Quadratic roots, wide range

Average Accuracy: 17.9% → 99.9%

Time: 13.6s

Precision: binary64

Cost: 13632.00

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

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

Derivation

1. Initial program 17.9%

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

2. Simplified 17.9%

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

   [Start] 17.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 17.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 18.3%

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

4. Simplified 18.3%

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

   [Start] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 18.3	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 18.3%

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

6. Simplified 99.5%

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

   [Start] 18.3	\[ \frac{-\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{1}{a \cdot -2} \]
   associate-*l/ [=>] 18.3	\[ \color{blue}{\frac{\left(-\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]
   fma-udef [=>] 18.4	\[ \frac{\left(-\left(b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)}\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   associate--r+ [=>] 99.5	\[ \frac{\left(-\color{blue}{\left(\left(b \cdot b - b \cdot b\right) - c \cdot \left(a \cdot 4\right)\right)}\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   +-inverses [=>] 99.5	\[ \frac{\left(-\left(\color{blue}{0} - c \cdot \left(a \cdot 4\right)\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   neg-sub0 [<=] 99.5	\[ \frac{\left(-\color{blue}{\left(-c \cdot \left(a \cdot 4\right)\right)}\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   associate-*r* [=>] 99.5	\[ \frac{\left(-\left(-\color{blue}{\left(c \cdot a\right) \cdot 4}\right)\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   distribute-rgt-neg-in [=>] 99.5	\[ \frac{\left(-\color{blue}{\left(c \cdot a\right) \cdot \left(-4\right)}\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   metadata-eval [=>] 99.5	\[ \frac{\left(-\left(c \cdot a\right) \cdot \color{blue}{-4}\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   distribute-rgt-neg-in [=>] 99.5	\[ \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(--4\right)\right)} \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   metadata-eval [=>] 99.5	\[ \frac{\left(\left(c \cdot a\right) \cdot \color{blue}{4}\right) \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   associate-*r* [<=] 99.5	\[ \frac{\color{blue}{\left(c \cdot \left(a \cdot 4\right)\right)} \cdot \frac{1}{a \cdot -2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   *-commutative [=>] 99.5	\[ \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{1}{\color{blue}{-2 \cdot a}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   associate-/r* [=>] 99.5	\[ \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \color{blue}{\frac{\frac{1}{-2}}{a}}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   metadata-eval [=>] 99.5	\[ \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{\color{blue}{-0.5}}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   *-commutative [=>] 99.5	\[ \frac{\left(c \cdot \left(a \cdot 4\right)\right) \cdot \frac{-0.5}{a}}{b + \sqrt{\mathsf{fma}\left(c, \color{blue}{-4 \cdot a}, b \cdot b\right)}} \]

7. Taylor expanded in c around 0 99.9%

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

8. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	7744.00

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

Alternative 2
Accuracy	95.3%
Cost	1024.00

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

Alternative 3
Accuracy	90.3%
Cost	256.00

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

Alternative 4
Accuracy	1.7%
Cost	192.00

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

Reproduce

herbie shell --seed 2023096 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))