quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.6% → 85.2%

Time: 8.9s

Alternatives: 8

Speedup: 1.7×

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: 52.6% 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: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}, \frac{-b\_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.06e+61)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.15e-37)
     (fma (/ 1.0 a) (sqrt (fma (- c) a (* b_2 b_2))) (/ (- b_2) a))
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.06e+61) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.15e-37) {
		tmp = fma((1.0 / a), sqrt(fma(-c, a, (b_2 * b_2))), (-b_2 / a));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.06e+61)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.15e-37)
		tmp = fma(Float64(1.0 / a), sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))), Float64(Float64(-b_2) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.06e+61], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.15e-37], N[(N[(1.0 / a), $MachinePrecision] * N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[((-b$95$2) / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.0599999999999999e61

   1. Initial program 59.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.2

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -1.0599999999999999e61 < b_2 < 1.15e-37

   1. Initial program 77.8%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2} \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b\_2}^{2}} + 1\right)} \cdot {b\_2}^{2}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{c}{{b\_2}^{2}}}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{c}{{b\_2}^{2}} \cdot a}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right)\right) \cdot a} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right), a, 1\right)} \cdot {b\_2}^{2}}}{a} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{\mathsf{neg}\left({b\_2}^{2}\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{-1 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(-1 \cdot b\_2\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      14. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      16. lower-neg.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      17. unpow2 N/A

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

      18. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 38.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 38.0

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

   7. Applied rewrites 39.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{a} \cdot \left(\sqrt{-1 \cdot \left(a \cdot c\right) + \color{blue}{{b\_2}^{2}}} - b\_2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

       \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-a, \color{blue}{c}, b\_2 \cdot b\_2\right)} - b\_2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-neg N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 77.8

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

   12. Applied rewrites 77.8%

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

   if 1.15e-37 < b_2

   1. Initial program 21.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.4

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a}, \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}, \frac{-b\_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.06e+61)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.15e-37)
     (/ (- (sqrt (fma (- c) a (* b_2 b_2))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.06e+61) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.15e-37) {
		tmp = (sqrt(fma(-c, a, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.06e+61)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.15e-37)
		tmp = Float64(Float64(sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.06e+61], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.15e-37], N[(N[(N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.0599999999999999e61

   1. Initial program 59.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.2

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -1.0599999999999999e61 < b_2 < 1.15e-37

   1. Initial program 77.8%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2} \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b\_2}^{2}} + 1\right)} \cdot {b\_2}^{2}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{c}{{b\_2}^{2}}}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{c}{{b\_2}^{2}} \cdot a}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right)\right) \cdot a} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right), a, 1\right)} \cdot {b\_2}^{2}}}{a} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{\mathsf{neg}\left({b\_2}^{2}\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{-1 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(-1 \cdot b\_2\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      14. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      16. lower-neg.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      17. unpow2 N/A

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

      18. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 38.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 38.0

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

   7. Applied rewrites 39.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{a} \cdot \left(\sqrt{-1 \cdot \left(a \cdot c\right) + \color{blue}{{b\_2}^{2}}} - b\_2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 77.6%

       \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-a, \color{blue}{c}, b\_2 \cdot b\_2\right)} - b\_2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lower-/.f64 77.8

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

   12. Applied rewrites 77.8%

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

   if 1.15e-37 < b_2

   1. Initial program 21.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.4

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{+61}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 5.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.6e-124)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 5.9e-38) (/ (- (sqrt (* (- a) c)) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-124) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.9e-38) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_2 <= (-7.6d-124)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 5.9d-38) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-124) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.9e-38) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.6e-124:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 5.9e-38:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.6e-124)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 5.9e-38)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.6e-124)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 5.9e-38)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.6e-124], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.9e-38], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.60000000000000025e-124

   1. Initial program 69.1%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 80.5

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 80.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -7.60000000000000025e-124 < b_2 < 5.89999999999999983e-38

   1. Initial program 74.8%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2} \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b\_2}^{2}} + 1\right)} \cdot {b\_2}^{2}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{c}{{b\_2}^{2}}}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{c}{{b\_2}^{2}} \cdot a}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right)\right) \cdot a} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right), a, 1\right)} \cdot {b\_2}^{2}}}{a} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{\mathsf{neg}\left({b\_2}^{2}\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{-1 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(-1 \cdot b\_2\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      14. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      16. lower-neg.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      17. unpow2 N/A

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

      18. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 19.4

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 19.4

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

   7. Applied rewrites 21.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \frac{1}{a} \cdot \left(\sqrt{-1 \cdot \color{blue}{\left(a \cdot c\right)}} - b\_2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.3%

       \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\left(-a\right) \cdot \color{blue}{c}} - b\_2\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lower-/.f64 70.3

