Quadratic roots, medium range

Average Error: 43.5 → 0.2

Time: 16.0s

Precision: binary64

Cost: 13696

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 43.5

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

2. Simplified 43.5

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

   [Start] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 43.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 43.0

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

4. Simplified 42.9

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

   [Start] 43.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/l/ [=>] 43.0	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   fma-def [<=] 42.9	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 42.9	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [=>] 42.9	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   distribute-lft-neg-in [<=] 42.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   rem-square-sqrt [=>] 42.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   fma-def [<=] 42.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right)}}{a \cdot 2} \]
   +-commutative [=>] 42.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)}}{a \cdot 2} \]
   fma-def [=>] 42.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)}}{a \cdot 2} \]

5. Taylor expanded in b around 0 0.4

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

6. Applied egg-rr 0.5

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

7. Simplified 0.4

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

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

8. Taylor expanded in a around 0 0.2

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

9. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	7680

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

Alternative 2
Error	6.1
Cost	1216

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

Alternative 3
Error	12.3
Cost	256

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

Reproduce

herbie shell --seed 2023073 
(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)))