Quadratic roots, medium range

Average Error: 43.8 → 0.4

Time: 16.4s

Precision: binary64

Cost: 13888

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 43.8

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

2. Simplified 43.8

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

   [Start] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 43.3

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

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 [<=] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   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 39.9

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

8. Simplified 0.4

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

   [Start] 39.9	\[ e^{\mathsf{log1p}\left(\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)} - 1 \]
   expm1-def [=>] 10.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)\right)} \]
   expm1-log1p [=>] 0.5	\[ \color{blue}{\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}} \]
   associate-*r/ [=>] 0.4	\[ \color{blue}{\frac{\frac{-4}{a \cdot 2} \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}} \]
   associate-*l/ [=>] 0.2	\[ \frac{\color{blue}{\frac{-4 \cdot \left(c \cdot a\right)}{a \cdot 2}}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-*r* [=>] 0.2	\[ \frac{\frac{\color{blue}{\left(-4 \cdot c\right) \cdot a}}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-*l/ [<=] 0.2	\[ \frac{\color{blue}{\frac{-4 \cdot c}{a \cdot 2} \cdot a}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-/l* [=>] 0.4	\[ \frac{\color{blue}{\frac{-4}{\frac{a \cdot 2}{c}}} \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-/r/ [=>] 0.4	\[ \frac{\color{blue}{\left(\frac{-4}{a \cdot 2} \cdot c\right)} \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-commutative [=>] 0.4	\[ \frac{\left(\frac{-4}{\color{blue}{2 \cdot a}} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   associate-/r* [=>] 0.4	\[ \frac{\left(\color{blue}{\frac{\frac{-4}{2}}{a}} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   metadata-eval [=>] 0.4	\[ \frac{\left(\frac{\color{blue}{-2}}{a} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-commutative [=>] 0.4	\[ \frac{\left(\frac{-2}{a} \cdot c\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot -4}, b \cdot b\right)}} \]

9. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	7744

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

Alternative 2
Error	5.9
Cost	7232

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

Alternative 3
Error	5.9
Cost	1344

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

Alternative 4
Error	5.9
Cost	1344

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

Alternative 5
Error	12.1
Cost	256

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))