Quadratic roots, narrow range

Average Error: 28.4 → 5.7

Time: 11.9s

Precision: binary64

Cost: 34432

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

Math

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

↓

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

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
   (* (* c (* (* c c) (pow b -5.0))) (* a a))
   (* -5.0 (/ (pow (* c a) 3.0) (/ (pow b 7.0) c))))
  (fma (* (/ c b) (/ c (* b b))) a (/ c b))))

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, ((c * ((c * c) * pow(b, -5.0))) * (a * a)), (-5.0 * (pow((c * a), 3.0) / (pow(b, 7.0) / c)))) - fma(((c / b) * (c / (b * b))), a, (c / b));
}

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

Derivation

1. Initial program 28.4

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

2. Taylor expanded in a around 0 5.7

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

3. Simplified 5.7

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

4. Taylor expanded in a around 0 5.7

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

5. Simplified 5.7

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

6. Applied egg-rr 5.7

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

7. Applied egg-rr 5.7

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

8. Final simplification 5.7

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

Alternatives

Alternative 1
Error	7.2
Cost	28036

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{c \cdot c}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	9.6
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.0025:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(c \cdot c\right) \cdot \left(a \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 3
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.0025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(c \cdot c\right) \cdot \left(a \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 4
Error	9.5
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.0025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(c \cdot c\right) \cdot \left(a \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 5
Error	9.6
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.0025:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \left(c \cdot c\right) \cdot \left(a \cdot {b}^{-3}\right)\\ \end{array} \]

Alternative 6
Error	11.8
Cost	7232

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

Alternative 7
Error	11.9
Cost	7168

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

Alternative 8
Error	11.8
Cost	832

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

Alternative 9
Error	22.9
Cost	256

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

Reproduce

herbie shell --seed 2022284 
(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)))