quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.0% → 63.2%

Time: 14.5s

Alternatives: 13

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

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

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

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

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Alternative 1: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(b\_2, \frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \left(\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)\right)}, a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.4e+37)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/
    (-
     (fma
      b_2
      (/
       (+ b_2 b_2)
       (* (fma b_2 b_2 0.0) (* (fma b_2 b_2 0.0) (fma b_2 b_2 0.0))))
      (* a c)))
    (* a (+ b_2 (sqrt (fma b_2 b_2 (* a c))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.4e+37) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -fma(b_2, ((b_2 + b_2) / (fma(b_2, b_2, 0.0) * (fma(b_2, b_2, 0.0) * fma(b_2, b_2, 0.0)))), (a * c)) / (a * (b_2 + sqrt(fma(b_2, b_2, (a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.4e+37)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(-fma(b_2, Float64(Float64(b_2 + b_2) / Float64(fma(b_2, b_2, 0.0) * Float64(fma(b_2, b_2, 0.0) * fma(b_2, b_2, 0.0)))), Float64(a * c))) / Float64(a * Float64(b_2 + sqrt(fma(b_2, b_2, Float64(a * c))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.4e+37], 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[((-N[(b$95$2 * N[(N[(b$95$2 + b$95$2), $MachinePrecision] / N[(N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision] * N[(N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision] * N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision]) / N[(a * N[(b$95$2 + N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.4000000000000001e37

   1. Initial program 60.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 4.4000000000000001e37 < b_2

   1. Initial program 10.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.8%

      \[\leadsto \color{blue}{\frac{-\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\mathsf{neg}\left(\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   6. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)}} \]
   7. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      2. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{\color{blue}{0}}{b\_2 - b\_2}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{\color{blue}{{0}^{3}}}{b\_2 - b\_2}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      4. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{{0}^{3}}{\color{blue}{0}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{{0}^{3}}{\color{blue}{{0}^{3}}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      6. cube-div N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{{\left(\frac{0}{0}\right)}^{3}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      7. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, {\left(\frac{\color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}{0}\right)}^{3}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      8. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, {\left(\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{b\_2 - b\_2}}\right)}^{3}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, {\color{blue}{\left(b\_2 + b\_2\right)}}^{3}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. flip3-+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, {\color{blue}{\left(\frac{{b\_2}^{3} + {b\_2}^{3}}{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}\right)}}^{3}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

   8. Applied egg-rr 60.8%

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \left(\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)\right)}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(b\_2, \frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \left(\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)\right)}, a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2, \frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.2e+43)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/
    (fma b_2 (/ (+ b_2 b_2) (* (fma b_2 b_2 0.0) (fma b_2 b_2 0.0))) (* a c))
    (* a (- (- b_2) (sqrt (fma b_2 b_2 (* a c))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.2e+43) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma(b_2, ((b_2 + b_2) / (fma(b_2, b_2, 0.0) * fma(b_2, b_2, 0.0))), (a * c)) / (a * (-b_2 - sqrt(fma(b_2, b_2, (a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.2e+43)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(fma(b_2, Float64(Float64(b_2 + b_2) / Float64(fma(b_2, b_2, 0.0) * fma(b_2, b_2, 0.0))), Float64(a * c)) / Float64(a * Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(a * c))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.2e+43], 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[(N[(b$95$2 * N[(N[(b$95$2 + b$95$2), $MachinePrecision] / N[(N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision] * N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(a * N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.20000000000000042e43

