Quadratic roots, wide range

Average Accuracy: 18.0% → 97.6%

Time: 12.8s

Precision: binary64

Cost: 40832

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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

Derivation

1. Initial program 17.0%

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

2. Simplified 16.9%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   Step-by-step derivation

   [Start] 17.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   /-rgt-identity [<=] 17.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
   metadata-eval [<=] 17.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
   associate-/l* [<=] 17.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{2 \cdot a}} \]
   associate-*r/ [<=] 17.0	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2 \cdot a}} \]
   +-commutative [=>] 17.0	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right)} \cdot \frac{--1}{2 \cdot a} \]
   unsub-neg [=>] 17.0	\[ \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \cdot \frac{--1}{2 \cdot a} \]
   fma-neg [=>] 16.9	\[ \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   associate-*l* [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   *-commutative [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   distribute-rgt-neg-in [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   metadata-eval [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b\right) \cdot \frac{--1}{2 \cdot a} \]
   associate-/r* [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \color{blue}{\frac{\frac{--1}{2}}{a}} \]
   metadata-eval [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\frac{\color{blue}{1}}{2}}{a} \]
   metadata-eval [=>] 16.9	\[ \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

3. Taylor expanded in a around 0 97.8%

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

4. Simplified 97.8%

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

   [Start] 97.8	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   +-commutative [=>] 97.8	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   mul-1-neg [=>] 97.8	\[ \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   unsub-neg [=>] 97.8	\[ \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

5. Taylor expanded in c around 0 97.8%

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

6. Simplified 97.8%

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

   [Start] 97.8	\[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   associate-*r/ [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{\color{blue}{\frac{-5 \cdot \left({c}^{4} \cdot {a}^{3}\right)}{{b}^{6}}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   associate-*r* [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{\frac{\color{blue}{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

7. Applied egg-rr 97.8%

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

   [Start] 97.8	\[ \left(\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   div-inv [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \color{blue}{\left({c}^{3} \cdot \frac{1}{{b}^{5}}\right)} \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   unpow3 [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \left(\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \frac{1}{{b}^{5}}\right) \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   associate-*l* [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \frac{1}{{b}^{5}}\right)\right)} \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   pow-flip [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \left(\left(c \cdot c\right) \cdot \left(c \cdot \color{blue}{{b}^{\left(-5\right)}}\right)\right) \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   metadata-eval [=>] 97.8	\[ \left(\mathsf{fma}\left(-2, \left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{\color{blue}{-5}}\right)\right) \cdot \left(a \cdot a\right), \frac{\frac{\left(-5 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{6}}}{b}\right) - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

8. Final simplification 97.8%

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

Alternatives

Alternative 1
Accuracy	96.9%
Cost	14528

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

Alternative 2
Accuracy	95.3%
Cost	7232

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

Alternative 3
Accuracy	94.7%
Cost	1600

\[\left(-2 \cdot \left(\frac{c \cdot c}{\frac{b \cdot b}{a} \cdot \frac{b}{a}} + \frac{c}{\frac{b}{a}}\right)\right) \cdot \frac{0.5}{a} \]

Alternative 4
Accuracy	90.3%
Cost	256

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

Reproduce

herbie shell --seed 2023159 
(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)))