quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.8% → 87.1%

Time: 9.8s

Alternatives: 10

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: 51.8% 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: 87.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{+109}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot \left(-c\right)\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.75e+109)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 8.5e-178)
     (/ (- (sqrt (fma b_2 b_2 (* a (- c)))) b_2) a)
     (if (<= b_2 7.8e-30)
       (/ (/ (* a c) (- (- b_2) (sqrt (fma c (- a) (* b_2 b_2))))) a)
       (/ (* c -0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.75e+109) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 8.5e-178) {
		tmp = (sqrt(fma(b_2, b_2, (a * -c))) - b_2) / a;
	} else if (b_2 <= 7.8e-30) {
		tmp = ((a * c) / (-b_2 - sqrt(fma(c, -a, (b_2 * 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 <= -1.75e+109)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 8.5e-178)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(a * Float64(-c)))) - b_2) / a);
	elseif (b_2 <= 7.8e-30)
		tmp = Float64(Float64(Float64(a * c) / Float64(Float64(-b_2) - sqrt(fma(c, Float64(-a), Float64(b_2 * 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, -1.75e+109], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.5e-178], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * (-c)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.8e-30], N[(N[(N[(a * c), $MachinePrecision] / N[((-b$95$2) - N[Sqrt[N[(c * (-a) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.74999999999999992e109

   1. Initial program 49.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -1.74999999999999992e109 < b_2 < 8.5000000000000001e-178

   1. Initial program 84.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 81.2

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

   5. Applied rewrites 81.2%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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 81.2

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

   7. Applied rewrites 81.2%

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

   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 84.2%

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

   if 8.5000000000000001e-178 < b_2 < 7.8000000000000007e-30

   1. Initial program 62.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.3

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

   5. Applied rewrites 62.3%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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 62.0%

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

   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 74.8

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

   10. Applied rewrites 74.8%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 74.8%

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

   if 7.8000000000000007e-30 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{+109}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, a \cdot \left(-c\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.75e+109)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4.15e-32)
     (/ (- (sqrt (fma b_2 b_2 (* a (- c)))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.75e+109) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.15e-32) {
		tmp = (sqrt(fma(b_2, b_2, (a * -c))) - 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 <= -1.75e+109)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4.15e-32)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(a * Float64(-c)))) - 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, -1.75e+109], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.15e-32], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * (-c)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.74999999999999992e109

   1. Initial program 49.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -1.74999999999999992e109 < b_2 < 4.15000000000000006e-32

   1. Initial program 78.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.0

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

   5. Applied rewrites 76.0%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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 76.0

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

   7. Applied rewrites 76.0%

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

   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 78.2%

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

   if 4.15000000000000006e-32 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 3: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{a \cdot \left(-c\right)} - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-69)
   (fma (/ b_2 a) -2.0 (* 0.5 (/ c b_2)))
   (if (<= b_2 4.15e-32)
     (* (/ 1.0 a) (- (sqrt (* a (- c))) b_2))
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-69) {
		tmp = fma((b_2 / a), -2.0, (0.5 * (c / b_2)));
	} else if (b_2 <= 4.15e-32) {
		tmp = (1.0 / a) * (sqrt((a * -c)) - b_2);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-69)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.15e-32)
		tmp = Float64(Float64(1.0 / a) * Float64(sqrt(Float64(a * Float64(-c))) - b_2));
	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, -4.2e-69], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.15e-32], N[(N[(1.0 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.1999999999999999e-69

   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

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   4. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 89.2%

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

   if -4.1999999999999999e-69 < b_2 < 4.15000000000000006e-32

   1. Initial program 71.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 71.6

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in c around inf

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

   10. Applied rewrites 66.3%

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

   if 4.15000000000000006e-32 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{a \cdot \left(-c\right)} - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b\_2}{a}, -2, 0.5 \cdot \frac{c}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-69)
   (fma (/ b_2 a) -2.0 (* 0.5 (/ c b_2)))
   (if (<= b_2 4.15e-32) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-69) {
		tmp = fma((b_2 / a), -2.0, (0.5 * (c / b_2)));
	} else if (b_2 <= 4.15e-32) {
		tmp = (sqrt((a * -c)) - 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 <= -4.2e-69)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.15e-32)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - 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, -4.2e-69], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.15e-32], N[(N[(N[Sqrt[N[(a * (-c)), $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 < -4.1999999999999999e-69

   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

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   4. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 89.2%

