Quadratic roots, narrow range

Average Accuracy: 55.7% → 98.8%

Time: 18.4s

Precision: binary64

Cost: 53184

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

Math

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

↓

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

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

Derivation

1. Initial program 55.7%

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

2. Simplified 55.7%

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

   [Start] 55.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 55.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 56.9%

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

4. Simplified 57.2%

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

   [Start] 56.9	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}^{2}}}}{-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/l/ [=>] 56.9	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}^{2}}}}}{a \cdot 2} \]
   associate-/r* [=>] 56.9	\[ \frac{\color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{\sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}^{2}}}}}{a \cdot 2} \]

5. Taylor expanded in b around 0 98.6%

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

6. Simplified 98.6%

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

   [Start] 98.6	\[ \frac{\frac{\frac{4 \cdot \left(c \cdot a\right)}{-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{\sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}^{2}}}}{a \cdot 2} \]
   *-commutative [=>] 98.6	\[ \frac{\frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{\sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}^{2}}}}{a \cdot 2} \]
   associate-*r* [<=] 98.6	\[ \frac{\frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{-\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{\sqrt[3]{{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}^{2}}}}{a \cdot 2} \]

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	94.9%
Cost	41924

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{b + \sqrt{t_0}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)}{b \cdot b - t_0}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}, \frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.6%
Cost	40960

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

Alternative 3
Accuracy	97.6%
Cost	40384

\[\begin{array}{l} t_0 := b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\\ \frac{\frac{\frac{c \cdot \left(a \cdot 4\right)}{-\sqrt[3]{t_0}}}{{t_0}^{0.6666666666666666}}}{a \cdot 2} \end{array} \]

Alternative 4
Accuracy	94.5%
Cost	35780

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

Alternative 5
Accuracy	89.6%
Cost	28292

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.11:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 6
Accuracy	89.5%
Cost	28228

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.11:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 7
Accuracy	89.3%
Cost	28164

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.11:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot \left(a \cdot -2\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 8
Accuracy	85.3%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2} \leq -0.11:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 9
Accuracy	85.3%
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -0.11:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 10
Accuracy	85.3%
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.11:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 11
Accuracy	84.6%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 80:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \frac{c \cdot c}{{b}^{3}}\\ \end{array} \]

Alternative 12
Accuracy	81.4%
Cost	7232

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

Alternative 13
Accuracy	81.2%
Cost	1600

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

Alternative 14
Accuracy	64.2%
Cost	256

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

Reproduce

herbie shell --seed 2023125 
(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)))