Quadratic roots, wide range

Average Error: 52.4 → 0.4

Time: 13.3s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

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 a) -4.0) (+ b (sqrt (fma c (* a -4.0) (* b b))))) (* a 2.0)))

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

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

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

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

   [Start] 52.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 52.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 52.2

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

4. Taylor expanded in b around 0 0.4

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

5. Simplified 0.4

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

   [Start] 0.4	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 0.4	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]

6. Applied egg-rr 0.5

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

7. Simplified 0.4

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

   [Start] 0.5	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{a \cdot 2} \]
   pow-sqr [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}}}{a \cdot 2} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.5}}}}{a \cdot 2} \]
   unpow1/2 [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	14016

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

Alternative 2
Error	2.1
Cost	8448

\[\frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2} \]

Alternative 3
Error	3.0
Cost	1024

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

Alternative 4
Error	6.2
Cost	256

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

Alternative 5
Error	62.9
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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)))