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

   12. Applied rewrites 70.3%

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

   if 5.89999999999999983e-38 < b_2

   1. Initial program 21.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 88.4

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 5.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -2.0 b_2) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-2.0d0) * b_2) / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-2.0 * b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.9%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 63.6

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 63.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 65.2

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (- b_2) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = -b_2 / a
    else
        tmp = (-0.5d0) * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -b_2 / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -b_2 / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[((-b$95$2) / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 18.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. sqrt-prod N/A

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

   6. Applied rewrites 27.5%

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

   7. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 24.2

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

   9. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 65.2

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (- b_2) a) (* (/ -0.5 b_2) c)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = -b_2 / a
    else
        tmp = ((-0.5d0) / b_2) * c
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -b_2 / a
	else:
		tmp = (-0.5 / b_2) * c
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(-0.5 / b_2) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -b_2 / a;
	else
		tmp = (-0.5 / b_2) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[((-b$95$2) / a), $MachinePrecision], N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 18.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. sqrt-prod N/A

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

   6. Applied rewrites 27.5%

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

   7. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 24.2

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

   9. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 65.2

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.0%

      \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -5e-310) (/ (- b_2) a) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = -b_2 / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -b_2 / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -b_2 / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[((-b$95$2) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.9%

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

   4. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 18.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. sqrt-prod N/A

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

   6. Applied rewrites 27.5%

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

   7. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 24.2

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

   9. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 39.1%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2} \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b\_2}^{2}} + 1\right)} \cdot {b\_2}^{2}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{c}{{b\_2}^{2}}}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{c}{{b\_2}^{2}} \cdot a}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right)\right) \cdot a} + 1\right) \cdot {b\_2}^{2}}}{a} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right), a, 1\right)} \cdot {b\_2}^{2}}}{a} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{\mathsf{neg}\left({b\_2}^{2}\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{-1 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(-1 \cdot b\_2\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      14. mul-1-neg N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      16. lower-neg.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

      17. unpow2 N/A

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

      18. lower-*.f64 21.9

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

   5. Applied rewrites 21.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 22.0

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 22.0

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

   7. Applied rewrites 22.4%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{\left(\mathsf{fma}\left(a, \frac{c}{\left(-b\_2\right) \cdot b\_2}, 1\right) \cdot b\_2\right) \cdot b\_2} - b\_2\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 15.5%

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

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \frac{b\_2}{a}} \]
   11. Step-by-step derivation

       1. distribute-lft1-in N/A

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

       2. metadata-eval N/A

          \[\leadsto \color{blue}{0} \cdot \frac{b\_2}{a} \]

       3. mul0-lft 19.7

          \[\leadsto \color{blue}{0} \]

   12. Applied rewrites 19.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 11.0% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in b_2 around inf

   \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{{b\_2}^{2} \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)}}}{a} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(1 + -1 \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot {b\_2}^{2}}}}{a} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\left(-1 \cdot \frac{a \cdot c}{{b\_2}^{2}} + 1\right)} \cdot {b\_2}^{2}}}{a} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)} + 1\right) \cdot {b\_2}^{2}}}{a} \]

   5. associate-/l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{c}{{b\_2}^{2}}}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

   6. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\left(\mathsf{neg}\left(\color{blue}{\frac{c}{{b\_2}^{2}} \cdot a}\right)\right) + 1\right) \cdot {b\_2}^{2}}}{a} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right)\right) \cdot a} + 1\right) \cdot {b\_2}^{2}}}{a} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{c}{{b\_2}^{2}}\right), a, 1\right)} \cdot {b\_2}^{2}}}{a} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{\mathsf{neg}\left({b\_2}^{2}\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{\frac{c}{-1 \cdot {b\_2}^{2}}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{-1 \cdot \color{blue}{\left(b\_2 \cdot b\_2\right)}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(-1 \cdot b\_2\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   14. mul-1-neg N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot b\_2}}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   16. lower-neg.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(\frac{c}{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot b\_2}, a, 1\right) \cdot {b\_2}^{2}}}{a} \]

   17. unpow2 N/A

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

   18. lower-*.f64 37.6

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

5. Applied rewrites 37.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 37.6

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 37.6

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

7. Applied rewrites 38.1%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{\left(\mathsf{fma}\left(a, \frac{c}{\left(-b\_2\right) \cdot b\_2}, 1\right) \cdot b\_2\right) \cdot b\_2} - b\_2\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

9. Applied rewrites 34.7%

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

10. Taylor expanded in c around 0

    \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a} + \frac{b\_2}{a}} \]
11. Step-by-step derivation

    1. distribute-lft1-in N/A

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

    2. metadata-eval N/A

       \[\leadsto \color{blue}{0} \cdot \frac{b\_2}{a} \]

    3. mul0-lft 11.0

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 11.0%

    \[\leadsto \color{blue}{0} \]
13. 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 2024236 
(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))