Quadratic roots, medium range

Average Error: 43.8 → 2.9

Time: 23.8s

Precision: binary64

Cost: 48896

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

Math

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

↓

\[\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right), \frac{\left(\frac{c}{b} \cdot \frac{c \cdot -2}{b \cdot b}\right) \cdot \left(-2 \cdot \left(c \cdot \frac{c}{b}\right)\right)}{b \cdot b}\right), \frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \left(\frac{c}{b} + \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\right) \]

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 a 3.0) b)
    (fma
     16.0
     (* (* c c) (* (* c c) (pow b -6.0)))
     (/
      (* (* (/ c b) (/ (* c -2.0) (* b b))) (* -2.0 (* c (/ c b))))
      (* b b))))
   (/ (* -2.0 (pow c 3.0)) (/ (pow b 5.0) (* a a))))
  (+ (/ c b) (/ c (/ (/ (pow b 3.0) a) c)))))

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(a, 3.0) / b) * fma(16.0, ((c * c) * ((c * c) * pow(b, -6.0))), ((((c / b) * ((c * -2.0) / (b * b))) * (-2.0 * (c * (c / b)))) / (b * b)))), ((-2.0 * pow(c, 3.0)) / (pow(b, 5.0) / (a * a)))) - ((c / b) + (c / ((pow(b, 3.0) / a) / c)));
}

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(-0.25, Float64(Float64((a ^ 3.0) / b) * fma(16.0, Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))), Float64(Float64(Float64(Float64(c / b) * Float64(Float64(c * -2.0) / Float64(b * b))) * Float64(-2.0 * Float64(c * Float64(c / b)))) / Float64(b * b)))), Float64(Float64(-2.0 * (c ^ 3.0)) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(Float64(c / b) + Float64(c / Float64(Float64((b ^ 3.0) / a) / c))))
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[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(16.0 * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * -2.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(c / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $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 43.7

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
   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	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
   unsub-neg [=>] 43.8	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]
   fma-neg [=>] 43.7	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a} \]
   *-commutative [=>] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-rgt-neg-in [=>] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-lft-neg-in [=>] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)} - b}{2 \cdot a} \]
   *-commutative [<=] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]
   metadata-eval [=>] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]
   *-commutative [=>] 43.7	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in a around 0 2.9

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

4. Simplified 2.9

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

   [Start] 2.9	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   associate-+r+ [=>] 2.9	\[ \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
   distribute-lft-out [=>] 2.9	\[ \color{blue}{-1 \cdot \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   mul-1-neg [=>] 2.9	\[ \color{blue}{\left(-\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)\right)} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   +-commutative [=>] 2.9	\[ \left(-\color{blue}{\left(\frac{c}{b} + \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   associate-/l* [=>] 2.9	\[ \left(-\left(\frac{c}{b} + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}\right)\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   unpow2 [=>] 2.9	\[ \left(-\left(\frac{c}{b} + \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}}\right)\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   associate-/l* [=>] 2.9	\[ \left(-\left(\frac{c}{b} + \color{blue}{\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}}\right)\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) \]
   fma-def [=>] 2.9	\[ \left(-\left(\frac{c}{b} + \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\right)\right) + \color{blue}{\mathsf{fma}\left(-0.25, \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}, -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]

5. Applied egg-rr 2.9

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

6. Applied egg-rr 2.9

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

7. Applied egg-rr 2.9

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

8. Final simplification 2.9

   \[\leadsto \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right), \frac{\left(\frac{c}{b} \cdot \frac{c \cdot -2}{b \cdot b}\right) \cdot \left(-2 \cdot \left(c \cdot \frac{c}{b}\right)\right)}{b \cdot b}\right), \frac{-2 \cdot {c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \left(\frac{c}{b} + \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\right) \]

Alternatives

Alternative 1
Error	3.9
Cost	20736

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

Alternative 2
Error	5.9
Cost	7232

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

Alternative 3
Error	12.1
Cost	256

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

Alternative 4
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023187 
(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)))