Quadratic roots, medium range

Average Accuracy: 31.3% → 99.3%

Time: 21.9s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

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 b b (* c (* -4.0 a)))))) (* 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(b, b, (c * (-4.0 * a)))))) / (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(b, b, Float64(c * Float64(-4.0 * a)))))) / 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[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

2. Simplified 31.3%

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

   [Start] 31.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 31.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 32.1%

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

   [Start] 31.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   +-commutative [=>] 31.3	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   flip-+ [=>] 31.3	\[ \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 32.2	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   sqr-neg [=>] 32.2	\[ \frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - \color{blue}{b \cdot b}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate--l- [=>] 32.2	\[ \frac{\frac{\color{blue}{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   +-commutative [<=] 32.2	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   fma-def [=>] 32.1	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate-*l* [=>] 32.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   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 [=>] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}}{a \cdot 2} \]
   sqr-neg [=>] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \sqrt{\color{blue}{b \cdot b}}}}{a \cdot 2} \]
   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 [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{b}}}{a \cdot 2} \]
   unsub-neg [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}}{a \cdot 2} \]
   +-commutative [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

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. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	90.9%
Cost	7232

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

Alternative 2
Accuracy	81.4%
Cost	256

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

Alternative 3
Accuracy	1.6%
Cost	192

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

Reproduce

herbie shell --seed 2023138 
(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)))