   1. Initial program 59.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 5.20000000000000042e43 < b_2

   1. Initial program 10.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.8%

      \[\leadsto \color{blue}{\frac{-\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\mathsf{neg}\left(\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   6. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)}} \]
   7. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      2. difference-of-squares N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{\color{blue}{\left(b\_2 + b\_2\right) \cdot \left(b\_2 - b\_2\right)}}{b\_2 - b\_2}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      3. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{\left(b\_2 + b\_2\right) \cdot \color{blue}{0}}{b\_2 - b\_2}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      4. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{\left(b\_2 + b\_2\right) \cdot 0}{\color{blue}{0}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\left(b\_2 + b\_2\right) \cdot \frac{0}{0}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      6. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \left(b\_2 + b\_2\right) \cdot \frac{\color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}{0}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      7. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \left(b\_2 + b\_2\right) \cdot \frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{b\_2 - b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      8. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \left(b\_2 + b\_2\right) \cdot \color{blue}{\left(b\_2 + b\_2\right)}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. flip3-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{{b\_2}^{3} + {b\_2}^{3}}{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}} \cdot \left(b\_2 + b\_2\right), a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. flip3-+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{{b\_2}^{3} + {b\_2}^{3}}{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)} \cdot \color{blue}{\frac{{b\_2}^{3} + {b\_2}^{3}}{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      11. frac-times N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{\left({b\_2}^{3} + {b\_2}^{3}\right) \cdot \left({b\_2}^{3} + {b\_2}^{3}\right)}{\left(b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)\right) \cdot \left(b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 - b\_2 \cdot b\_2\right)\right)}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

   8. Applied egg-rr 57.0%

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2, \frac{b\_2 + b\_2}{\mathsf{fma}\left(b\_2, b\_2, 0\right) \cdot \mathsf{fma}\left(b\_2, b\_2, 0\right)}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2 + b\_2, \frac{1}{\mathsf{fma}\left(b\_2, b\_2, 0\right)}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 4.1e+45)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/
    (fma (+ b_2 b_2) (/ 1.0 (fma b_2 b_2 0.0)) (* a c))
    (* a (- (- b_2) (sqrt (fma b_2 b_2 (* a c))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 4.1e+45) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma((b_2 + b_2), (1.0 / fma(b_2, b_2, 0.0)), (a * c)) / (a * (-b_2 - sqrt(fma(b_2, b_2, (a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 4.1e+45)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(fma(Float64(b_2 + b_2), Float64(1.0 / fma(b_2, b_2, 0.0)), Float64(a * c)) / Float64(a * Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(a * c))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 4.1e+45], 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[(N[(N[(b$95$2 + b$95$2), $MachinePrecision] * N[(1.0 / N[(b$95$2 * b$95$2 + 0.0), $MachinePrecision]), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(a * N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.10000000000000012e45

   1. Initial program 59.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 4.10000000000000012e45 < b_2

   1. Initial program 10.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.8%

      \[\leadsto \color{blue}{\frac{-\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\mathsf{neg}\left(\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   6. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)}} \]
   7. Step-by-step derivation

   8. Applied egg-rr 52.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2 + b\_2, \frac{1}{\mathsf{fma}\left(b\_2, b\_2, 0\right)}, a \cdot c\right)}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 4.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2 + b\_2, \frac{1}{\mathsf{fma}\left(b\_2, b\_2, 0\right)}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2, \frac{1}{b\_2 + b\_2}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.6e+45)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/
    (fma b_2 (/ 1.0 (+ b_2 b_2)) (* a c))
    (* a (- (- b_2) (sqrt (fma b_2 b_2 (* a c))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.6e+45) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma(b_2, (1.0 / (b_2 + b_2)), (a * c)) / (a * (-b_2 - sqrt(fma(b_2, b_2, (a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.6e+45)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(fma(b_2, Float64(1.0 / Float64(b_2 + b_2)), Float64(a * c)) / Float64(a * Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(a * c))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.6e+45], 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[(N[(b$95$2 * N[(1.0 / N[(b$95$2 + b$95$2), $MachinePrecision]), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(a * N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.6000000000000004e45

   1. Initial program 59.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 7.6000000000000004e45 < b_2

   1. Initial program 10.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.8%

      \[\leadsto \color{blue}{\frac{-\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\mathsf{neg}\left(\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   6. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)}} \]
   7. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      2. clear-num N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{1}{\frac{b\_2 - b\_2}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      3. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\frac{\color{blue}{0}}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      4. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\frac{\color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{0}}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      6. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{b\_2 - b\_2}}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      7. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\color{blue}{b\_2 + b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{1}{b\_2 + b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. +-lowering-+.f64 48.4

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \frac{1}{\color{blue}{b\_2 + b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]

