Quadratic roots, narrow range

Average Accuracy: 55.3% → 99.3%

Time: 17.8s

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

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 55.3%

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

2. Simplified 55.3%

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

   [Start] 55.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 55.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 56.5%

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

   [Start] 55.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   +-commutative [=>] 55.3	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   flip-+ [=>] 55.3	\[ \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 56.8	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   sqr-neg [=>] 56.8	\[ \frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - \color{blue}{b \cdot b}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate--l- [=>] 56.8	\[ \frac{\frac{\color{blue}{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   +-commutative [<=] 56.8	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   fma-def [=>] 56.5	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate-*l* [=>] 56.5	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}}{a \cdot 2} \]
   sqrt-unprod [=>] 1.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}}{a \cdot 2} \]
   sqr-neg [=>] 1.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \sqrt{\color{blue}{b \cdot b}}}}{a \cdot 2} \]
   sqrt-prod [=>] 1.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 1.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{b}}}{a \cdot 2} \]
   unsub-neg [<=] 1.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}}{a \cdot 2} \]
   +-commutative [<=] 1.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

4. Taylor expanded in b around 0 99.3%

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

5. Applied egg-rr 99.2%

   \[\leadsto \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}}{a \cdot 2} \]
   Proof

   [Start] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   pow1/2 [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.5}}}}{a \cdot 2} \]
   sqr-pow [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}}{a \cdot 2} \]
   metadata-eval [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.25}} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}{a \cdot 2} \]
   metadata-eval [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.25}}}}{a \cdot 2} \]

6. Simplified 99.3%

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

   [Start] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{a \cdot 2} \]
   pow-sqr [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}}}{a \cdot 2} \]
   metadata-eval [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.5}}}}{a \cdot 2} \]
   unpow1/2 [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 99.3	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

7. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	89.1%
Cost	21060

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.159:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	89.1%
Cost	15876

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.159:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.159:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(c \cdot a\right)}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	14016

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

Alternative 5
Accuracy	81.9%
Cost	1344

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

Alternative 6
Accuracy	81.9%
Cost	1344

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

Alternative 7
Accuracy	64.4%
Cost	256

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

Reproduce

herbie shell --seed 2023138 
(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)))