Quadratic roots, narrow range

Average Error: 28.3 → 0.5

Time: 18.2s

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

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/ (/ (* c (* -4.0 a)) (+ b (sqrt (fma (* c -4.0) a (* b b))))) (* 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 ((c * (-4.0 * a)) / (b + sqrt(fma((c * -4.0), a, (b * b))))) / (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(c * Float64(-4.0 * a)) / Float64(b + sqrt(fma(Float64(c * -4.0), a, Float64(b * b))))) / 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[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.3

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

2. Simplified 28.3

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

   [Start] 28.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 28.4

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

4. Applied egg-rr 27.4

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

5. Simplified 27.2

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

   [Start] 27.4	\[ \frac{\frac{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 27.2	\[ \frac{\frac{\color{blue}{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   +-commutative [<=] 27.2	\[ \frac{\frac{\color{blue}{\left(-4 \cdot \left(c \cdot a\right) + b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   associate-*r* [=>] 27.2	\[ \frac{\frac{\left(\color{blue}{\left(-4 \cdot c\right) \cdot a} + b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   fma-def [=>] 27.2	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 27.2	\[ \frac{\frac{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 27.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{b \cdot b + -4 \cdot \left(c \cdot a\right)}}}}{a \cdot 2} \]
   +-commutative [<=] 27.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + b \cdot b}}}}{a \cdot 2} \]
   associate-*r* [=>] 27.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{\left(-4 \cdot c\right) \cdot a} + b \cdot b}}}{a \cdot 2} \]
   fma-def [=>] 27.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}{a \cdot 2} \]
   *-commutative [=>] 27.2	\[ \frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, b \cdot b\right)}}}{a \cdot 2} \]

6. Taylor expanded in c around 0 0.5

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

7. Simplified 0.5

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

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

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	9.6
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 6.8:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 2
Error	9.6
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3
Error	12.1
Cost	7232

\[\frac{-c}{b} - a \cdot \frac{c}{\frac{{b}^{3}}{c}} \]

Alternative 4
Error	23.0
Cost	256

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

Reproduce

herbie shell --seed 2023054 
(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)))