   8. Applied egg-rr 48.4%

      \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{\frac{1}{b\_2 + b\_2}}, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b\_2, \frac{1}{b\_2 + b\_2}, a \cdot c\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b\_2 + b\_2\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.7e+103)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/ (fma c a (+ b_2 b_2)) (* a (- (- b_2) (sqrt (fma b_2 b_2 (* a c))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.7e+103) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma(c, a, (b_2 + b_2)) / (a * (-b_2 - sqrt(fma(b_2, b_2, (a * c)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.7e+103)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(fma(c, a, Float64(b_2 + b_2)) / Float64(a * Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(a * c))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.7e+103], 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[(N[(c * a + N[(b$95$2 + b$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.70000000000000033e103

   1. Initial program 58.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 5.70000000000000033e103 < b_2

   1. Initial program 6.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\frac{-\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   5. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)}} \]

      5. associate-+r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{\mathsf{neg}\left(a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\mathsf{neg}\left(\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   6. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b\_2 \cdot \left(b\_2 + b\_2\right)}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot a} + b\_2 \cdot \left(b\_2 + b\_2\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      3. flip-+ N/A

         \[\leadsto \frac{c \cdot a + b\_2 \cdot \color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      4. +-inverses N/A

         \[\leadsto \frac{c \cdot a + b\_2 \cdot \frac{\color{blue}{0}}{b\_2 - b\_2}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      5. +-inverses N/A

         \[\leadsto \frac{c \cdot a + b\_2 \cdot \frac{0}{\color{blue}{0}}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{c \cdot a + \color{blue}{\frac{b\_2 \cdot 0}{0}}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      7. +-inverses N/A

         \[\leadsto \frac{c \cdot a + \frac{b\_2 \cdot \color{blue}{\left(b\_2 - b\_2\right)}}{0}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      8. distribute-lft-out-- N/A

         \[\leadsto \frac{c \cdot a + \frac{\color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}{0}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. +-inverses N/A

         \[\leadsto \frac{c \cdot a + \frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{b\_2 - b\_2}}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. flip-+ N/A

          \[\leadsto \frac{c \cdot a + \color{blue}{\left(b\_2 + b\_2\right)}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, a, b\_2 + b\_2\right)}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      12. +-lowering-+.f64 45.9

          \[\leadsto \frac{\mathsf{fma}\left(c, a, \color{blue}{b\_2 + b\_2}\right)}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]

   8. Applied egg-rr 45.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, a, b\_2 + b\_2\right)}}{\left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right) \cdot \left(-a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b\_2 + b\_2\right)}{a \cdot \left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 10^{+155}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(a, c, b\_2 \cdot b\_2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1e+155)
   (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
   (/ (fma a c (+ b_2 b_2)) (* a (+ b_2 (sqrt (fma a c (* b_2 b_2))))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1e+155) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = fma(a, c, (b_2 + b_2)) / (a * (b_2 + sqrt(fma(a, c, (b_2 * b_2)))));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1e+155)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(fma(a, c, Float64(b_2 + b_2)) / Float64(a * Float64(b_2 + sqrt(fma(a, c, Float64(b_2 * b_2))))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1e+155], 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[(N[(a * c + N[(b$95$2 + b$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b$95$2 + N[Sqrt[N[(a * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.00000000000000001e155

   1. Initial program 55.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.00000000000000001e155 < b_2

   1. Initial program 1.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b\_2, b\_2, \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{b\_2 \cdot b\_2 + \left(b\_2 \cdot b\_2 + a \cdot c\right)}{a}}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(b\_2 \cdot b\_2 + b\_2 \cdot b\_2\right) + a \cdot c}}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{\frac{\color{blue}{b\_2 \cdot \left(b\_2 + b\_2\right)} + a \cdot c}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, \color{blue}{b\_2 + b\_2}, a \cdot c\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, \color{blue}{a \cdot c}\right)}{a}}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{a}}{\color{blue}{b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}}} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{a}}{b\_2 + \color{blue}{\sqrt{b\_2 \cdot b\_2 + a \cdot c}}} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{a}}{b\_2 + \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]

