Quadratic roots, medium range

Average Accuracy: 32.0% → 99.7%

Time: 18.3s

Precision: binary64

Cost: 13760

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

Math

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

↓

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

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) (+ b (sqrt (fma b b (* (* c -4.0) a)))))))

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) / (b + sqrt(fma(b, b, ((c * -4.0) * a)))));
}

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

Derivation

1. Initial program 32.0%

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

2. Simplified 32.0%

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

   [Start] 32.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 32.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 32.7%

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

   [Start] 32.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   +-commutative [=>] 32.0	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]
   flip-+ [=>] 32.0	\[ \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}}{a \cdot 2} \]
   add-sqr-sqrt [<=] 32.9	\[ \frac{\frac{\color{blue}{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   sqr-neg [=>] 32.9	\[ \frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - \color{blue}{b \cdot b}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate--l- [=>] 32.9	\[ \frac{\frac{\color{blue}{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   +-commutative [<=] 32.9	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   fma-def [=>] 32.7	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   associate-*l* [=>] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right)}}{a \cdot 2} \]
   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 [=>] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}}{a \cdot 2} \]
   sqr-neg [=>] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \sqrt{\color{blue}{b \cdot b}}}}{a \cdot 2} \]
   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 [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{b}}}{a \cdot 2} \]
   unsub-neg [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}}{a \cdot 2} \]
   +-commutative [<=] 0.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

4. Simplified 32.7%

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

   [Start] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   *-commutative [=>] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}{a \cdot 2} \]
   fma-def [<=] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}}{a \cdot 2} \]
   +-commutative [=>] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}}}{a \cdot 2} \]
   fma-def [=>] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]

5. Applied egg-rr 18.8%

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

   [Start] 32.7	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2} \]
   expm1-log1p-u [=>] 22.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\right)\right)} \]
   expm1-udef [=>] 18.8	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\right)} - 1} \]
   associate-/l/ [=>] 18.8	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}\right)} - 1 \]
   associate-*l* [=>] 18.8	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot 4\right)}\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\right)} - 1 \]
   *-commutative [=>] 18.8	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}}\right)} - 1 \]

6. Simplified 99.6%

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

   [Start] 18.8	\[ e^{\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)} - 1 \]
   expm1-def [=>] 22.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}\right)\right)} \]
   expm1-log1p [=>] 32.7	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
   *-lft-identity [<=] 32.7	\[ \frac{\color{blue}{1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
   *-commutative [=>] 32.7	\[ \frac{\color{blue}{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]
   associate-*r* [=>] 32.7	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}{\color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot a\right) \cdot 2}} \]
   *-commutative [<=] 32.7	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right) \cdot 1}{\color{blue}{\left(a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\right)} \cdot 2} \]
   times-frac [=>] 32.7	\[ \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \frac{1}{2}} \]
   metadata-eval [=>] 32.7	\[ \frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \color{blue}{0.5} \]
   *-commutative [=>] 32.7	\[ \color{blue}{0.5 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]
   associate-/r* [=>] 32.7	\[ 0.5 \cdot \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

7. Taylor expanded in c around 0 99.7%

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

8. Simplified 99.7%

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

   [Start] 99.7	\[ 0.5 \cdot \frac{-4 \cdot c}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]
   *-commutative [=>] 99.7	\[ 0.5 \cdot \frac{\color{blue}{c \cdot -4}}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}} \]

9. Taylor expanded in c around 0 99.7%

   \[\leadsto 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\color{blue}{{b}^{2} + -4 \cdot \left(c \cdot a\right)}}} \]

10. Simplified 99.7%

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

    [Start] 99.7	\[ 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \]
    unpow2 [=>] 99.7	\[ 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\color{blue}{b \cdot b} + -4 \cdot \left(c \cdot a\right)}} \]
    fma-def [=>] 99.7	\[ 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}} \]
    associate-*r* [=>] 99.7	\[ 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)}} \]
    *-commutative [=>] 99.7	\[ 0.5 \cdot \frac{c \cdot -4}{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right)} \cdot a\right)}} \]

11. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7488

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

Alternative 2
Accuracy	90.7%
Cost	1088

\[0.5 \cdot \frac{c \cdot -4}{b + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \]

Alternative 3
Accuracy	80.8%
Cost	256

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

Reproduce

herbie shell --seed 2023126 
(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)))