Quadratic roots, narrow range

Average Error: 28.4 → 0.3

Time: 16.7s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (* -2.0 (/ c (+ b (sqrt (fma c (* a -4.0) (* b 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 -2.0 * (c / (b + sqrt(fma(c, (a * -4.0), (b * 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(-2.0 * Float64(c / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * 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[(-2.0 * N[(c / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Simplified 28.4

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

   [Start] 28.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 28.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 27.6

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

4. Simplified 27.4

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

   [Start] 27.6	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/l/ [=>] 27.6	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   /-rgt-identity [<=] 27.6	\[ \frac{\frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}{1}}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   /-rgt-identity [=>] 27.6	\[ \frac{\frac{\color{blue}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 27.4	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 27.4	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [=>] 27.4	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{\left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\right) \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   distribute-lft-neg-in [<=] 27.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   rem-square-sqrt [=>] 27.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   *-lft-identity [<=] 27.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{1 \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]
   *-lft-identity [=>] 27.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]

5. Taylor expanded in b around 0 0.4

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

6. Applied egg-rr 0.5

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

7. Simplified 0.4

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

   [Start] 0.5	\[ \frac{\frac{-4}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \cdot c}{a \cdot 2} \]
   associate-*l/ [=>] 0.4	\[ \frac{\color{blue}{\frac{-4 \cdot c}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}}}}{a \cdot 2} \]

8. Applied egg-rr 24.7

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot \left(a \cdot \frac{0.5}{a}\right)\right)} - 1} \]

9. Simplified 0.3

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

   [Start] 24.7	\[ e^{\mathsf{log1p}\left(\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot \left(a \cdot \frac{0.5}{a}\right)\right)} - 1 \]
   expm1-def [=>] 10.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot \left(a \cdot \frac{0.5}{a}\right)\right)\right)} \]
   expm1-log1p [=>] 0.4	\[ \color{blue}{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot \left(a \cdot \frac{0.5}{a}\right)} \]
   associate-*r* [=>] 0.4	\[ \color{blue}{\left(\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot a\right) \cdot \frac{0.5}{a}} \]
   *-commutative [<=] 0.4	\[ \color{blue}{\left(a \cdot \frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right)} \cdot \frac{0.5}{a} \]
   associate-*r/ [=>] 0.3	\[ \color{blue}{\frac{\left(a \cdot \frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}\right) \cdot 0.5}{a}} \]
   associate-*l/ [<=] 0.3	\[ \color{blue}{\frac{a \cdot \frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{a} \cdot 0.5} \]
   *-commutative [=>] 0.3	\[ \frac{\color{blue}{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot a}}{a} \cdot 0.5 \]
   associate-/l* [=>] 0.3	\[ \color{blue}{\frac{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{\frac{a}{a}}} \cdot 0.5 \]
   *-inverses [=>] 0.3	\[ \frac{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{\color{blue}{1}} \cdot 0.5 \]
   associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \cdot 0.5}{1}} \]
   associate-/l* [=>] 0.3	\[ \color{blue}{\frac{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{\frac{1}{0.5}}} \]
   metadata-eval [=>] 0.3	\[ \frac{\frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}{\color{blue}{2}} \]
   associate-/l/ [=>] 0.3	\[ \color{blue}{\frac{-4 \cdot c}{2 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}\right)}} \]
   times-frac [=>] 0.3	\[ \color{blue}{\frac{-4}{2} \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}} \]

10. Final simplification 0.3

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

Alternatives

Alternative 1
Error	9.7
Cost	7756

\[\begin{array}{l} t_0 := \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 15:\\ \;\;\;\;\frac{c \cdot a}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2} \cdot \frac{-4}{a \cdot 2}\\ \mathbf{elif}\;b \leq 75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot -4}{-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}}}{a \cdot 2}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	7756

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq 0.29:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 15:\\ \;\;\;\;\frac{c \cdot a}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2} \cdot \frac{-4}{a \cdot 2}\\ \mathbf{elif}\;b \leq 75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot -4}{-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}}}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7744

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

Alternative 4
Error	11.7
Cost	1216

\[\frac{\frac{c \cdot -4}{-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}}}{a \cdot 2} \]

Alternative 5
Error	12.0
Cost	1024

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

Alternative 6
Error	12.1
Cost	832

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

Alternative 7
Error	23.0
Cost	256

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

Alternative 8
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023027 
(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)))