Quadratic roots, medium range

Average Error: 43.8 → 0.4

Time: 13.4s

Precision: binary64

Cost: 14016

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 43.8

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

2. Simplified 43.8

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

   [Start] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 45.0

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

4. Simplified 45.0

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

   [Start] 45.0	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{\frac{1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   associate-/r* [<=] 45.0	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   associate-*r/ [=>] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}}{a \cdot 2} \]
   *-commutative [=>] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   *-commutative [=>] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{\frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 44.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 44.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [=>] 44.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot 1}{b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}{a \cdot 2} \]
   fma-def [<=] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot 1}{b - \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}}{a \cdot 2} \]
   +-commutative [=>] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot 1}{b - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}}{a \cdot 2} \]
   fma-def [=>] 45.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot 1}{b - \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}}{a \cdot 2} \]

5. Applied egg-rr 43.5

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

6. Simplified 0.4

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

   [Start] 43.5	\[ \frac{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(-\frac{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)}{a \cdot 2} \]
   sub-neg [<=] 43.5	\[ \frac{\color{blue}{\frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]
   div-sub [<=] 43.3	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]
   fma-udef [=>] 43.2	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(c \cdot a\right) \cdot 4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   associate--r+ [=>] 0.4	\[ \frac{\frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot 4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   +-inverses [=>] 0.4	\[ \frac{\frac{\color{blue}{0} - \left(c \cdot a\right) \cdot 4}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   neg-sub0 [<=] 0.4	\[ \frac{\frac{\color{blue}{-\left(c \cdot a\right) \cdot 4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   distribute-rgt-neg-in [=>] 0.4	\[ \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \left(-4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\left(c \cdot a\right) \cdot \color{blue}{-4}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   associate-*r* [<=] 0.4	\[ \frac{\frac{\color{blue}{c \cdot \left(a \cdot -4\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	5.9
Cost	1024

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

Alternative 2
Error	12.0
Cost	256

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

Alternative 3
Error	63.0
Cost	192

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

Reproduce

herbie shell --seed 2023045 
(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)))