Quadratic roots, medium range

Average Error: 44.0 → 2.8

Time: 13.0s

Precision: binary64

Cost: 40768

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

Math

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

↓

\[\mathsf{fma}\left(-2, \left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), -5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right) \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (-
  (fma
   -2.0
   (* (* c (* (* c c) (pow b -5.0))) (* a a))
   (* -5.0 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0))))
  (fma (* (/ c (* b b)) (/ c b)) a (/ c 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 fma(-2.0, ((c * ((c * c) * pow(b, -5.0))) * (a * a)), (-5.0 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)))) - fma(((c / (b * b)) * (c / b)), a, (c / 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(fma(-2.0, Float64(Float64(c * Float64(Float64(c * c) * (b ^ -5.0))) * Float64(a * a)), Float64(-5.0 * Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)))) - fma(Float64(Float64(c / Float64(b * b)) * Float64(c / b)), a, Float64(c / 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 * N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.0

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

2. Taylor expanded in a around 0 2.8

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

3. Simplified 2.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)} \]

4. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right)} \cdot \left(a \cdot a\right), \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right) \]

5. Applied egg-rr 2.8

   \[\leadsto \mathsf{fma}\left(-2, \left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \frac{{a}^{3} \cdot -0.25}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right) - \mathsf{fma}\left(\color{blue}{\frac{c}{b \cdot b} \cdot \frac{c}{b}}, a, \frac{c}{b}\right) \]

6. Taylor expanded in a around 0 2.8

   \[\leadsto \mathsf{fma}\left(-2, \left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), \color{blue}{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}}\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right) \]

7. Final simplification 2.8

   \[\leadsto \mathsf{fma}\left(-2, \left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right), -5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right) \]

Alternatives

Alternative 1
Error	3.8
Cost	27008

\[\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot {c}^{3}\right)}{{b}^{5}} - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right) \]

Alternative 2
Error	5.8
Cost	7232

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

Alternative 3
Error	5.8
Cost	832

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

Alternative 4
Error	11.9
Cost	256

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

Alternative 5
Error	62.0
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2022228 
(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)))