Quadratic roots, narrow range

Average Error: 28.5 → 0.5

Time: 18.2s

Precision: binary64

Cost: 33856

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\\ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{\mathsf{fma}\left(t_0, t_0, b\right)}}{a \cdot 2} \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma b b (* c (* a -4.0))) 0.25)))
   (/ (/ (/ (* 4.0 (* c a)) -1.0) (fma t_0 t_0 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) {
	double t_0 = pow(fma(b, b, (c * (a * -4.0))), 0.25);
	return (((4.0 * (c * a)) / -1.0) / fma(t_0, t_0, 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)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0))) ^ 0.25
	return Float64(Float64(Float64(Float64(4.0 * Float64(c * a)) / -1.0) / fma(t_0, t_0, 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_] := Block[{t$95$0 = N[Power[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, N[(N[(N[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / N[(t$95$0 * t$95$0 + b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $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 28.5

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

   [Start] 28.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 27.7

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

4. Taylor expanded in b around 0 0.4

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

5. Applied egg-rr 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	10.3
Cost	29644

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.008:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{elif}\;t_0 \leq -6.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \mathbf{elif}\;t_0 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	14144

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

Alternative 3
Error	6.5
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 1.85:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Error	6.5
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 1.85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 5
Error	6.6
Cost	8708

\[\begin{array}{l} \mathbf{if}\;b \leq 1.85:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{b + \left(b + -2 \cdot \left(a \cdot \frac{c}{b} + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 27:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-1}}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \end{array} \]

Alternative 7
Error	11.7
Cost	1472

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

Alternative 8
Error	11.6
Cost	1472

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

Alternative 9
Error	22.8
Cost	256

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

Alternative 10
Error	63.0
Cost	192

\[\frac{b}{a} \]

Reproduce

herbie shell --seed 2023039 
(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)))