quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.5% → 86.2%

Time: 10.1s

Alternatives: 9

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: 52.5% 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: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\left(-b\_2\right) - \sqrt{c \cdot \left(\frac{b\_2 \cdot b\_2}{c} - a\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \frac{c}{b\_2}, \frac{-0.5}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+101)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.5e-69)
     (/ (- (sqrt (fma b_2 b_2 (* c (- a)))) b_2) a)
     (if (<= b_2 3.5e+84)
       (/ (/ (* a c) (- (- b_2) (sqrt (* c (- (/ (* b_2 b_2) c) a))))) a)
       (* c (fma -0.125 (* (/ (/ a b_2) b_2) (/ c b_2)) (/ -0.5 b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+101) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-69) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) - b_2) / a;
	} else if (b_2 <= 3.5e+84) {
		tmp = ((a * c) / (-b_2 - sqrt((c * (((b_2 * b_2) / c) - a))))) / a;
	} else {
		tmp = c * fma(-0.125, (((a / b_2) / b_2) * (c / b_2)), (-0.5 / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+101)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.5e-69)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) - b_2) / a);
	elseif (b_2 <= 3.5e+84)
		tmp = Float64(Float64(Float64(a * c) / Float64(Float64(-b_2) - sqrt(Float64(c * Float64(Float64(Float64(b_2 * b_2) / c) - a))))) / a);
	else
		tmp = Float64(c * fma(-0.125, Float64(Float64(Float64(a / b_2) / b_2) * Float64(c / b_2)), Float64(-0.5 / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+101], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-69], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.5e+84], N[(N[(N[(a * c), $MachinePrecision] / N[((-b$95$2) - N[Sqrt[N[(c * N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] / c), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.125 * N[(N[(N[(a / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision] * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -4.99999999999999989e101

   1. Initial program 64.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -4.99999999999999989e101 < b_2 < 3.5000000000000001e-69

   1. Initial program 83.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 80.1

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

   5. Applied rewrites 80.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 83.9%

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

   if 3.5000000000000001e-69 < b_2 < 3.4999999999999999e84

   1. Initial program 56.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 52.4%

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

   8. Taylor expanded in b_2 around 0

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

      1. lower-*.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if 3.4999999999999999e84 < b_2

   1. Initial program 13.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot c}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\frac{a \cdot c}{{b\_2}^{3}}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\color{blue}{a \cdot c}}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 89.5

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

   5. Applied rewrites 89.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.125, \frac{a \cdot c}{b\_2 \cdot \left(b\_2 \cdot b\_2\right)}, \frac{-0.5}{b\_2}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.0%

      \[\leadsto c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \color{blue}{\frac{c}{b\_2}}, \frac{-0.5}{b\_2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\left(-b\_2\right) - \sqrt{c \cdot \left(\frac{b\_2 \cdot b\_2}{c} - a\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \frac{c}{b\_2}, \frac{-0.5}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \frac{c}{b\_2}, \frac{-0.5}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+101)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.5e-23)
     (/ (- (sqrt (fma b_2 b_2 (* c (- a)))) b_2) a)
     (* c (fma -0.125 (* (/ (/ a b_2) b_2) (/ c b_2)) (/ -0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+101) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-23) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) - b_2) / a;
	} else {
		tmp = c * fma(-0.125, (((a / b_2) / b_2) * (c / b_2)), (-0.5 / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+101)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.5e-23)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) - b_2) / a);
	else
		tmp = Float64(c * fma(-0.125, Float64(Float64(Float64(a / b_2) / b_2) * Float64(c / b_2)), Float64(-0.5 / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+101], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-23], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.125 * N[(N[(N[(a / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision] * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999989e101

   1. Initial program 64.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -4.99999999999999989e101 < b_2 < 3.49999999999999993e-23

   1. Initial program 84.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 84.4%

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

   if 3.49999999999999993e-23 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot c}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\frac{a \cdot c}{{b\_2}^{3}}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\color{blue}{a \cdot c}}{{b\_2}^{3}}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)\right) \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(-0.125, \frac{a \cdot c}{b\_2 \cdot \left(b\_2 \cdot b\_2\right)}, \frac{-0.5}{b\_2}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.7%

      \[\leadsto c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \color{blue}{\frac{c}{b\_2}}, \frac{-0.5}{b\_2}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(-0.125, \frac{\frac{a}{b\_2}}{b\_2} \cdot \frac{c}{b\_2}, \frac{-0.5}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(a, -0.125 \cdot \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+101)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.5e-23)
     (/ (- (sqrt (fma b_2 b_2 (* c (- a)))) b_2) a)
     (* c (/ (fma a (* -0.125 (/ c (* b_2 b_2))) -0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+101) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-23) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) - b_2) / a;
	} else {
		tmp = c * (fma(a, (-0.125 * (c / (b_2 * b_2))), -0.5) / b_2);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+101)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.5e-23)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) - b_2) / a);
	else
		tmp = Float64(c * Float64(fma(a, Float64(-0.125 * Float64(c / Float64(b_2 * b_2))), -0.5) / b_2));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+101], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-23], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(N[(a * N[(-0.125 * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999989e101

