Quadratic roots, wide range

Average Accuracy: 18.0% → 97.6%

Time: 14.7s

Precision: binary64

Cost: 41152

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

Math

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

↓

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

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
   -0.25
   (* (/ (pow a 3.0) b) (* (* (* c c) (* (* c c) (pow b -6.0))) 20.0))
   (* -2.0 (* (* (* c c) (* c (pow b -5.0))) (* a a))))
  (fma (/ c (/ (pow b 3.0) c)) 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(a, 3.0) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 20.0)), (-2.0 * (((c * c) * (c * pow(b, -5.0))) * (a * a)))) - fma((c / (pow(b, 3.0) / c)), 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(-0.25, Float64(Float64((a ^ 3.0) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 20.0)), Float64(-2.0 * Float64(Float64(Float64(c * c) * Float64(c * (b ^ -5.0))) * Float64(a * a)))) - fma(Float64(c / Float64((b ^ 3.0) / c)), 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[(-0.25 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.0%

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

2. Simplified 18.0%

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

   [Start] 18.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 18.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]
   +-commutative [=>] 18.0	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   unsub-neg [=>] 18.0	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]
   fma-neg [=>] 18.0	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]
   associate-*l* [=>] 18.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]
   *-commutative [=>] 18.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]
   distribute-rgt-neg-in [=>] 18.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]
   metadata-eval [=>] 18.0	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

3. Taylor expanded in a around 0 97.6%

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

4. Simplified 97.6%

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

   [Start] 97.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   associate-+r+ [=>] 97.6	\[ \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right) + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
   +-commutative [=>] 97.6	\[ \color{blue}{\left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right)} \]
   distribute-lft-out [=>] 97.6	\[ \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \color{blue}{-1 \cdot \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)} \]
   mul-1-neg [=>] 97.6	\[ \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + \color{blue}{\left(-\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)\right)} \]
   unsub-neg [=>] 97.6	\[ \color{blue}{\left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) - \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)} \]

5. Applied egg-rr 97.6%

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

   [Start] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   div-inv [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\color{blue}{\left({c}^{3} \cdot \frac{1}{{b}^{5}}\right)} \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   unpow3 [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\left(\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \frac{1}{{b}^{5}}\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   associate-*l* [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \frac{1}{{b}^{5}}\right)\right)} \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   pow-flip [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot \color{blue}{{b}^{\left(-5\right)}}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{\color{blue}{-5}}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]

6. Applied egg-rr 97.6%

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

   [Start] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   div-inv [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left({c}^{4} \cdot \frac{1}{{b}^{6}}\right)} \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [<=] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\color{blue}{\left(2 + 2\right)}} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [<=] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\left(\color{blue}{\sqrt{4}} + 2\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [<=] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\left(\sqrt{4} + \color{blue}{\sqrt{4}}\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   pow-prod-up [<=] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{\left({c}^{\left(\sqrt{4}\right)} \cdot {c}^{\left(\sqrt{4}\right)}\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   pow-prod-down [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{{\left(c \cdot c\right)}^{\left(\sqrt{4}\right)}} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left({\left(c \cdot c\right)}^{\color{blue}{2}} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   pow2 [<=] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   associate-*l* [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{1}{{b}^{6}}\right)\right)} \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   pow-flip [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{{b}^{\left(-6\right)}}\right)\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
   metadata-eval [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{\color{blue}{-6}}\right)\right) \cdot 20\right), -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]

7. Final simplification 97.6%

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

Alternatives

Alternative 1
Accuracy	96.7%
Cost	20736

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

Alternative 2
Accuracy	95.1%
Cost	1024

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

Alternative 3
Accuracy	90.3%
Cost	256

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

Alternative 4
Accuracy	1.7%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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)))