Quadratic roots, narrow range

Average Error: 28.5 → 5.7

Time: 12.1s

Precision: binary64

Cost: 34624

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

Math

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

↓

\[\mathsf{fma}\left(-2, \left(\frac{a}{\frac{{b}^{5}}{a}} \cdot c\right) \cdot \left(c \cdot c\right), \frac{-0.25}{\frac{b}{\left(c \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{20}{{b}^{6}}}}\right) - \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right) \]

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
   -2.0
   (* (* (/ a (/ (pow b 5.0) a)) c) (* c c))
   (/ -0.25 (/ b (* (* c (pow (* a c) 3.0)) (/ 20.0 (pow b 6.0))))))
  (+ (/ c b) (* a (/ (* c c) (pow b 3.0))))))

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 fma(-2.0, (((a / (pow(b, 5.0) / a)) * c) * (c * c)), (-0.25 / (b / ((c * pow((a * c), 3.0)) * (20.0 / pow(b, 6.0)))))) - ((c / b) + (a * ((c * c) / pow(b, 3.0))));
}

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(fma(-2.0, Float64(Float64(Float64(a / Float64((b ^ 5.0) / a)) * c) * Float64(c * c)), Float64(-0.25 / Float64(b / Float64(Float64(c * (Float64(a * c) ^ 3.0)) * Float64(20.0 / (b ^ 6.0)))))) - Float64(Float64(c / b) + Float64(a * Float64(Float64(c * c) / (b ^ 3.0)))))
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[(-2.0 * N[(N[(N[(a / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[(b / N[(N[(c * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.5

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

2. Simplified 28.4

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

3. Applied egg-rr 29.3

   \[\leadsto \left(\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{1.5}}} - b\right) \cdot \frac{0.5}{a} \]

4. Taylor expanded in a around 0 5.7

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

5. Simplified 5.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{-0.25}{\frac{b}{{a}^{3} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right)}}\right) - \left(\frac{c}{b} + \frac{c \cdot c}{{b}^{3}} \cdot a\right)} \]

6. Taylor expanded in a around 0 5.7

   \[\leadsto \mathsf{fma}\left(-2, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{-0.25}{\frac{b}{\color{blue}{20 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{6}}}}}\right) - \left(\frac{c}{b} + \frac{c \cdot c}{{b}^{3}} \cdot a\right) \]

7. Simplified 5.7

   \[\leadsto \mathsf{fma}\left(-2, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{-0.25}{\frac{b}{\color{blue}{\left(c \cdot {\left(c \cdot a\right)}^{3}\right) \cdot \frac{20}{{b}^{6}}}}}\right) - \left(\frac{c}{b} + \frac{c \cdot c}{{b}^{3}} \cdot a\right) \]

8. Applied egg-rr 5.7

   \[\leadsto \mathsf{fma}\left(-2, \color{blue}{\left(\frac{a}{\frac{{b}^{5}}{a}} \cdot c\right) \cdot \left(c \cdot c\right)}, \frac{-0.25}{\frac{b}{\left(c \cdot {\left(c \cdot a\right)}^{3}\right) \cdot \frac{20}{{b}^{6}}}}\right) - \left(\frac{c}{b} + \frac{c \cdot c}{{b}^{3}} \cdot a\right) \]

9. Final simplification 5.7

   \[\leadsto \mathsf{fma}\left(-2, \left(\frac{a}{\frac{{b}^{5}}{a}} \cdot c\right) \cdot \left(c \cdot c\right), \frac{-0.25}{\frac{b}{\left(c \cdot {\left(a \cdot c\right)}^{3}\right) \cdot \frac{20}{{b}^{6}}}}\right) - \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right) \]

Alternatives

Alternative 1
Error	6.8
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.023:\\ \;\;\;\;\frac{t_0 - b \cdot b}{b + \sqrt{t_0}} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 2
Error	7.0
Cost	28164

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.023:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \left(\frac{c}{b} + a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 3
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.00172:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.00172:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.00172:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 6
Error	10.3
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 235:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 7
Error	11.7
Cost	7232

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

Alternative 8
Error	11.8
Cost	7168

\[c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]

Alternative 9
Error	22.8
Cost	256

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

Reproduce

herbie shell --seed 2022217 
(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)))