   1. Initial program 64.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -4.99999999999999989e101 < b_2 < 3.49999999999999993e-23

   1. Initial program 84.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 84.4%

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

   if 3.49999999999999993e-23 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 10.6

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

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{3}} - \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. div-sub N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}}{b\_2}} \]

      9. lower-/.f64 N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}}{b\_2}} \]

   8. Applied rewrites 85.5%

      \[\leadsto \color{blue}{c \cdot \frac{\mathsf{fma}\left(a, -0.125 \cdot \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(a, -0.125 \cdot \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+101)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.5e-23)
     (/ (- (sqrt (fma b_2 b_2 (* c (- a)))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+101) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-23) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+101)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.5e-23)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+101], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-23], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999989e101

   1. Initial program 64.9%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -4.99999999999999989e101 < b_2 < 3.49999999999999993e-23

   1. Initial program 84.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 80.9

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

   5. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 84.4%

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

   if 3.49999999999999993e-23 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 10.6

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

   5. Applied rewrites 10.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 9.3%

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

   8. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.3

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

   10. Applied rewrites 85.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, \frac{c \cdot 0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+18)
   (fma (/ b_2 a) -2.0 (/ (* c 0.5) b_2))
   (if (<= b_2 3.5e-23) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+18) {
		tmp = fma((b_2 / a), -2.0, ((c * 0.5) / b_2));
	} else if (b_2 <= 3.5e-23) {
		tmp = (sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+18)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 3.5e-23)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+18], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-23], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4e18

   1. Initial program 71.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 58.7

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

   5. Applied rewrites 58.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 3.6%

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

   8. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-neg.f64 96.2

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

   10. Applied rewrites 96.2%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 96.3%

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

   if -4e18 < b_2 < 3.49999999999999993e-23

   1. Initial program 82.2%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if 3.49999999999999993e-23 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 10.6

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

   5. Applied rewrites 10.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 9.3%

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

   8. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.3

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

   10. Applied rewrites 85.3%

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

Alternative 6: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+18)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 3.5e-23) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+18) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-23) {
		tmp = (sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / 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 <= (-4d+18)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 3.5d-23) then
        tmp = (sqrt((c * -a)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / 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 <= -4e+18) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 3.5e-23) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e+18:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 3.5e-23:
		tmp = (math.sqrt((c * -a)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+18)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 3.5e-23)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e+18)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 3.5e-23)
		tmp = (sqrt((c * -a)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+18], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-23], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4e18

   1. Initial program 71.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -4e18 < b_2 < 3.49999999999999993e-23

   1. Initial program 82.2%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 74.3

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

   5. Applied rewrites 74.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if 3.49999999999999993e-23 < b_2

   1. Initial program 19.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 10.6

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

   5. Applied rewrites 10.6%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 9.3%

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

   8. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.3

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

   10. Applied rewrites 85.3%

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

Alternative 7: 67.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-300}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-300) (* -2.0 (/ b_2 a)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c * -0.5) / 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 <= 9d-300) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c * (-0.5d0)) / 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 <= 9e-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-300:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-300)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-300)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-300], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 9.0000000000000001e-300

   1. Initial program 77.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 9.0000000000000001e-300 < b_2

   1. Initial program 38.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 32.4

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

   5. Applied rewrites 32.4%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 62.6

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

   10. Applied rewrites 62.6%

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

Alternative 8: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-300}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-300) (* -2.0 (/ b_2 a)) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = c * (-0.5 / 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 <= 9d-300) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = c * ((-0.5d0) / 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 <= 9e-300) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-300:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-300)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-300)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-300], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 9.0000000000000001e-300

   1. Initial program 77.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if 9.0000000000000001e-300 < b_2

   1. Initial program 38.4%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 9: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

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

Derivation

1. Initial program 58.6%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 34.5

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

5. Applied rewrites 34.5%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
6. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024226 
(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))