Quadratic roots, narrow range

Average Error: 55.8% → 99.5%

Time: 25.2s

Precision: binary64

Cost: 14144.00

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

\[\frac{\frac{4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a}{-1}}{a \cdot 2} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ (* 4.0 c) (+ b (sqrt (fma c (* a -4.0) (* b b))))) (/ a -1.0))
  (* 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 (((4.0 * c) / (b + sqrt(fma(c, (a * -4.0), (b * b))))) * (a / -1.0)) / (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(4.0 * c) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))) * Float64(a / -1.0)) / 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[(4.0 * c), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / -1.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8

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

2. Simplified 55.8

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

   [Start] 55.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 55.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 57.1

   \[\leadsto \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{-1}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]

4. Taylor expanded in b around 0 99.3

   \[\leadsto \frac{\frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{-1}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]

5. Applied egg-rr 99.2

   \[\leadsto \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}}{a \cdot 2} \]

6. Simplified 99.3

   \[\leadsto \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]
   Proof

   [Start] 99.2	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}}}{a \cdot 2} \]
   pow-sqr [=>] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}}}{a \cdot 2} \]
   metadata-eval [=>] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{\color{blue}{0.5}}}}{a \cdot 2} \]
   unpow1/2 [=>] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \color{blue}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 99.3	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

7. Applied egg-rr 99.5

   \[\leadsto \frac{\color{blue}{\frac{4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a}{-1}}}{a \cdot 2} \]

8. Final simplification 99.5

   \[\leadsto \frac{\frac{4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a}{-1}}{a \cdot 2} \]

Alternatives

Alternative 1
Error	99.3%
Cost	7872.00

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}{a \cdot 2} \]

Alternative 2
Error	85.2%
Cost	7492.00

\[\begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	81.8%
Cost	1472.00

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}{a \cdot 2} \]

Alternative 4
Error	81.8%
Cost	1472.00

\[\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2} \]

Alternative 5
Error	81.3%
Cost	1024.00

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

Alternative 6
Error	81.1%
Cost	960.00

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

Alternative 7
Error	64.0%
Cost	256.00

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

Reproduce

herbie shell --seed 2023093 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))