quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.3% → 87.0%

Time: 7.5s

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: 51.3% 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: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 1.16 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+137)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 1.16e-154)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (if (<= b_2 2.6e+141)
       (/ (/ (* c a) a) (- (- b_2) (sqrt (fma b_2 b_2 (* (- a) c)))))
       (* (/ c b_2) -0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+137) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 1.16e-154) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else if (b_2 <= 2.6e+141) {
		tmp = ((c * a) / a) / (-b_2 - sqrt(fma(b_2, b_2, (-a * c))));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e+137)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 1.16e-154)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	elseif (b_2 <= 2.6e+141)
		tmp = Float64(Float64(Float64(c * a) / a) / Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(Float64(-a) * c)))));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e+137], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.16e-154], 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], If[LessEqual[b$95$2, 2.6e+141], N[(N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision] / N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[((-a) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.8e137

   1. Initial program 43.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 95.6%

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

   if -1.8e137 < b_2 < 1.16000000000000004e-154

   1. Initial program 81.5%

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

   if 1.16000000000000004e-154 < b_2 < 2.5999999999999999e141

   1. Initial program 42.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 42.9

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

   4. Applied rewrites 42.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 33.7%

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

   8. Applied rewrites 79.8%

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

      1. lift-+.f64 N/A

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

      2. +-lft-identity 79.8

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 79.8

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

   10. Applied rewrites 79.8%

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

   if 2.5999999999999999e141 < b_2

   1. Initial program 2.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 1.16 \cdot 10^{-154}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b\_2 \leq 2.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b\_2 \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{a \cdot c}{\left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+137)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 9.5e-21)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (if (<= b_2 3.6e+138)
       (/ (* a c) (* (- (- b_2) (sqrt (fma b_2 b_2 (* c (- a))))) a))
       (* (/ c b_2) -0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+137) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 9.5e-21) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else if (b_2 <= 3.6e+138) {
		tmp = (a * c) / ((-b_2 - sqrt(fma(b_2, b_2, (c * -a)))) * a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e+137)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 9.5e-21)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	elseif (b_2 <= 3.6e+138)
		tmp = Float64(Float64(a * c) / Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(c * Float64(-a))))) * a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e+137], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.5e-21], 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], If[LessEqual[b$95$2, 3.6e+138], N[(N[(a * c), $MachinePrecision] / N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.8e137

   1. Initial program 43.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 95.6%

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

   if -1.8e137 < b_2 < 9.4999999999999994e-21

   1. Initial program 75.0%

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

   if 9.4999999999999994e-21 < b_2 < 3.6000000000000001e138

   1. Initial program 40.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 40.4

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

   4. Applied rewrites 40.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 91.4

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

   9. Applied rewrites 91.4%

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

   if 3.6000000000000001e138 < b_2

   1. Initial program 2.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b\_2 \leq 3.6 \cdot 10^{+138}:\\ \;\;\;\;\frac{a \cdot c}{\left(\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+137)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 9e+31)
     (/ (+ (- b_2) (sqrt (fma b_2 b_2 (* (- a) c)))) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+137) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 9e+31) {
		tmp = (-b_2 + sqrt(fma(b_2, b_2, (-a * c)))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e+137)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 9e+31)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(fma(b_2, b_2, Float64(Float64(-a) * c)))) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e+137], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9e+31], N[(N[((-b$95$2) + N[Sqrt[N[(b$95$2 * b$95$2 + N[((-a) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.8e137

   1. Initial program 43.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 95.6%

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

   if -1.8e137 < b_2 < 8.9999999999999992e31

   1. Initial program 73.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 73.6

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

   4. Applied rewrites 73.6%

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

   if 8.9999999999999992e31 < b_2

   1. Initial program 13.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 13.6

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

   4. Applied rewrites 13.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 6.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 91.9

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

   9. Applied rewrites 91.9%

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

   11. Applied rewrites 91.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+137)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 9e+31)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e+137) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 9e+31) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e+137)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 9e+31)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e+137], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9e+31], 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], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.8e137

   1. Initial program 43.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 95.6%

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

   if -1.8e137 < b_2 < 8.9999999999999992e31

   1. Initial program 73.6%

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

   if 8.9999999999999992e31 < b_2

   1. Initial program 13.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 13.6

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

   4. Applied rewrites 13.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 6.5%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 91.9

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

   9. Applied rewrites 91.9%

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

   11. Applied rewrites 91.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 1.75 \cdot 10^{-68}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-92)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 1.75e-68)
     (/ (+ (- b_2) (sqrt (* c (- a)))) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-92) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 1.75e-68) {
		tmp = (-b_2 + sqrt((c * -a))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-92)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 1.75e-68)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-92], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.75e-68], N[(N[((-b$95$2) + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1999999999999997e-92

   1. Initial program 62.3%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 86.7%

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

   if -3.1999999999999997e-92 < b_2 < 1.75000000000000006e-68

   1. Initial program 73.3%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 1.75000000000000006e-68 < b_2

   1. Initial program 20.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 20.0

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

   4. Applied rewrites 20.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 14.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.6

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

   9. Applied rewrites 81.6%

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

   11. Applied rewrites 81.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b\_2 \leq 1.75 \cdot 10^{-68}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 68.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 66.6%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 30.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 30.7

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

   4. Applied rewrites 30.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 26.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.7

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

   9. Applied rewrites 66.7%

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

   11. Applied rewrites 66.8%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2}{a} \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -2.0 (/ b_2 a)) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 68.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 30.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 30.7

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

   4. Applied rewrites 30.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   6. Applied rewrites 26.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.7

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

   9. Applied rewrites 66.7%

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

   11. Applied rewrites 66.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -2.0 (/ b_2 a)) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 68.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 30.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.7

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

   5. Applied rewrites 66.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.25e+39) (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.25e+39) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 1.25d+39) then
        tmp = (-2.0d0) * (b_2 / a)
    else
        tmp = 0.5d0 * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.25e+39) {
		tmp = -2.0 * (b_2 / a);
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.25e+39:
		tmp = -2.0 * (b_2 / a)
	else:
		tmp = 0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.25e+39)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	else
		tmp = Float64(0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.25e+39)
		tmp = -2.0 * (b_2 / a);
	else
		tmp = 0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.25e+39], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.25000000000000004e39

   1. Initial program 64.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 47.1

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

   5. Applied rewrites 47.1%

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

   if 1.25000000000000004e39 < b_2

   1. Initial program 13.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 21.1%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 10.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return 0.5 * (c / 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 = 0.5d0 * (c / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 33.9

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

5. Applied rewrites 33.9%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 8.4%

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

9. Final simplification 8.4%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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))