Quadratic roots, narrow range

Average Error: 28.6 → 0.5

Time: 16.6s

Precision: binary64

Cost: 14016

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/ (/ (* (* -4.0 c) a) (+ b (sqrt (fma b b (* c (* -4.0 a)))))) (* a 2.0)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	return (((-4.0 * c) * a) / (b + sqrt(fma(b, b, (c * (-4.0 * a)))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-4.0 * c) * a) / Float64(b + sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6

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

2. Simplified 28.6

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

   [Start] 28.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 27.8

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

4. Taylor expanded in b around 0 0.5

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

5. Simplified 0.5

   \[\leadsto \frac{\frac{\color{blue}{\left(-4 \cdot c\right) \cdot a}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   Proof

   [Start] 0.5	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   associate-*r* [=>] 0.5	\[ \frac{\frac{\color{blue}{\left(-4 \cdot c\right) \cdot a}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	7.7
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00205:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-4 \cdot c\right) \cdot a}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 2
Error	7.7
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(-4 \cdot a\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.00205:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-4 \cdot c\right) \cdot a}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	7.8
Cost	15876

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.00205:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-4 \cdot c\right) \cdot a}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Error	10.1
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.0002:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-4 \cdot c\right) \cdot a}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	7492

\[\begin{array}{l} t_0 := \left(-4 \cdot c\right) \cdot a\\ \mathbf{if}\;b \leq 0.225:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{t_0 + b \cdot b} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 6
Error	11.6
Cost	1344

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

Alternative 7
Error	22.8
Cost	256

\[\frac{-c}{b} \]

Reproduce

herbie shell --seed 2023059 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))