Quadratic roots, wide range

Average Error: 52.7 → 0.1

Time: 15.7s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/ (* c -2.0) (+ b (sqrt (fma b b (* -4.0 (* c a)))))))

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 (c * -2.0) / (b + sqrt(fma(b, b, (-4.0 * (c * a)))));
}

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(c * -2.0) / Float64(b + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))))
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[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

   [Start] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   /-rgt-identity [<=] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
   metadata-eval [<=] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
   *-commutative [=>] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\frac{\color{blue}{a \cdot 2}}{--1}} \]
   associate-/l* [=>] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{\frac{a}{\frac{--1}{2}}}} \]
   associate-/l* [<=] 52.7	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{2}}{a}} \]
   associate-*r/ [<=] 52.7	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{\frac{--1}{2}}{a}} \]
   /-rgt-identity [<=] 52.7	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{--1}{2}}{a} \]
   metadata-eval [<=] 52.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{--1}{2}}{a} \]

3. Applied egg-rr 52.5

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

4. Taylor expanded in a around 0 0.1

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

5. Simplified 0.1

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

   [Start] 0.1	\[ \frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]
   *-commutative [=>] 0.1	\[ \frac{\color{blue}{c \cdot -2}}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	13632

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

Alternative 2
Error	0.5
Cost	8256

\[\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \frac{-0.5}{a} \]

Alternative 3
Error	0.4
Cost	8256

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \frac{\left(b \cdot b - b \cdot b\right) + t_0}{\left(b + \sqrt{b \cdot b + t_0}\right) \cdot \left(a \cdot 2\right)} \end{array} \]

Alternative 4
Error	1.9
Cost	8064

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

Alternative 5
Error	2.3
Cost	1984

\[\frac{1}{\frac{a}{0.5} \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{0.5}{b} + \left(-0.5 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}\right)\right)} \]

Alternative 6
Error	2.9
Cost	960

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

Alternative 7
Error	2.9
Cost	960

\[\frac{c \cdot -2}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2} \]

Alternative 8
Error	3.1
Cost	576

\[\frac{1}{\frac{a}{b} - \frac{b}{c}} \]

Alternative 9
Error	6.1
Cost	256

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

Alternative 10
Error	62.9
Cost	192

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

Reproduce

herbie shell --seed 2023187 
(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)))