quad2p (problem 3.2.1, positive)

Average Accuracy: 42.3% → 76.5%

Time: 16.1s

Precision: binary64

Cost: 13704

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}^{-1}\\ \end{array} \]

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 -2e+152)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 9.5e-77)
     (/ (- (sqrt (- (* b_2 b_2) (* 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 <= -2e+152) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 9.5e-77) {
		tmp = (sqrt(((b_2 * b_2) - (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 <= -2e+152)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 9.5e-77)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - 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, -2e+152], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 9.5e-77], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $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 < -2.0000000000000001e152

   1. Initial program 0.7%

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

   2. Simplified 0.7%

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

      [Start] 0.7	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 0.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 0.7	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 58.1%

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

   4. Simplified 58.1%

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

      [Start] 58.1	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 58.1	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -2.0000000000000001e152 < b_2 < 9.5000000000000005e-77

   1. Initial program 76.2%

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

   2. Simplified 76.2%

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

      [Start] 76.2	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 76.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 76.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if 9.5000000000000005e-77 < b_2

   1. Initial program 16.8%

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

   2. Simplified 16.8%

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

      [Start] 16.8	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 16.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 16.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 25.1%

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

      [Start] 16.8	\[ \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \]
      clear-num [=>] 16.8	\[ \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}} \]
      inv-pow [=>] 16.8	\[ \color{blue}{{\left(\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)}^{-1}} \]
      sub-neg [=>] 16.8	\[ {\left(\frac{a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-a \cdot c\right)}} - b_2}\right)}^{-1} \]
      add-sqr-sqrt [=>] 14.0	\[ {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b_2}\right)}^{-1} \]
      hypot-def [=>] 25.1	\[ {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2}\right)}^{-1} \]
      *-commutative [=>] 25.1	\[ {\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b_2}\right)}^{-1} \]
      distribute-rgt-neg-in [=>] 25.1	\[ {\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b_2}\right)}^{-1} \]

   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 85.3%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}}^{-1} \]
      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 [=>] 85.3	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{-1} \cdot c}\right)\right)}^{-1} \]
      times-frac [=>] 85.3	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b_2}{c}}\right)\right)}^{-1} \]
      metadata-eval [=>] 85.3	\[ {\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{-2} \cdot \frac{b_2}{c}\right)\right)}^{-1} \]

3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.4%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.35 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Accuracy	71.6%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{c \cdot 0.5}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Accuracy	58.9%
Cost	964

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{c \cdot 0.5}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Accuracy	39.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 6 \cdot 10^{-239}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	58.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 7.4 \cdot 10^{-242}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Accuracy	58.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.8 \cdot 10^{-238}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Accuracy	15.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 8
Accuracy	11.2%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023153 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))