Average Error: 33.7 → 10.9
Time: 52.8s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.661687511607583 \cdot 10^{-109}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*} + b\right) \cdot \frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.7
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.661687511607583e-109

    1. Initial program 51.0

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

    2. Simplified51.0

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

    4. Applied *-un-lft-identity51.0

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

    5. Applied div-inv51.0

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

    6. Applied times-frac51.0

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

    7. Simplified51.0

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

    8. Simplified51.0

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

    9. Taylor expanded around -inf 10.9

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

    10. Simplified10.9

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

    if -4.661687511607583e-109 < b < 1.7760913692530284e+28

    1. Initial program 13.3

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

    2. Simplified13.2

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

    4. Applied *-un-lft-identity13.2

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

    5. Applied div-inv13.2

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

    6. Applied times-frac13.4

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

    7. Simplified13.4

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

    8. Simplified13.4

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

    10. Applied associate-*r/13.3

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

    if 1.7760913692530284e+28 < b

    1. Initial program 33.8

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

    2. Simplified33.8

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

    3. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.661687511607583 \cdot 10^{-109}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*} + b\right) \cdot \frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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