Quadratic roots, medium range

Average Error: 43.8 → 0.4

Time: 13.7s

Precision: binary64

Cost: 14144

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

Math

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

↓

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

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)) -1.0) (+ b (sqrt (fma b b (* c (* a -4.0))))))
  (* 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)) / -1.0) / (b + sqrt(fma(b, b, (c * (a * -4.0)))))) / (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(Float64(4.0 * Float64(c * a)) / -1.0) / Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))) / 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[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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

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

4. Taylor expanded in b around 0 0.4

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

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	3.9
Cost	8576

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2} \]

Alternative 2
Error	5.9
Cost	1472

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a \cdot 2} \]

Alternative 3
Error	5.9
Cost	1472

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2} \]

Alternative 4
Error	12.0
Cost	256

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

Reproduce

herbie shell --seed 2023054 
(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)))