Quadratic roots, narrow range

Average Error: 28.1 → 0.4

Time: 18.0s

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

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 28.1

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

2. Simplified 28.1

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

   [Start] 28.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 27.3

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

4. Taylor expanded in b around 0 0.4

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

5. Applied egg-rr 0.5

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

6. Simplified 0.4

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

   [Start] 0.5	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{a \cdot 2} \]
   pow-sqr [=>] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}}}{a \cdot 2} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.5}}}}{a \cdot 2} \]
   unpow1/2 [=>] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 0.5	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 0.5	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

7. Applied egg-rr 0.4

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

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	9.1
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.022:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}{a \cdot 2}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	14016

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

Alternative 3
Error	0.5
Cost	7744

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

Alternative 4
Error	9.7
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 1.865:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}{a \cdot 2}\\ \end{array} \]

Alternative 5
Error	11.6
Cost	1344

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

Alternative 6
Error	11.6
Cost	1344

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

Alternative 7
Error	12.0
Cost	1024

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

Alternative 8
Error	23.1
Cost	256

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

Alternative 9
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023056 
(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)))