Quadratic roots, narrow range

Average Accuracy: 55.5% → 99.3%

Time: 16.6s

Precision: binary64

Cost: 13888

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

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

Derivation

1. Initial program 55.5%

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

2. Simplified 55.5%

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

   [Start] 55.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   *-commutative [=>] 55.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Applied egg-rr 56.6%

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

4. Taylor expanded in b around 0 99.3%

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

5. Applied egg-rr 98.0%

   \[\leadsto \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

6. Simplified 99.2%

   \[\leadsto \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{1.5}}}}}{a \cdot 2} \]
   Proof

   [Start] 98.0	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{a \cdot 2} \]
   unpow1/3 [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}}}}{a \cdot 2} \]
   fma-def [<=] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt[3]{{\color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}}^{1.5}}}}{a \cdot 2} \]
   +-commutative [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt[3]{{\color{blue}{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right)}}^{1.5}}}}{a \cdot 2} \]
   fma-def [=>] 99.2	\[ \frac{\frac{-4 \cdot \left(c \cdot a\right)}{b + \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}}^{1.5}}}}{a \cdot 2} \]

7. Applied egg-rr 61.5%

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

8. Simplified 99.3%

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

   [Start] 61.5	\[ e^{\mathsf{log1p}\left(\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)} - 1 \]
   expm1-def [=>] 83.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\right)\right)} \]
   expm1-log1p [=>] 99.2	\[ \color{blue}{\frac{-4}{a \cdot 2} \cdot \frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]
   *-commutative [=>] 99.2	\[ \color{blue}{\frac{c \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{-4}{a \cdot 2}} \]
   associate-/l* [=>] 99.2	\[ \color{blue}{\frac{c}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}}} \cdot \frac{-4}{a \cdot 2} \]
   associate-*l/ [=>] 99.2	\[ \color{blue}{\frac{c \cdot \frac{-4}{a \cdot 2}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}}} \]
   *-commutative [=>] 99.2	\[ \frac{c \cdot \frac{-4}{\color{blue}{2 \cdot a}}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]
   associate-/r* [=>] 99.2	\[ \frac{c \cdot \color{blue}{\frac{\frac{-4}{2}}{a}}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]
   metadata-eval [=>] 99.2	\[ \frac{c \cdot \frac{\color{blue}{-2}}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]
   metadata-eval [<=] 99.2	\[ \frac{c \cdot \frac{\color{blue}{-4 \cdot 0.5}}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]
   associate-*r/ [<=] 99.2	\[ \frac{c \cdot \color{blue}{\left(-4 \cdot \frac{0.5}{a}\right)}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}} \]
   associate-*r/ [<=] 99.2	\[ \color{blue}{c \cdot \frac{-4 \cdot \frac{0.5}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}}} \]
   associate-/r/ [=>] 99.2	\[ c \cdot \color{blue}{\left(\frac{-4 \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot a\right)} \]
   associate-*l/ [=>] 99.3	\[ c \cdot \color{blue}{\frac{\left(-4 \cdot \frac{0.5}{a}\right) \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]
   associate-*r/ [=>] 99.3	\[ c \cdot \frac{\color{blue}{\frac{-4 \cdot 0.5}{a}} \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]
   metadata-eval [=>] 99.3	\[ c \cdot \frac{\frac{\color{blue}{-2}}{a} \cdot a}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

9. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	85.4%
Cost	14788

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.0025:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(c \cdot a\right)}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	7744

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \frac{\frac{t_0}{b + \sqrt{b \cdot b + t_0}}}{a \cdot 2} \end{array} \]

Alternative 3
Accuracy	85.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.18:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(c \cdot a\right)}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	82.0%
Cost	1344

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

Alternative 5
Accuracy	82.1%
Cost	1344

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

Alternative 6
Accuracy	64.4%
Cost	256

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

Reproduce

herbie shell --seed 2023122 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))