Quadratic roots, wide range

Average Error: 52.3 → 0.2

Time: 14.6s

Precision: binary64

Cost: 14400

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

\[0.5 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right) + 0 \cdot \left(b \cdot b\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (*
  0.5
  (/
   (/ (+ (* c (* -4.0 a)) (* 0.0 (* b b))) a)
   (+ b (sqrt (fma c (* -4.0 a) (* 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 0.5 * ((((c * (-4.0 * a)) + (0.0 * (b * b))) / a) / (b + sqrt(fma(c, (-4.0 * a), (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(0.5 * Float64(Float64(Float64(Float64(c * Float64(-4.0 * a)) + Float64(0.0 * Float64(b * b))) / a) / Float64(b + sqrt(fma(c, Float64(-4.0 * a), 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[(0.5 * N[(N[(N[(N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(-4.0 * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

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

   [Start] 52.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 52.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 52.1

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

4. Simplified 52.1

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

   [Start] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 52.1	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 58.8

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1} \]

6. Simplified 0.2

   \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right) + 0 \cdot \left(b \cdot b\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}} \]
   Proof

   [Start] 58.8	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1 \]
   expm1-def [=>] 56.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)\right)} \]
   expm1-log1p [=>] 52.1	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
   *-lft-identity [<=] 52.1	\[ \frac{\color{blue}{1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
   associate-*l/ [<=] 52.1	\[ \color{blue}{\frac{1}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)} \]
   associate-/r* [=>] 52.1	\[ \color{blue}{\frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}} \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \]

7. Final simplification 0.2

   \[\leadsto 0.5 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right) + 0 \cdot \left(b \cdot b\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

Alternatives

Alternative 1
Error	0.4
Cost	14016

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

Alternative 2
Error	3.0
Cost	7232

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

Alternative 3
Error	3.1
Cost	832

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

Alternative 4
Error	3.1
Cost	576

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

Alternative 5
Error	6.3
Cost	256

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

Alternative 6
Error	62.9
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))