Average Error: 33.2 → 6.6
Time: 1.8m
Precision: 64
Internal Precision: 3200
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -4.525036217684268 \cdot 10^{+158}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le -5.266140246506573 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}{c}}\\ \mathbf{if}\;b/2 \le 6.320733100371715 \cdot 10^{+112}:\\ \;\;\;\;\frac{-b/2}{a} - \frac{\sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b/2 < -4.525036217684268e+158

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 14.0

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}}{a}\]

    3. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{c}{b/2} \cdot \frac{-1}{2}}\]

    if -4.525036217684268e+158 < b/2 < -5.266140246506573e-287

    1. Initial program 34.8

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

    3. Applied flip--34.9

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

    4. Applied simplify16.4

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

    6. Applied clear-num16.7

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

    7. Applied simplify8.6

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

    if -5.266140246506573e-287 < b/2 < 6.320733100371715e+112

    1. Initial program 8.8

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

    3. Applied div-sub8.8

      \[\leadsto \color{blue}{\frac{-b/2}{a} - \frac{\sqrt{b/2 \cdot b/2 - a \cdot c}}{a}}\]

    if 6.320733100371715e+112 < b/2

    1. Initial program 47.3

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

    3. Applied flip--61.8

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

    4. Applied simplify62.0

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

    6. Applied clear-num62.0

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

    7. Applied simplify61.9

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

    8. Taylor expanded around inf 3.0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (a b/2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))