Quadratic roots, wide range

Average Accuracy: 17.7% → 97.6%

Time: 16.6s

Precision: binary64

Cost: 40704

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

Math

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

↓

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

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
   -0.25
   (/ (* (pow (* a c) 4.0) (/ 20.0 (pow b 7.0))) a)
   (- (/ (* (pow c 3.0) -2.0) (/ (pow b 5.0) (* a a))) (/ 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(-0.25, ((pow((a * c), 4.0) * (20.0 / pow(b, 7.0))) / a), (((pow(c, 3.0) * -2.0) / (pow(b, 5.0) / (a * a))) - (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(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) * Float64(20.0 / (b ^ 7.0))) / a), Float64(Float64(Float64((c ^ 3.0) * -2.0) / Float64((b ^ 5.0) / Float64(a * a))) - 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[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

2. Simplified 17.7%

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

   [Start] 17.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 17.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in b around inf 97.6%

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

4. Simplified 97.6%

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

   [Start] 97.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   +-commutative [=>] 97.6	\[ \color{blue}{\left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   mul-1-neg [=>] 97.6	\[ \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   unsub-neg [=>] 97.6	\[ \color{blue}{\left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

5. Applied egg-rr 97.3%

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

   [Start] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   expm1-log1p-u [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   expm1-udef [=>] 97.3	\[ \mathsf{fma}\left(-0.25, \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} - 1}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   distribute-rgt-out [=>] 97.3	\[ \mathsf{fma}\left(-0.25, e^{\mathsf{log1p}\left(\frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)} - 1, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   times-frac [=>] 97.3	\[ \mathsf{fma}\left(-0.25, e^{\mathsf{log1p}\left(\color{blue}{\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}}\right)} - 1, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   pow-prod-down [=>] 97.3	\[ \mathsf{fma}\left(-0.25, e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(c \cdot a\right)}^{4}}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)} - 1, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   metadata-eval [=>] 97.3	\[ \mathsf{fma}\left(-0.25, e^{\mathsf{log1p}\left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{\color{blue}{20}}{{b}^{7}}\right)} - 1, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

6. Simplified 97.6%

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

   [Start] 97.3	\[ \mathsf{fma}\left(-0.25, e^{\mathsf{log1p}\left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)} - 1, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   expm1-def [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)\right)}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   expm1-log1p [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   associate-*l/ [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{7}}}{a}}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]
   *-commutative [=>] 97.6	\[ \mathsf{fma}\left(-0.25, \frac{{\color{blue}{\left(a \cdot c\right)}}^{4} \cdot \frac{20}{{b}^{7}}}{a}, \frac{{c}^{3} \cdot -2}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c}{\frac{{b}^{3}}{c}} \cdot a \]

7. Final simplification 97.6%

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

Alternatives

Alternative 1
Accuracy	96.8%
Cost	14528

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

Alternative 2
Accuracy	95.3%
Cost	7232

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

Alternative 3
Accuracy	94.8%
Cost	1728

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

Alternative 4
Accuracy	90.5%
Cost	256

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

Alternative 5
Accuracy	1.7%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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)))