Quadratic roots, wide range

Average Error: 52.8 → 1.3

Time: 9.7s

Precision: binary64

Cost: 34560

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

Math

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

↓

\[\mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right)\right) \]

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
  -0.25
  (/ (* (pow (* c a) 4.0) 20.0) (* a (pow b 7.0)))
  (-
   (* -2.0 (* (* c (* (* c c) (pow b -5.0))) (* a a)))
   (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(-0.25, ((pow((c * a), 4.0) * 20.0) / (a * pow(b, 7.0))), ((-2.0 * ((c * ((c * c) * pow(b, -5.0))) * (a * a))) - 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 fma(-0.25, Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / Float64(a * (b ^ 7.0))), Float64(Float64(-2.0 * Float64(Float64(c * Float64(Float64(c * c) * (b ^ -5.0))) * Float64(a * a))) - 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[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] - N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8

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

2. Taylor expanded in b around inf 1.3

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

3. Simplified 1.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right)} \]

4. Applied egg-rr 1.3

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]

5. Applied egg-rr 1.3

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\color{blue}{\left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]

6. Applied egg-rr 1.3

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\color{blue}{\frac{c}{b \cdot b} \cdot \frac{c}{b}}, a, \frac{c}{b}\right)\right) \]

7. Final simplification 1.3

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \left(\left(c \cdot \left(\left(c \cdot c\right) \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}, a, \frac{c}{b}\right)\right) \]

Alternatives

Alternative 1
Error	1.8
Cost	20736

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

Alternative 2
Error	2.8
Cost	7360

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

Alternative 3
Error	2.8
Cost	832

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

Alternative 4
Error	6.0
Cost	256

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

Reproduce

herbie shell --seed 2022217 
(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)))