Average Error: 43.6 → 3.1
Time: 8.2s
Precision: binary64
\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[-\mathsf{fma}\left(2, \frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}} \cdot \frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-\mathsf{fma}\left(2, \frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}} \cdot \frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (fma
   2.0
   (*
    (/ (pow c 3.0) (pow (cbrt b) 5.0))
    (/ (* a a) (pow (* (cbrt b) (cbrt b)) 5.0)))
   (fma
    5.0
    (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))
    (+ (/ c b) (/ (* c (* c a)) (pow b 3.0)))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	return -fma(2.0, ((pow(c, 3.0) / pow(cbrt(b), 5.0)) * ((a * a) / pow((cbrt(b) * cbrt(b)), 5.0))), fma(5.0, ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)), ((c / b) + ((c * (c * a)) / pow(b, 3.0)))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded in b around inf 3.1

    \[\leadsto \color{blue}{-\left(2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)\right)\right)} \]

  3. Simplified3.1

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)} \]

  4. Applied add-log-exp_binary644.6

    \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\log \left(e^{\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}}\right)}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  5. Applied add-cube-cbrt_binary644.6

    \[\leadsto -\mathsf{fma}\left(2, \log \left(e^{\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{\color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)}}^{5}}}\right), \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  6. Applied unpow-prod-down_binary644.6

    \[\leadsto -\mathsf{fma}\left(2, \log \left(e^{\frac{\left(a \cdot a\right) \cdot {c}^{3}}{\color{blue}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5} \cdot {\left(\sqrt[3]{b}\right)}^{5}}}}\right), \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  7. Applied times-frac_binary644.6

    \[\leadsto -\mathsf{fma}\left(2, \log \left(e^{\color{blue}{\frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}} \cdot \frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}}}}\right), \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  8. Applied exp-prod_binary649.5

    \[\leadsto -\mathsf{fma}\left(2, \log \color{blue}{\left({\left(e^{\frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}}\right)}^{\left(\frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}}\right)}\right)}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  9. Applied log-pow_binary649.1

    \[\leadsto -\mathsf{fma}\left(2, \color{blue}{\frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}} \cdot \log \left(e^{\frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}}\right)}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  10. Simplified3.1

    \[\leadsto -\mathsf{fma}\left(2, \frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}} \cdot \color{blue}{\frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

  11. Final simplification3.1

    \[\leadsto -\mathsf{fma}\left(2, \frac{{c}^{3}}{{\left(\sqrt[3]{b}\right)}^{5}} \cdot \frac{a \cdot a}{{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right) \]

Reproduce

herbie shell --seed 2022081 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))