Quadratic roots, wide range

Average Accuracy: 17.5% → 99.5%

Time: 16.6s

Precision: binary64

Cost: 13760

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 17.5%

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

2. Simplified 17.5%

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

   [Start] 17.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 17.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 17.9%

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

   [Start] 17.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   +-commutative [=>] 17.5	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   flip-+ [=>] 17.5	\[ \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 18.0	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   sqr-neg [=>] 18.0	\[ \frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - \color{blue}{b \cdot b}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate--l- [=>] 18.0	\[ \frac{\frac{\color{blue}{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   +-commutative [<=] 18.0	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   fma-def [=>] 17.9	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate-*l* [=>] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}}{a \cdot 2} \]
   sqrt-unprod [=>] 0.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}}{a \cdot 2} \]
   sqr-neg [=>] 0.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \sqrt{\color{blue}{b \cdot b}}}}{a \cdot 2} \]
   sqrt-prod [=>] 1.6	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 0.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{b}}}{a \cdot 2} \]
   unsub-neg [<=] 0.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}}{a \cdot 2} \]
   +-commutative [<=] 0.4	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

4. Simplified 17.9%

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

   [Start] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 8.1%

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

   [Start] 17.9	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   expm1-log1p-u [=>] 11.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\right)\right)} \]
   expm1-udef [=>] 8.1	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\right)} - 1} \]
   associate-/l/ [=>] 8.1	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}\right)} - 1 \]
   associate-*l* [=>] 8.1	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot 4\right)}\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]
   *-commutative [=>] 8.1	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}}\right)} - 1 \]

6. Simplified 99.4%

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

   [Start] 8.1	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1 \]
   expm1-def [=>] 11.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)\right)} \]
   expm1-log1p [=>] 17.9	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
   associate-/r* [=>] 17.9	\[ \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}} \]
   associate-/r* [=>] 17.9	\[ \color{blue}{\frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a}}{2}} \]
   *-lft-identity [<=] 17.9	\[ \frac{\frac{\frac{\color{blue}{1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a}}{2} \]
   associate-*l/ [<=] 17.9	\[ \frac{\frac{\color{blue}{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}{a}}{2} \]
   *-commutative [<=] 17.9	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a}}{2} \]
   associate-*l/ [<=] 17.9	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{2} \]
   associate-*r/ [<=] 17.9	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a} \cdot \frac{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2}} \]

7. Taylor expanded in c around 0 99.5%

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

8. Simplified 99.5%

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

   [Start] 99.5	\[ \left(-4 \cdot c\right) \cdot \frac{0.5}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-commutative [=>] 99.5	\[ \color{blue}{\left(c \cdot -4\right)} \cdot \frac{0.5}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

9. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	95.6%
Cost	1024

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

Alternative 2
Accuracy	90.7%
Cost	256

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

Reproduce

herbie shell --seed 2023136 
(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)))