Average Error: 33.4 → 9.7

Time: 24.6s

Precision: 64

Internal Precision: 3392

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.91564606410173 \cdot 10^{+122}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.820248000579309 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

The X axis uses an exponential scale — Bits error versus `a`

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In
Out

Enter valid numbers for all inputs

Target

Original	33.4
Target	20.8
Herbie	9.7

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

Derivation

Split input into 3 regimes
if b < -2.91564606410173e+122
1. Initial program 50.0
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Initial simplification50.0
  \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
3. Taylor expanded around -inf 2.6
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
if -2.91564606410173e+122 < b < 5.820248000579309e-42
1. Initial program 13.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Initial simplification13.6
  \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
3. Using strategy rm
4. Applied div-sub13.6
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}}\]
if 5.820248000579309e-42 < b
1. Initial program 53.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Initial simplification53.5
  \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
3. Using strategy rm
4. Applied div-sub54.4
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}}\]
5. Taylor expanded around inf 7.3
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
6. Simplified7.3
  \[\leadsto \color{blue}{\frac{-c}{b}}\]
Recombined 3 regimes into one program.
Final simplification9.7
\[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.91564606410173 \cdot 10^{+122}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.820248000579309 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	33.5	9.7	5.6	27.9	85.1%

herbie shell --seed 2018285 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Error

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Target

Derivation

`if b < -2.91564606410173e+122`

`if -2.91564606410173e+122 < b < 5.820248000579309e-42`

`if 5.820248000579309e-42 < b`

Runtime