Quadratic roots, wide range

Average Accuracy: 17.1% → 97.7%

Time: 16.9s

Precision: binary64

Cost: 40768

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

Math

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

↓

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

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 a)
  (/ (* (pow (* a c) 4.0) 20.0) (pow b 7.0))
  (-
   (* -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 / a), ((pow((a * c), 4.0) * 20.0) / pow(b, 7.0)), ((-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 fma(Float64(-0.25 / a), Float64(Float64((Float64(a * c) ^ 4.0) * 20.0) / (b ^ 7.0)), Float64(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 / a), $MachinePrecision] * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] - N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * a + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

2. Simplified 17.1%

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

   [Start] 17.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 17.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]
   +-commutative [=>] 17.1	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   unsub-neg [=>] 17.1	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]
   fma-neg [=>] 17.1	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]
   associate-*l* [=>] 17.1	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]
   *-commutative [=>] 17.1	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]
   distribute-rgt-neg-in [=>] 17.1	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]
   metadata-eval [=>] 17.1	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

3. Taylor expanded in b around inf 97.7%

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

4. Simplified 97.7%

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

   [Start] 97.7	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   +-commutative [=>] 97.7	\[ \color{blue}{\left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   associate-+l+ [=>] 97.7	\[ \color{blue}{-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   associate-*r/ [=>] 97.7	\[ \color{blue}{\frac{-0.25 \cdot \left({\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]
   times-frac [=>] 97.7	\[ \color{blue}{\frac{-0.25}{a} \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) \]
   +-commutative [<=] 97.7	\[ \frac{-0.25}{a} \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
   fma-def [=>] 97.7	\[ \color{blue}{\mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}, -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

5. Applied egg-rr 97.3%

   \[\leadsto \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)} - 1\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   Proof

   [Start] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   expm1-log1p-u [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   expm1-udef [=>] 97.3	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\left(e^{\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)} - 1\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   pow-prod-down [=>] 97.3	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot a\right)}^{4}}\right)} - 1\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

6. Simplified 97.7%

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

   [Start] 97.3	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(e^{\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)} - 1\right) \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   expm1-def [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)\right)} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   expm1-log1p [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

7. Applied egg-rr 97.7%

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

   [Start] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   div-inv [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\color{blue}{\left({c}^{3} \cdot \frac{1}{{b}^{5}}\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   unpow3 [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \frac{1}{{b}^{5}}\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   associate-*l* [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \frac{1}{{b}^{5}}\right)\right)} \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   pow-flip [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot \color{blue}{{b}^{\left(-5\right)}}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]
   metadata-eval [=>] 97.7	\[ \mathsf{fma}\left(\frac{-0.25}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{7}}, -2 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot {b}^{\color{blue}{-5}}\right)\right) \cdot \left(a \cdot a\right)\right) - \mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\right) \]

8. Final simplification 97.7%

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

Alternatives

Alternative 1
Accuracy	96.9%
Cost	20736

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

Alternative 2
Accuracy	95.5%
Cost	1024

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

Alternative 3
Accuracy	90.9%
Cost	256

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

Alternative 4
Accuracy	3.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023146 
(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)))