Quadratic roots, narrow range

Average Error: 28.4 → 0.3

Time: 19.8s

Precision: binary64

Cost: 13760

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

Math

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

↓

\[0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}} \]

FPCore

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

↓

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

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

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

Derivation

1. Initial program 28.4

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

2. Simplified 28.4

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

   [Start] 28.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 29.0

   \[\leadsto \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{a \cdot 2} \]

4. Applied egg-rr 15.0

   \[\leadsto \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}}}{a \cdot 2} \]

5. Simplified 15.0

   \[\leadsto \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\color{blue}{\frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}}}}}{a \cdot 2} \]
   Proof

   [Start] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c \cdot \left(a \cdot -4\right)}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}}{a \cdot 2} \]
   associate-/l* [=>] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\color{blue}{\frac{c}{\frac{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot -4}}}}}}{a \cdot 2} \]
   *-commutative [=>] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c}{\frac{-\left(b + \sqrt{b \cdot b - \color{blue}{\left(a \cdot 4\right)} \cdot c}\right)}{a \cdot -4}}}}}{a \cdot 2} \]
   associate-*r* [<=] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c}{\frac{-\left(b + \sqrt{b \cdot b - \color{blue}{a \cdot \left(4 \cdot c\right)}}\right)}{a \cdot -4}}}}}{a \cdot 2} \]
   fma-neg [=>] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c}{\frac{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -a \cdot \left(4 \cdot c\right)\right)}}\right)}{a \cdot -4}}}}}{a \cdot 2} \]
   *-commutative [=>] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, -a \cdot \color{blue}{\left(c \cdot 4\right)}\right)}\right)}{a \cdot -4}}}}}{a \cdot 2} \]
   distribute-rgt-neg-in [=>] 15.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(-c \cdot 4\right)}\right)}\right)}{a \cdot -4}}}}}{a \cdot 2} \]

6. Applied egg-rr 15.0

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(\frac{1}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}\right)}}{a \cdot 2} \]

7. Simplified 0.4

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

   [Start] 15.0	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(\frac{1}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}\right)}{a \cdot 2} \]
   associate-*l/ [=>] 15.0	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \color{blue}{\frac{1 \cdot \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}{a \cdot 2} \]
   *-lft-identity [=>] 15.0	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{\color{blue}{\frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}}}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}{a \cdot 2} \]
   associate-*r/ [=>] 15.0	\[ \frac{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right) \cdot \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}{a \cdot 2} \]
   associate-*l/ [<=] 14.9	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot \frac{c}{\frac{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)}{a \cdot -4}}}}{a \cdot 2} \]
   associate-/r/ [=>] 14.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}{b \cdot b - \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} \cdot \color{blue}{\left(\frac{c}{-\left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-c \cdot 4\right)\right)}\right)} \cdot \left(a \cdot -4\right)\right)}}{a \cdot 2} \]

8. Applied egg-rr 24.8

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{a \cdot 2}\right)} - 1} \]

9. Simplified 0.3

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

   [Start] 24.8	\[ e^{\mathsf{log1p}\left(\frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{a \cdot 2}\right)} - 1 \]
   expm1-def [=>] 10.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{a \cdot 2}\right)\right)} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{\frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{a \cdot 2}} \]
   *-commutative [=>] 0.4	\[ \frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{\color{blue}{2 \cdot a}} \]
   associate-/r* [=>] 0.4	\[ \color{blue}{\frac{\frac{-4 \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{2}}{a}} \]
   associate-/l* [=>] 0.5	\[ \frac{\color{blue}{\frac{-4}{\frac{2}{a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}}}}}{a} \]
   associate-/r/ [=>] 0.4	\[ \frac{\color{blue}{\frac{-4}{2} \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}}{a} \]
   metadata-eval [=>] 0.4	\[ \frac{\color{blue}{-2} \cdot \left(a \cdot \frac{\frac{c \cdot \left(-4 \cdot \left(c \cdot a\right)\right)}{-4 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}\right)}{a} \]

10. Final simplification 0.3

    \[\leadsto 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}} \]

Alternatives

Alternative 1
Error	0.4
Cost	13632

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

Alternative 2
Error	7.0
Cost	8964

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right) + b \cdot b\\ \mathbf{if}\;b \leq 48:\\ \;\;\;\;\frac{t_0 - b \cdot b}{\frac{b + \sqrt{t_0}}{\frac{0.5}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot \left(b - b\right) + c \cdot \left(-4 \cdot a\right)}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	9.1
Cost	8388

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ t_1 := t_0 + b \cdot b\\ \mathbf{if}\;b \leq 51:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{t_1 - b \cdot b}{b + \sqrt{t_1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 4
Error	9.1
Cost	8388

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ t_1 := t_0 + b \cdot b\\ \mathbf{if}\;b \leq 51:\\ \;\;\;\;\frac{\left(t_1 - b \cdot b\right) \cdot \frac{0.5}{a}}{b + \sqrt{t_1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 5
Error	9.1
Cost	8388

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ t_1 := t_0 + b \cdot b\\ \mathbf{if}\;b \leq 54:\\ \;\;\;\;\frac{t_1 - b \cdot b}{\frac{b + \sqrt{t_1}}{\frac{0.5}{a}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 6
Error	9.4
Cost	7748

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq 56:\\ \;\;\;\;\frac{0.5}{a} \cdot \frac{1}{\frac{1}{\sqrt{t_0 + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	7748

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq 51:\\ \;\;\;\;\frac{1}{\left(a \cdot 2\right) \cdot \frac{1}{\sqrt{t_0 + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 8
Error	9.4
Cost	7620

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq 51:\\ \;\;\;\;\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{t_0 + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 9
Error	9.4
Cost	7492

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq 51:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{t_0 + b \cdot b} - b\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 10
Error	9.4
Cost	7492

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq 56:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{t_0 + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t_0}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \end{array} \]

Alternative 11
Error	11.6
Cost	1344

\[0.5 \cdot \frac{-4 \cdot \left(c \cdot a\right)}{a \cdot \left(b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)} \]

Alternative 12
Error	22.9
Cost	256

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

Reproduce

herbie shell --seed 2023011 
(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)))