r150085 = r150076 + r150084;
        double r150086 = 2.0;
        double r150087 = r150086 * r150079;
        double r150088 = r150085 / r150087;
        return r150088;
}
↓
double f(double a, double b, double c) {
        double r150089 = b;
        double r150090 = -5.874864305580093e+54;
        bool r150091 = r150089 <= r150090;
        double r150092 = 1.0;
        double r150093 = 1.0;
        double r150094 = c;
        double r150095 = r150094 / r150089;
        double r150096 = a;
        double r150097 = r150089 / r150096;
        double r150098 = r150095 - r150097;
        double r150099 = r150093 * r150098;
        double r150100 = r150092 * r150099;
        double r150101 = -9.847501781463365e+24;
        bool r150102 = r150089 <= r150101;
        double r150103 = 4.0;
        double r150104 = r150089 * r150089;
        double r150105 = r150103 * r150096;
        double r150106 = r150105 * r150094;
        double r150107 = r150104 - r150106;
        double r150108 = -r150107;
        double r150109 = fma(r150089, r150089, r150108);
        double r150110 = r150096 * r150094;
        double r150111 = r150109 / r150110;
        double r150112 = r150103 / r150111;
        double r150113 = -r150089;
        double r150114 = sqrt(r150107);
        double r150115 = r150113 + r150114;
        double r150116 = r150112 * r150115;
        double r150117 = 2.0;
        double r150118 = r150117 * r150096;
        double r150119 = r150116 / r150118;
        double r150120 = -3.097985126053574e-61;
        bool r150121 = r150089 <= r150120;
        double r150122 = 6.972127475937741e+134;
        bool r150123 = r150089 <= r150122;
        double r150124 = r150113 - r150114;
        double r150125 = r150092 / r150124;
        double r150126 = r150117 / r150103;
        double r150127 = r150126 * r150092;
        double r150128 = r150127 / r150094;
        double r150129 = r150125 / r150128;
        double r150130 = r150092 * r150129;
        double r150131 = -1.0;
        double r150132 = r150131 * r150095;
        double r150133 = r150092 * r150132;
        double r150134 = r150123 ? r150130 : r150133;
        double r150135 = r150121 ? r150100 : r150134;
        double r150136 = r150102 ? r150119 : r150135;
        double r150137 = r150091 ? r150100 : r150136;
        return r150137;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.7
Target21.9
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.874864305580093e+54 or -9.847501781463365e+24 < b < -3.097985126053574e-61

    1. Initial program 32.6

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

    3. Applied flip-+57.0

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

    4. Simplified56.5

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num56.5

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified56.5

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

    9. Applied div-inv56.5

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

    10. Simplified56.3

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

    11. Taylor expanded around -inf 10.1

      \[\leadsto 1 \cdot \color{blue}{\left(1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\right)}\]

    12. Simplified10.1

      \[\leadsto 1 \cdot \color{blue}{\left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)}\]

    if -5.874864305580093e+54 < b < -9.847501781463365e+24

    1. Initial program 5.2

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

    3. Applied flip-+50.9

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

    4. Simplified50.9

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied flip--50.9

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    7. Applied associate-/r/50.9

      \[\leadsto \frac{\color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    8. Simplified49.3

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

    if -3.097985126053574e-61 < b < 6.972127475937741e+134

    1. Initial program 26.9

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

    3. Applied flip-+29.7

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

    4. Simplified17.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num17.9

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified17.0

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

    9. Applied div-inv17.0

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

    10. Simplified12.8

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

    if 6.972127475937741e+134 < b

    1. Initial program 62.0

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

    3. Applied flip-+62.0

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

    4. Simplified37.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num37.2

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified36.7

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

    9. Applied div-inv36.7

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

    10. Simplified36.3

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

    11. Taylor expanded around inf 2.0

      \[\leadsto 1 \cdot \color{blue}{\left(-1 \cdot \frac{c}{b}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.87486430558009272 \cdot 10^{54}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le -9.8475017814633646 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{4}{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.09798512605357415 \cdot 10^{-61}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le 6.9721274759377412 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{\frac{2}{4} \cdot 1}{c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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