Average Error: 33.4 → 9.8
Time: 2.5m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -2.4186527003658756 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 2.58526379711982 \cdot 10^{-66}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{1}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- b) < -2.4186527003658756e+74

    1. Initial program 39.2

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

    2. Taylor expanded around inf 9.8

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

    3. Applied simplify5.0

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

    if -2.4186527003658756e+74 < (- b) < 2.58526379711982e-66

    1. Initial program 13.0

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

    3. Applied add-cube-cbrt13.2

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

    4. Applied fma-neg13.2

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

    if 2.58526379711982e-66 < (- b)

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 46.8

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

    3. Applied simplify8.4

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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)))