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

   if -4.1999999999999999e-69 < b_2 < 4.15000000000000006e-32

   1. Initial program 71.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 71.6

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

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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. +-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 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in c around inf

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

   10. Applied rewrites 66.3%

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

   if 4.15000000000000006e-32 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 5: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(b\_2, \frac{-2}{a}, c \cdot \frac{0.5}{b\_2}\right)\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-69)
   (fma b_2 (/ -2.0 a) (* c (/ 0.5 b_2)))
   (if (<= b_2 4.15e-32) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-69) {
		tmp = fma(b_2, (-2.0 / a), (c * (0.5 / b_2)));
	} else if (b_2 <= 4.15e-32) {
		tmp = (sqrt((a * -c)) - 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 <= -4.2e-69)
		tmp = fma(b_2, Float64(-2.0 / a), Float64(c * Float64(0.5 / b_2)));
	elseif (b_2 <= 4.15e-32)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - 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, -4.2e-69], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.15e-32], N[(N[(N[Sqrt[N[(a * (-c)), $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 < -4.1999999999999999e-69

   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

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   4. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 88.9%

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

   if -4.1999999999999999e-69 < b_2 < 4.15000000000000006e-32

   1. Initial program 71.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 71.6

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

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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. +-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 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in c around inf

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

   10. Applied rewrites 66.3%

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

   if 4.15000000000000006e-32 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 6: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 4.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-69)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 4.15e-32) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-69) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.15e-32) {
		tmp = (sqrt((a * -c)) - 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 <= (-4.2d-69)) then
        tmp = (-2.0d0) * (b_2 / a)
    else if (b_2 <= 4.15d-32) then
        tmp = (sqrt((a * -c)) - 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 <= -4.2e-69) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 4.15e-32) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-69:
		tmp = -2.0 * (b_2 / a)
	elif b_2 <= 4.15e-32:
		tmp = (math.sqrt((a * -c)) - 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 <= -4.2e-69)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 4.15e-32)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - 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 <= -4.2e-69)
		tmp = -2.0 * (b_2 / a);
	elseif (b_2 <= 4.15e-32)
		tmp = (sqrt((a * -c)) - 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, -4.2e-69], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.15e-32], N[(N[(N[Sqrt[N[(a * (-c)), $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 < -4.1999999999999999e-69

   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 88.8

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

   5. Applied rewrites 88.8%

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

   if -4.1999999999999999e-69 < b_2 < 4.15000000000000006e-32

   1. Initial program 71.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 71.6

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

   5. Applied rewrites 71.6%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{c \cdot \mathsf{fma}\left(b\_2, \frac{b\_2}{c}, -a\right)}}}{a} \]
   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. +-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 71.6

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

   7. Applied rewrites 71.6%

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

   8. Taylor expanded in c around inf

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

   10. Applied rewrites 66.3%

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

   if 4.15000000000000006e-32 < b_2

   1. Initial program 8.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 96.4

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

   8. Applied rewrites 96.4%

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

Alternative 7: 68.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   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 67.4

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

   5. Applied rewrites 67.4%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 31.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 27.5

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. 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 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 8: 68.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2e-310) (* -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 <= 2e-310) {
		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 <= 2d-310) 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 <= 2e-310) {
		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 <= 2e-310:
		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 <= 2e-310)
		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 <= 2e-310)
		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, 2e-310], 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 < 1.999999999999994e-310

   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 67.4

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

   5. Applied rewrites 67.4%

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

   if 1.999999999999994e-310 < b_2

   1. Initial program 31.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

Alternative 9: 43.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.1 \cdot 10^{+56}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.1e+56) (* -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 <= 2.1e+56) {
		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 <= 2.1d+56) 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 <= 2.1e+56) {
		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 <= 2.1e+56:
		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 <= 2.1e+56)
		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 <= 2.1e+56)
		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, 2.1e+56], 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 < 2.10000000000000017e56

   1. Initial program 65.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 45.4

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

   5. Applied rewrites 45.4%

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

   if 2.10000000000000017e56 < b_2

   1. Initial program 9.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
   4. 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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in b_2 around 0

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

   8. Applied rewrites 39.0%

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

Alternative 10: 10.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 * (0.5d0 / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]
4. 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. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 33.5

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

5. Applied rewrites 33.5%

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

6. Taylor expanded in b_2 around 0

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

8. Applied rewrites 12.2%

   \[\leadsto c \cdot \color{blue}{\frac{0.5}{b\_2}} \]
9. 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 2024233 
(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))