Quadratic roots, narrow range

Average Error: 44.25% → 0.45%

Time: 18.2s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Derivation

1. Initial program 44.25

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

2. Simplified 44.25

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

   [Start] 44.25	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 44.25	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 43.05

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

4. Simplified 42.72

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

   [Start] 43.05	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/l/ [=>] 43.05	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   fma-def [<=] 42.73	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 42.73	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [=>] 42.73	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   distribute-lft-neg-in [<=] 42.73	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   rem-square-sqrt [=>] 42.71	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   fma-def [<=] 42.72	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right)}}{a \cdot 2} \]
   +-commutative [=>] 42.72	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)}}{a \cdot 2} \]
   fma-def [=>] 42.72	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)}}{a \cdot 2} \]

5. Taylor expanded in b around 0 0.72

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

6. Applied egg-rr 0.69

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

7. Applied egg-rr 38.04

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(\left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \cdot \frac{0.5}{a}\right)\right)} - 1} \]

8. Simplified 0.45

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

   [Start] 38.04	\[ e^{\mathsf{log1p}\left(a \cdot \left(\left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \cdot \frac{0.5}{a}\right)\right)} - 1 \]
   expm1-def [=>] 15.84	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(\left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \cdot \frac{0.5}{a}\right)\right)\right)} \]
   expm1-log1p [=>] 0.72	\[ \color{blue}{a \cdot \left(\left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \cdot \frac{0.5}{a}\right)} \]
   *-commutative [=>] 0.72	\[ a \cdot \color{blue}{\left(\frac{0.5}{a} \cdot \left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)\right)} \]
   associate-*r* [=>] 0.59	\[ \color{blue}{\left(a \cdot \frac{0.5}{a}\right) \cdot \left(-4 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)} \]
   associate-*r* [=>] 0.59	\[ \color{blue}{\left(\left(a \cdot \frac{0.5}{a}\right) \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}} \]
   *-commutative [=>] 0.59	\[ \left(\color{blue}{\left(\frac{0.5}{a} \cdot a\right)} \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-*l/ [=>] 0.45	\[ \left(\color{blue}{\frac{0.5 \cdot a}{a}} \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-/l* [=>] 0.45	\[ \left(\color{blue}{\frac{0.5}{\frac{a}{a}}} \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-inverses [=>] 0.45	\[ \left(\frac{0.5}{\color{blue}{1}} \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   metadata-eval [=>] 0.45	\[ \left(\color{blue}{0.5} \cdot -4\right) \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   metadata-eval [=>] 0.45	\[ \color{blue}{-2} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

9. Final simplification 0.45

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

Alternatives

Alternative 1
Error	0.79%
Cost	7744

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

Alternative 2
Error	0.69%
Cost	7744

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

Alternative 3
Error	15.26%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 102:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Error	18.28%
Cost	1344

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

Alternative 5
Error	18.78%
Cost	1024

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

Alternative 6
Error	35.91%
Cost	256

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

Alternative 7
Error	98.4%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(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)))