      12. *-lowering-*.f64 0.0

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{a \cdot c}\right)}} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(b\_2, b\_2 + b\_2, a \cdot c\right)}{a}}{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}} \]
   7. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot \left(b\_2 + b\_2\right) + a \cdot c}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b\_2 \cdot \left(b\_2 + b\_2\right) + a \cdot c}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b\_2 \cdot \left(b\_2 + b\_2\right)}}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c, b\_2 \cdot \left(b\_2 + b\_2\right)\right)}}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      5. flip-+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 \cdot \color{blue}{\frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{b\_2 - b\_2}}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      6. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 \cdot \frac{\color{blue}{0}}{b\_2 - b\_2}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      7. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 \cdot \frac{0}{\color{blue}{0}}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, c, \color{blue}{\frac{b\_2 \cdot 0}{0}}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      9. +-inverses N/A

         \[\leadsto \frac{\mathsf{fma}\left(a, c, \frac{b\_2 \cdot \color{blue}{\left(b\_2 - b\_2\right)}}{0}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      10. distribute-lft-out-- N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, \frac{\color{blue}{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}}{0}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      11. +-inverses N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, \frac{b\_2 \cdot b\_2 - b\_2 \cdot b\_2}{\color{blue}{b\_2 - b\_2}}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      12. flip-+ N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, \color{blue}{b\_2 + b\_2}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, \color{blue}{b\_2 + b\_2}\right)}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right) \cdot a} \]

      14. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{\color{blue}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{\color{blue}{a \cdot \left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}} \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \color{blue}{\left(b\_2 + \sqrt{b\_2 \cdot b\_2 + a \cdot c}\right)}} \]

      17. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \color{blue}{\sqrt{b\_2 \cdot b\_2 + a \cdot c}}\right)} \]

      18. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\color{blue}{a \cdot c + b\_2 \cdot b\_2}}\right)} \]

      19. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\color{blue}{\mathsf{fma}\left(a, c, b\_2 \cdot b\_2\right)}}\right)} \]

      20. *-lowering-*.f64 50.5

          \[\leadsto \frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(a, c, \color{blue}{b\_2 \cdot b\_2}\right)}\right)} \]

   8. Applied egg-rr 50.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(a, c, b\_2 \cdot b\_2\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 10^{+155}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b\_2 + b\_2\right)}{a \cdot \left(b\_2 + \sqrt{\mathsf{fma}\left(a, c, b\_2 \cdot b\_2\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\\ \mathbf{if}\;b\_2 \leq -1.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{t\_0} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left|t\_0\right|}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (fma b_2 b_2 (* a c))))
   (if (<= b_2 -1.4e-92)
     (/ (- (sqrt t_0) b_2) a)
     (/ (+ b_2 (sqrt (fabs t_0))) a))))

C

double code(double a, double b_2, double c) {
	double t_0 = fma(b_2, b_2, (a * c));
	double tmp;
	if (b_2 <= -1.4e-92) {
		tmp = (sqrt(t_0) - b_2) / a;
	} else {
		tmp = (b_2 + sqrt(fabs(t_0))) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = fma(b_2, b_2, Float64(a * c))
	tmp = 0.0
	if (b_2 <= -1.4e-92)
		tmp = Float64(Float64(sqrt(t_0) - b_2) / a);
	else
		tmp = Float64(Float64(b_2 + sqrt(abs(t_0))) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -1.4e-92], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 + N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.4e-92

   1. Initial program 60.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}}{a} \]

   if -1.4e-92 < b_2

   1. Initial program 39.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 1.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} + b\_2}}{a} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{b\_2 \cdot b\_2 + a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 + a \cdot c}}} + b\_2}{a} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{\left(b\_2 \cdot b\_2 + a \cdot c\right) \cdot \left(b\_2 \cdot b\_2 + a \cdot c\right)}}} + b\_2}{a} \]

      3. rem-sqrt-square N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left|b\_2 \cdot b\_2 + a \cdot c\right|}} + b\_2}{a} \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left|b\_2 \cdot b\_2 + a \cdot c\right|}} + b\_2}{a} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left|\color{blue}{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}\right|} + b\_2}{a} \]

