Average Error: 33.9 → 10.5
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.794908999297654 \cdot 10^{-113}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 1.3722158195955077 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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

Original33.9
Target21.0
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.794908999297654e-113

    1. Initial program 50.7

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

    2. Taylor expanded around -inf 47.5

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

    3. Applied simplify11.3

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

    if -2.794908999297654e-113 < b < 1.3722158195955077e+141

    1. Initial program 11.9

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

    if 1.3722158195955077e+141 < b

    1. Initial program 56.1

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

    2. Taylor expanded around inf 11.0

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

    3. Applied simplify2.6

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

  4. Applied simplify10.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -2.794908999297654 \cdot 10^{-113}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 1.3722158195955077 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018206 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))