Quadratic roots, medium range

Average Error: 43.6 → 0.5

Time: 15.6s

Precision: binary64

Cost: 14144

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (/ (+ b (sqrt (fma b b (* c (* a -4.0))))) (* -4.0 (* c a))))
  (* 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) {
	return (1.0 / ((b + sqrt(fma(b, b, (c * (a * -4.0))))) / (-4.0 * (c * a)))) / (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)
	return Float64(Float64(1.0 / Float64(Float64(b + sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))) / Float64(-4.0 * Float64(c * a)))) / 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_] := N[(N[(1.0 / N[(N[(b + N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.6

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

2. Simplified 43.5

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

   [Start] 43.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   +-commutative [=>] 43.6	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
   unsub-neg [=>] 43.6	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]
   fma-neg [=>] 43.5	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{2 \cdot a} \]
   *-commutative [=>] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-rgt-neg-in [=>] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)}\right)} - b}{2 \cdot a} \]
   distribute-lft-neg-in [=>] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(\left(-4\right) \cdot a\right)}\right)} - b}{2 \cdot a} \]
   *-commutative [<=] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-4\right)\right)}\right)} - b}{2 \cdot a} \]
   metadata-eval [=>] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-4}\right)\right)} - b}{2 \cdot a} \]
   *-commutative [=>] 43.5	\[ \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 43.6

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

4. Applied egg-rr 43.1

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

5. Taylor expanded in b around 0 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	4.3
Cost	8192

\[\frac{\frac{1}{-0.5 \cdot \frac{b}{c \cdot a} + \left(0.5 \cdot \frac{1}{b} + 0.5 \cdot \frac{c \cdot a}{{b}^{3}}\right)}}{a \cdot 2} \]

Alternative 2
Error	4.3
Cost	8128

\[\frac{\frac{1}{\mathsf{fma}\left(-0.5, \frac{\frac{b}{c}}{a}, \frac{0.5}{b}\right) + 0.5 \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{a}}}}{a \cdot 2} \]

Alternative 3
Error	6.3
Cost	1024

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

Alternative 4
Error	12.3
Cost	256

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

Alternative 5
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023057 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))