      6. *-lowering-*.f64 35.4

         \[\leadsto \frac{\sqrt{\left|\mathsf{fma}\left(b\_2, b\_2, \color{blue}{a \cdot c}\right)\right|} + b\_2}{a} \]

   6. Applied egg-rr 35.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right|}} + b\_2}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left|\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)\right|}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.5e-94)
   (/ (- (sqrt (fma b_2 b_2 (* a c))) b_2) a)
   (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-94) {
		tmp = (sqrt(fma(b_2, b_2, (a * c))) - b_2) / a;
	} else {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.5e-94)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.5e-94], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.49999999999999998e-94

   1. Initial program 60.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 52.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}}{a} \]

   if -3.49999999999999998e-94 < b_2

   1. Initial program 39.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{\left(0 - b\_2\right)} + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - b\_2 \cdot b\_2}{0 + b\_2}} + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{0} - b\_2 \cdot b\_2}{0 + b\_2} + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   4. Applied egg-rr 35.4%

      \[\leadsto \frac{\color{blue}{b\_2} + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 10.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 10^{-277}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1e-277)
   (/ (+ b_2 (sqrt (fma c a (* b_2 b_2)))) a)
   (/ (- b_2 (sqrt (fma b_2 b_2 (* a c)))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1e-277) {
		tmp = (b_2 + sqrt(fma(c, a, (b_2 * b_2)))) / a;
	} else {
		tmp = (b_2 - sqrt(fma(b_2, b_2, (a * c)))) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1e-277)
		tmp = Float64(Float64(b_2 + sqrt(fma(c, a, Float64(b_2 * b_2)))) / a);
	else
		tmp = Float64(Float64(b_2 - sqrt(fma(b_2, b_2, Float64(a * c)))) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1e-277], N[(N[(b$95$2 + N[Sqrt[N[(c * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 - N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 9.99999999999999969e-278

   1. Initial program 65.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 9.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} + b\_2}}{a} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{a \cdot c + b\_2 \cdot b\_2}} + b\_2}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot a} + b\_2 \cdot b\_2} + b\_2}{a} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}} + b\_2}{a} \]

      4. *-lowering-*.f64 9.6

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, a, \color{blue}{b\_2 \cdot b\_2}\right)} + b\_2}{a} \]

   6. Applied egg-rr 9.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}} + b\_2}{a} \]

   if 9.99999999999999969e-278 < b_2

   1. Initial program 26.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   4. Applied egg-rr 5.5%

      \[\leadsto \frac{\color{blue}{b\_2 - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 10^{-277}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 - \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
}

Python

def code(a, b_2, c):
	return (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a

Julia

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
end

Wolfram

code[a_, b$95$2_, c_] := 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]

Derivation

1. Initial program 48.1%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Final simplification 48.1%

   \[\leadsto \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a} \]
4. Add Preprocessing

Alternative 11: 26.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}{a} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.1%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

4. Applied egg-rr 24.4%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} - b\_2}}{a} \]
5. Add Preprocessing

Alternative 12: 8.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 + \sqrt{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}}{a} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.1%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

4. Applied egg-rr 6.1%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} + b\_2}}{a} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sqrt{\color{blue}{a \cdot c + b\_2 \cdot b\_2}} + b\_2}{a} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot a} + b\_2 \cdot b\_2} + b\_2}{a} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}} + b\_2}{a} \]

   4. *-lowering-*.f64 6.2

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, a, \color{blue}{b\_2 \cdot b\_2}\right)} + b\_2}{a} \]

6. Applied egg-rr 6.2%

   \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}} + b\_2}{a} \]

7. Final simplification 6.2%

   \[\leadsto \frac{b\_2 + \sqrt{\mathsf{fma}\left(c, a, b\_2 \cdot b\_2\right)}}{a} \]
8. Add Preprocessing

Alternative 13: 8.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}{a} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.1%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

4. Applied egg-rr 6.1%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)} + b\_2}}{a} \]

5. Final simplification 6.1%

   \[\leadsto \frac{b\_2 + \sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot c\right)}}{a} \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{t\_1 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b\_2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))