quad2p (problem 3.2.1, positive)

Average Accuracy: 48.4% → 83.9%

Time: 13.4s

Precision: binary64

Cost: 20552

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.3 \cdot 10^{+104}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}^{-1}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.3e+104)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.6e-110)
     (/ (- (sqrt (+ (fma b_2 b_2 (* c (- a))) (fma a (- c) (* a c)))) b_2) a)
     (pow (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c))) -1.0))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.3e+104) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.6e-110) {
		tmp = (sqrt((fma(b_2, b_2, (c * -a)) + fma(a, -c, (a * c)))) - b_2) / a;
	} else {
		tmp = pow(fma(0.5, (a / b_2), (-2.0 * (b_2 / c))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.3e+104)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.6e-110)
		tmp = Float64(Float64(sqrt(Float64(fma(b_2, b_2, Float64(c * Float64(-a))) + fma(a, Float64(-c), Float64(a * c)))) - b_2) / a);
	else
		tmp = fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.3e+104], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.6e-110], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision] + N[(a * (-c) + N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[Power[N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.29999999999999985e104

   1. Initial program 28.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 28.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof

      [Start] 28.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 28.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 28.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 95.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

   if -3.29999999999999985e104 < b_2 < 1.60000000000000014e-110

   1. Initial program 81.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof

      [Start] 81.6	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 81.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 81.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 81.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)}} - b_2}{a} \]
      Proof

      [Start] 81.6	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      prod-diff [=>] 81.6	\[ \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}} - b_2}{a} \]
      distribute-rgt-neg-in [=>] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]
      *-commutative [<=] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)} - b_2}{a} \]
      fma-udef [=>] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}} - b_2}{a} \]
      distribute-lft-neg-in [<=] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)} - b_2}{a} \]
      *-commutative [<=] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)} - b_2}{a} \]
      distribute-rgt-neg-in [=>] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)} - b_2}{a} \]
      fma-def [=>] 81.6	\[ \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}} - b_2}{a} \]

   if 1.60000000000000014e-110 < b_2

   1. Initial program 20.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 20.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Proof

      [Start] 20.6	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 20.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 20.6	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}\right)}^{-1}} \]
      Proof

      [Start] 20.6	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      clear-num [=>] 20.6	\[ \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}} \]
      inv-pow [=>] 20.6	\[ \color{blue}{{\left(\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)}^{-1}} \]
      sub-neg [=>] 20.6	\[ {\left(\frac{a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-a \cdot c\right)}} - b_2}\right)}^{-1} \]
      add-sqr-sqrt [=>] 17.5	\[ {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b_2}\right)}^{-1} \]
      hypot-def [=>] 28.2	\[ {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2}\right)}^{-1} \]
      *-commutative [=>] 28.2	\[ {\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2}\right)}^{-1} \]
      distribute-rgt-neg-in [=>] 28.2	\[ {\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2}\right)}^{-1} \]

   4. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto {\color{blue}{\left(0.5 \cdot \frac{a}{b_2} + 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}}^{-1} \]

   5. Simplified 82.2%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}}^{-1} \]
      Proof

      [Start] 0.0	\[ {\left(0.5 \cdot \frac{a}{b_2} + 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}^{-1} \]
      fma-def [=>] 0.0	\[ {\color{blue}{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)}}^{-1} \]
      associate-*r/ [=>] 0.0	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2 \cdot b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)\right)}^{-1} \]
      *-commutative [=>] 0.0	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)\right)}^{-1} \]
      unpow2 [=>] 0.0	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)\right)}^{-1} \]
      rem-square-sqrt [=>] 82.2	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{-1} \cdot c}\right)\right)}^{-1} \]
      times-frac [=>] 82.2	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b_2}{c}}\right)\right)}^{-1} \]
      metadata-eval [=>] 82.2	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{-2} \cdot \frac{b_2}{c}\right)\right)}^{-1} \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.3 \cdot 10^{+104}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.9%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.1 \cdot 10^{+104}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}^{-1}\\ \end{array} \]

Alternative 2
Accuracy	84.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.3 \cdot 10^{+104}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Accuracy	78.8%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Accuracy	64.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.3 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	29.0%
Cost	320

\[-2 \cdot \frac{b_2}{a} \]

Reproduce

herbie shell --seed 2023151 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))