quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.6% → 85.9%

Time: 8.7s

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: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e+135)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.85e-44)
     (- (/ (sqrt (fma a (- c) (* b_2 b_2))) a) (/ b_2 a))
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e+135) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.85e-44) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) / a) - (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e+135)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.85e-44)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.2e+135], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-44], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.2e135

   1. Initial program 48.5%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\_2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2 \cdot b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2}}{b\_2} \cdot \frac{c}{b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{b\_2} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right)} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b\_2}}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(-b\_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \frac{c}{b\_2}, \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{-0.5}{b\_2}, \frac{c}{b\_2}, \mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b\_2}^{4}}, -0.125, \frac{2}{a}\right)\right)} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b\_2}}, \frac{b\_2}{a} \cdot -2\right) \]

   if -5.2e135 < b_2 < 1.85e-44

   1. Initial program 77.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 77.3

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

   4. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 77.3

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 77.3

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

   6. Applied rewrites 77.3%

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

   if 1.85e-44 < b_2

   1. Initial program 7.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e+135)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.85e-44)
     (/ (- (sqrt (fma a (- c) (* b_2 b_2))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e+135) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.85e-44) {
		tmp = (sqrt(fma(a, -c, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e+135)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.85e-44)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-c), Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.2e+135], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-44], N[(N[(N[Sqrt[N[(a * (-c) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.2e135

   1. Initial program 48.5%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\_2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2 \cdot b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2}}{b\_2} \cdot \frac{c}{b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{b\_2} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right)} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b\_2}}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(-b\_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \frac{c}{b\_2}, \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)} \]

   5. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{-0.5}{b\_2}, \frac{c}{b\_2}, \mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b\_2}^{4}}, -0.125, \frac{2}{a}\right)\right)} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b\_2}}, \frac{b\_2}{a} \cdot -2\right) \]

   if -5.2e135 < b_2 < 1.85e-44

   1. Initial program 77.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 77.3

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

   4. Applied rewrites 77.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 77.3

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 77.3

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

   6. Applied rewrites 77.3%

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

   if 1.85e-44 < b_2

   1. Initial program 7.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-89)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.85e-44) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-89) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.85e-44) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e-89)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.85e-44)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e-89], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-44], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.4999999999999999e-89

   1. Initial program 73.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\_2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2 \cdot b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2}}{b\_2} \cdot \frac{c}{b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{b\_2} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right)} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b\_2}}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(-b\_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \frac{c}{b\_2}, \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)} \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{-0.5}{b\_2}, \frac{c}{b\_2}, \mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b\_2}^{4}}, -0.125, \frac{2}{a}\right)\right)} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 89.7%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b\_2}}, \frac{b\_2}{a} \cdot -2\right) \]

   if -4.4999999999999999e-89 < b_2 < 1.85e-44

   1. Initial program 64.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 64.9

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 64.9

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 64.9

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.6

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

   9. Applied rewrites 62.6%

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

   if 1.85e-44 < b_2

   1. Initial program 7.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-89}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-89)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 1.85e-44) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-89) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.85e-44) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-4.5d-89)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 1.85d-44) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-89) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 1.85e-44) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.5e-89:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 1.85e-44:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e-89)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 1.85e-44)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.5e-89)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 1.85e-44)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e-89], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-44], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.4999999999999999e-89

   1. Initial program 73.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -4.4999999999999999e-89 < b_2 < 1.85e-44

   1. Initial program 64.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 64.9

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

   4. Applied rewrites 64.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 64.9

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 64.9

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

   6. Applied rewrites 64.9%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.6

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

   9. Applied rewrites 62.6%

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

   if 1.85e-44 < b_2

   1. Initial program 7.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 5: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{-265}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e-265) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-265) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= 7.2d-265) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-265) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e-265:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.2e-265)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 7.2e-265)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.2e-265], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.2000000000000004e-265

   1. Initial program 73.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 7.2000000000000004e-265 < b_2

   1. Initial program 26.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 70.0

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

   5. Applied rewrites 70.0%

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

Alternative 6: 67.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{-265}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{b\_2} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e-265) (* -2.0 (/ b_2 a)) (* (/ -0.5 b_2) c)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-265) {
		tmp = -2.0 * (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 <= 7.2d-265) then
        tmp = (-2.0d0) * (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 <= 7.2e-265) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 / b_2) * c;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e-265:
		tmp = -2.0 * (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 <= 7.2e-265)
		tmp = Float64(-2.0 * 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 <= 7.2e-265)
		tmp = -2.0 * (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, 7.2e-265], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.2000000000000004e-265

   1. Initial program 73.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 7.2000000000000004e-265 < b_2

   1. Initial program 26.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 69.8%

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

Alternative 7: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 660000000:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 660000000.0) (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 660000000.0) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	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 <= 660000000.0d0) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 660000000.0) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 660000000.0:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * 0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 660000000.0)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 660000000.0)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 660000000.0], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 6.6e8

   1. Initial program 68.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if 6.6e8 < b_2

   1. Initial program 8.1%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\_2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2 \cdot b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      7. times-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2}}{b\_2} \cdot \frac{c}{b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{b\_2} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right)} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b\_2}}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(-b\_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \frac{c}{b\_2}, \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)} \]

   5. Applied rewrites 2.4%

      \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{-0.5}{b\_2}, \frac{c}{b\_2}, \mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b\_2}^{4}}, -0.125, \frac{2}{a}\right)\right)} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 2.8%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b\_2}}, \frac{b\_2}{a} \cdot -2\right) \]

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 19.0%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 660000000:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ c b_2) 0.5))

C

double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

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 = (c / b_2) * 0.5d0
end function

Java

public static double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

Python

def code(a, b_2, c):
	return (c / b_2) * 0.5

Julia

function code(a, b_2, c)
	return Float64(Float64(c / b_2) * 0.5)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (c / b_2) * 0.5;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot b\_2\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   4. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-b\_2\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2} \cdot c}{{b\_2}^{2}}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2 \cdot b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   7. times-frac N/A

      \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\frac{\frac{-1}{2}}{b\_2} \cdot \frac{c}{b\_2}} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(-b\_2\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{b\_2} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \left(-b\_2\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b\_2}\right)\right)} \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \left(-b\_2\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{1}{b\_2}}\right)\right) \cdot \frac{c}{b\_2} + \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(-b\_2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \frac{c}{b\_2}, \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{4}} + 2 \cdot \frac{1}{a}\right)} \]

5. Applied rewrites 27.6%

   \[\leadsto \color{blue}{\left(-b\_2\right) \cdot \mathsf{fma}\left(\frac{-0.5}{b\_2}, \frac{c}{b\_2}, \mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{{b\_2}^{4}}, -0.125, \frac{2}{a}\right)\right)} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 38.0%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b\_2}}, \frac{b\_2}{a} \cdot -2\right) \]

9. Taylor expanded in a around inf

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

11. Applied rewrites 7.5%

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

12. Final simplification 7.5%

    \[\leadsto \frac{c}{b\_2} \cdot 0.5 \]
13. Add Preprocessing

Developer Target 1: 99.6% 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 2024304 
(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))