Quadratic roots, medium range

Average Accuracy: 31.1% → 95.6%

Time: 20.4s

Precision: binary64

Cost: 48256

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

Math

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

↓

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

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
    -0.25
    (*
     (/ (pow a 3.0) b)
     (fma
      16.0
      (/ (pow c 4.0) (pow b 6.0))
      (* 4.0 (* (* c c) (/ (* c c) (pow b 6.0))))))
    (* c (/ (* a (* a -2.0)) (/ (pow b 5.0) (* c c)))))
   (/ c b))
  (* a (* (/ c b) (/ c (* b 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(a, 3.0) / b) * fma(16.0, (pow(c, 4.0) / pow(b, 6.0)), (4.0 * ((c * c) * ((c * c) / pow(b, 6.0)))))), (c * ((a * (a * -2.0)) / (pow(b, 5.0) / (c * c))))) - (c / b)) - (a * ((c / b) * (c / (b * 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(Float64(fma(-0.25, Float64(Float64((a ^ 3.0) / b) * fma(16.0, Float64((c ^ 4.0) / (b ^ 6.0)), Float64(4.0 * Float64(Float64(c * c) * Float64(Float64(c * c) / (b ^ 6.0)))))), Float64(c * Float64(Float64(a * Float64(a * -2.0)) / Float64((b ^ 5.0) / Float64(c * c))))) - Float64(c / b)) - Float64(a * Float64(Float64(c / b) * Float64(c / Float64(b * 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[(N[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(a * N[(a * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Simplified 31.1%

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

   [Start] 31.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 31.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in a around 0 95.6%

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

4. Simplified 95.6%

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

   [Start] 95.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   +-commutative [=>] 95.6	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   mul-1-neg [=>] 95.6	\[ \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   unsub-neg [=>] 95.6	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

5. Applied egg-rr 95.6%

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

   [Start] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{{b}^{3}} \cdot a \]
   unpow3 [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\color{blue}{\left(b \cdot b\right) \cdot b}} \cdot a \]
   times-frac [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \color{blue}{\left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right)} \cdot a \]

6. Applied egg-rr 95.6%

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

   [Start] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(\left(a \cdot -2\right) \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   associate-/l* [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \color{blue}{\frac{\left(a \cdot -2\right) \cdot a}{\frac{{b}^{5}}{{c}^{3}}}}\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   unpow3 [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(a \cdot -2\right) \cdot a}{\frac{{b}^{5}}{\color{blue}{\left(c \cdot c\right) \cdot c}}}\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   associate-/r* [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\left(a \cdot -2\right) \cdot a}{\color{blue}{\frac{\frac{{b}^{5}}{c \cdot c}}{c}}}\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   associate-/r/ [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \color{blue}{\frac{\left(a \cdot -2\right) \cdot a}{\frac{{b}^{5}}{c \cdot c}} \cdot c}\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   *-commutative [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{\color{blue}{a \cdot \left(a \cdot -2\right)}}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]

7. Applied egg-rr 95.6%

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

   [Start] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   sqr-pow [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{\color{blue}{{c}^{\left(\frac{4}{2}\right)} \cdot {c}^{\left(\frac{4}{2}\right)}}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   metadata-eval [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{\color{blue}{2}} \cdot {c}^{\left(\frac{4}{2}\right)}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   pow2 [<=] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{\color{blue}{\left(c \cdot c\right)} \cdot {c}^{\left(\frac{4}{2}\right)}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   metadata-eval [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{\left(c \cdot c\right) \cdot {c}^{\color{blue}{2}}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   pow2 [<=] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{6}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   associate-/l* [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \color{blue}{\frac{c \cdot c}{\frac{{b}^{6}}{c \cdot c}}}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]
   associate-/r/ [=>] 95.6	\[ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \color{blue}{\left(\frac{c \cdot c}{{b}^{6}} \cdot \left(c \cdot c\right)\right)}\right), \frac{a \cdot \left(a \cdot -2\right)}{\frac{{b}^{5}}{c \cdot c}} \cdot c\right) - \frac{c}{b}\right) - \left(\frac{c}{b \cdot b} \cdot \frac{c}{b}\right) \cdot a \]

8. Final simplification 95.6%

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

Alternatives

Alternative 1
Accuracy	94.1%
Cost	20608

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

Alternative 2
Accuracy	94.1%
Cost	14528

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

Alternative 3
Accuracy	91.0%
Cost	1024

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

Alternative 4
Accuracy	81.6%
Cost	256

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

Reproduce

herbie shell --seed 2023143 
(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)))