Quadratic roots, narrow range

Average Error: 28.3 → 0.3

Time: 21.2s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

FPCore

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

↓

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

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 ((4.0 * (c * a)) / a) / (-2.0 * (b + sqrt(fma(b, b, (-4.0 * (c * a))))));
}

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(Float64(Float64(4.0 * Float64(c * a)) / a) / Float64(-2.0 * Float64(b + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))))))
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[(N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(-2.0 * N[(b + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.3

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

2. Simplified 28.3

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

   [Start] 28.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   +-commutative [=>] 28.3	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
   unsub-neg [=>] 28.3	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]
   fma-neg [=>] 28.3	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a} \]
   *-commutative [=>] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-rgt-neg-in [=>] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-lft-neg-in [=>] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)} - b}{2 \cdot a} \]
   *-commutative [<=] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]
   metadata-eval [=>] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]
   *-commutative [=>] 28.3	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 28.8

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

4. Applied egg-rr 27.6

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

5. Applied egg-rr 27.6

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

6. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\frac{0 - -4 \cdot \left(c \cdot a\right)}{a}}{-2 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
   Proof

   [Start] 27.6	\[ \left(-\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b\right)\right) \cdot \frac{1}{\left(a \cdot 2\right) \cdot \left(-\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\right)} \]
   associate-*r/ [=>] 27.6	\[ \color{blue}{\frac{\left(-\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b\right)\right) \cdot 1}{\left(a \cdot 2\right) \cdot \left(-\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\right)}} \]
   associate-*l* [=>] 27.6	\[ \frac{\left(-\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b\right)\right) \cdot 1}{\color{blue}{a \cdot \left(2 \cdot \left(-\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\right)\right)}} \]
   associate-/r* [=>] 27.6	\[ \color{blue}{\frac{\frac{\left(-\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - b \cdot b\right)\right) \cdot 1}{a}}{2 \cdot \left(-\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\right)}} \]

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	6.8
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2} \leq -1.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + c \cdot \frac{a \cdot a}{{b}^{3}}}\\ \end{array} \]

Alternative 2
Error	6.8
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2} \leq -1.5:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + c \cdot \frac{a \cdot a}{{b}^{3}}}\\ \end{array} \]

Alternative 3
Error	6.8
Cost	21060

\[\begin{array}{l} t_0 := c \cdot \left(-4 \cdot a\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b + t_0} - b}{a \cdot 2} \leq -1.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + c \cdot \frac{a \cdot a}{{b}^{3}}}\\ \end{array} \]

Alternative 4
Error	6.8
Cost	14980

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{a}{b} - \frac{b}{c}\right) + c \cdot \frac{a \cdot a}{{b}^{3}}}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	14016

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

Alternative 7
Error	9.4
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 16.5:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 8
Error	11.6
Cost	576

\[\frac{1}{\frac{a}{b} - \frac{b}{c}} \]

Alternative 9
Error	22.9
Cost	256

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

Alternative 10
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023187 
(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)))