quadp (p42, positive)

Percentage Accurate: 53.1% → 84.9%

Time: 10.0s

Alternatives: 10

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+47)
   (- (/ c b) (/ b a))
   (if (<= b 4.4e-21)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ (fma c (* c (/ a (* b b))) c) (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+47) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.4e-21) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = fma(c, (c * (a / (b * b))), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+47)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.4e-21)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+47], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-21], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000007e47

   1. Initial program 71.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -5.20000000000000007e47 < b < 4.4000000000000001e-21

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 4.4000000000000001e-21 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 91.6

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

   7. Applied rewrites 91.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+47)
   (- (/ c b) (/ b a))
   (if (<= b 4.4e-21)
     (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
     (/ (fma c (* c (/ a (* b b))) c) (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+47) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4.4e-21) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = fma(c, (c * (a / (b * b))), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+47)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4.4e-21)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+47], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-21], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.20000000000000007e47

   1. Initial program 71.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -5.20000000000000007e47 < b < 4.4000000000000001e-21

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 79.1%

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

   if 4.4000000000000001e-21 < b

   1. Initial program 12.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 3: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-115)
   (- (/ c b) (/ b a))
   (if (<= b 7.4e-65)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ (fma c (* c (/ a (* b b))) c) (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-115) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.4e-65) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = fma(c, (c * (a / (b * b))), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-115)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.4e-65)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-115], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.4e-65], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.7999999999999997e-115

   1. Initial program 78.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -7.7999999999999997e-115 < b < 7.4e-65

   1. Initial program 76.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if 7.4e-65 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 85.4

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

   7. Applied rewrites 85.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-115)
   (- (/ c b) (/ b a))
   (if (<= b 7.4e-65)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ (fma c (* c (/ a (* b b))) c) (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-115) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.4e-65) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = fma(c, (c * (a / (b * b))), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-115)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.4e-65)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-115], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.4e-65], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.7999999999999997e-115

   1. Initial program 78.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -7.7999999999999997e-115 < b < 7.4e-65

   1. Initial program 76.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 72.5

          \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Applied rewrites 72.5%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

   if 7.4e-65 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 85.4

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

   7. Applied rewrites 85.4%

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

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-115)
   (- (/ c b) (/ b a))
   (if (<= b 7.4e-65)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-115) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.4e-65) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d-115)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7.4d-65) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-115) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7.4e-65) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e-115:
		tmp = (c / b) - (b / a)
	elif b <= 7.4e-65:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-115)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7.4e-65)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e-115)
		tmp = (c / b) - (b / a);
	elseif (b <= 7.4e-65)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-115], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.4e-65], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.7999999999999997e-115

   1. Initial program 78.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -7.7999999999999997e-115 < b < 7.4e-65

   1. Initial program 76.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 72.5

          \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \color{blue}{\left(c \cdot -4\right)}}\right) \]

   7. Applied rewrites 72.5%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right) \]

   if 7.4e-65 < b

   1. Initial program 17.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 85.4

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

   5. Applied rewrites 85.4%

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

Alternative 6: 67.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (- (/ c b) (/ b a)) (/ c (- b))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 79.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. mul-1-neg N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 71.9

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

   5. Applied rewrites 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-frac-neg2 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. lift-/.f64 N/A

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

      12. *-lft-identity N/A

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

      13. lower-/.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if -1.9999999999999e-311 < b

   1. Initial program 32.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

Alternative 7: 67.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-311) (/ (- b) a) (/ c (- b))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-311], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9999999999999e-311

   1. Initial program 79.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -1.9999999999999e-311 < b

   1. Initial program 32.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

4. Final simplification 70.2%

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

Alternative 8: 43.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1900:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1900.0) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1900.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1900.0d0) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1900.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1900.0:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1900.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1900.0)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1900.0], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1900

   1. Initial program 74.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if 1900 < b

   1. Initial program 12.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in b around -inf

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

      1. lower-/.f64 30.3

         \[\leadsto \color{blue}{\frac{c}{b}} \]

   7. Applied rewrites 30.3%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1900:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 10.8% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 58.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 34.8%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot a}} \]

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 10.3

      \[\leadsto \color{blue}{\frac{c}{b}} \]

7. Applied rewrites 10.3%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Alternative 10: 2.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 58.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 34.8%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot a}} \]

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{a}} \]
6. Step-by-step derivation

   1. lower-/.f64 2.2

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

7. Applied rewrites 2.2%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "quadp (p42, 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 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))