Quadratic roots, narrow range

Average Error: 44.36% → 1.37%

Time: 21.8s

Precision: binary64

Cost: 46784

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

Math

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

↓

\[\begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\\ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-\sqrt[3]{{t_0}^{2}}}}{\sqrt[3]{t_0}}}{a \cdot 2} \end{array} \]

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 (+ b (sqrt (fma b b (* c (* a -4.0)))))))
   (/ (/ (/ (* 4.0 (* c a)) (- (cbrt (pow t_0 2.0)))) (cbrt t_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) {
	double t_0 = b + sqrt(fma(b, b, (c * (a * -4.0))));
	return (((4.0 * (c * a)) / -cbrt(pow(t_0, 2.0))) / cbrt(t_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)
	t_0 = Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))))
	return Float64(Float64(Float64(Float64(4.0 * Float64(c * a)) / Float64(-cbrt((t_0 ^ 2.0)))) / cbrt(t_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_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / (-N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision] / N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 44.36

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

2. Simplified 44.36

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

   [Start] 44.36	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 44.36	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 43.11

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

4. Taylor expanded in b around 0 1.37

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

5. Final simplification 1.37

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

Alternatives

Alternative 1
Error	7.97%
Cost	48260

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 2
Error	10.03%
Cost	42244

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ t_1 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_1} - b}{a \cdot 2} \leq -2.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-\sqrt[3]{\mathsf{fma}\left(-8, c \cdot a, 4 \cdot \left(b \cdot b\right)\right) + -4 \cdot \left(\frac{c}{\frac{b \cdot b}{c}} \cdot \left(a \cdot a\right)\right)}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(b, b, t_1\right)}}}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	10.31%
Cost	28292

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -2.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 4
Error	10.37%
Cost	28228

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -2.5:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 5
Error	10.47%
Cost	28164

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -2.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 6
Error	15.05%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.19:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	15.08%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.19:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	18.33%
Cost	7232

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

Alternative 9
Error	18.44%
Cost	832

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

Alternative 10
Error	35.73%
Cost	512

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

Alternative 11
Error	35.71%
Cost	256

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

Alternative 12
Error	98.4%
Cost	192

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

Reproduce

herbie shell --seed 2023102 
(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)))