quadp (p42, positive)

Percentage Accurate: 52.3% → 85.6%

Time: 10.6s

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: 52.3% 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: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e+152)
   (/ (- b) a)
   (if (<= b 2.4e-45)
     (/ (- (sqrt (fma (* -4.0 c) a (* b b))) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e+152) {
		tmp = -b / a;
	} else if (b <= 2.4e-45) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e+152)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.4e-45)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e+152], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.4e-45], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4999999999999997e152

   1. Initial program 29.0%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if -6.4999999999999997e152 < b < 2.3999999999999999e-45

   1. Initial program 83.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 83.9

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 83.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   8. Applied rewrites 83.9%

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

   if 2.3999999999999999e-45 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e+75)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 2.4e-45)
     (* (- (sqrt (fma -4.0 (* c a) (* b b))) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e+75) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 2.4e-45) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e+75)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 2.4e-45)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e+75], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-45], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999985e75

   1. Initial program 54.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -2.59999999999999985e75 < b < 2.3999999999999999e-45

   1. Initial program 80.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if 2.3999999999999999e-45 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b} \cdot a, -2, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-58)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 1.25e-64)
     (/ (- (sqrt (* (* -4.0 c) a)) b) (* 2.0 a))
     (/ (* 2.0 c) (- (- b) (fma (* (/ c b) a) -2.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-58) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 1.25e-64) {
		tmp = (sqrt(((-4.0 * c) * a)) - b) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b - fma(((c / b) * a), -2.0, b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-58)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 1.25e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - fma(Float64(Float64(c / b) * a), -2.0, b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-58], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-64], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000009e-58

   1. Initial program 68.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -3.60000000000000009e-58 < b < 1.25000000000000008e-64

   1. Initial program 75.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 68.9

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

   9. Applied rewrites 68.9%

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

   if 1.25000000000000008e-64 < b

   1. Initial program 19.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 10.7%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 30.9

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

   10. Applied rewrites 30.9%

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

   11. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 85.4

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

   13. Applied rewrites 85.4%

       \[\leadsto \frac{2 \cdot c}{\left(-b\right) - \color{blue}{\mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b} \cdot a, -2, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-58)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 1.25e-64)
     (/ (- (sqrt (* (* -4.0 c) a)) b) (* 2.0 a))
     (/ (* 2.0 c) (* (fma a (/ c b) (- b)) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-58) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 1.25e-64) {
		tmp = (sqrt(((-4.0 * c) * a)) - b) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (fma(a, (c / b), -b) * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-58)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 1.25e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(fma(a, Float64(c / b), Float64(-b)) * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-58], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-64], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000009e-58

   1. Initial program 68.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -3.60000000000000009e-58 < b < 1.25000000000000008e-64

   1. Initial program 75.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 68.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 68.9

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

   9. Applied rewrites 68.9%

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

   if 1.25000000000000008e-64 < b

   1. Initial program 19.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 11.0

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

   5. Applied rewrites 11.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   7. Applied rewrites 10.7%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 30.9

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

   10. Applied rewrites 30.9%

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

   11. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

       2. lower-*.f64 N/A

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

       3. sub-neg N/A

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

       4. associate-/l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-neg.f64 85.4

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

   13. Applied rewrites 85.4%

       \[\leadsto \frac{2 \cdot c}{\color{blue}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(a, \frac{c}{b}, -b\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-58)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 2.4e-45)
     (/ (- (sqrt (* (* -4.0 c) a)) b) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-58) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 2.4e-45) {
		tmp = (sqrt(((-4.0 * c) * a)) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-58)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 2.4e-45)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-58], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-45], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000009e-58

   1. Initial program 68.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -3.60000000000000009e-58 < b < 2.3999999999999999e-45

   1. Initial program 74.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 67.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 67.9

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

   9. Applied rewrites 67.9%

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

   if 2.3999999999999999e-45 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-58)
   (fma (/ c (* b b)) b (/ (- b) a))
   (if (<= b 2.4e-45)
     (* (- (sqrt (* (* -4.0 c) a)) b) (/ 0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-58) {
		tmp = fma((c / (b * b)), b, (-b / a));
	} else if (b <= 2.4e-45) {
		tmp = (sqrt(((-4.0 * c) * a)) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-58)
		tmp = fma(Float64(c / Float64(b * b)), b, Float64(Float64(-b) / a));
	elseif (b <= 2.4e-45)
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-58], N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[((-b) / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-45], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.60000000000000009e-58

   1. Initial program 68.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if -3.60000000000000009e-58 < b < 2.3999999999999999e-45

   1. Initial program 74.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.9

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

   5. Applied rewrites 67.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 67.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 67.7%

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

   if 2.3999999999999999e-45 < b

   1. Initial program 17.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b \cdot b}, b, \frac{-b}{a}\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot c\right) \cdot a} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.8e-299) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-299) {
		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 <= 3.8d-299) 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 <= 3.8e-299) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.8e-299:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.8e-299)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.8e-299)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.8e-299], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.8000000000000003e-299

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 3.8000000000000003e-299 < b

   1. Initial program 34.3%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 65.0

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

   5. Applied rewrites 65.0%

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

Alternative 8: 43.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 8.5e-19) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.5e-19) {
		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 <= 8.5d-19) 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 <= 8.5e-19) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.5e-19:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.5e-19)
		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 <= 8.5e-19)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5e-19], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.50000000000000003e-19

   1. Initial program 70.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if 8.50000000000000003e-19 < b

   1. Initial program 15.4%

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

   4. Applied rewrites 14.2%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 88.6

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

   7. Applied rewrites 88.6%

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

   9. Applied rewrites 29.5%

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

Alternative 9: 11.5% 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 53.3%

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

4. Applied rewrites 48.2%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 33.2

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

7. Applied rewrites 33.2%

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

9. Applied rewrites 11.5%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
10. 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 53.3%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 26.8%

   \[\leadsto \frac{\frac{0 - b \cdot b}{0 + b}}{a} \]

9. Applied rewrites 2.7%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
10. 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 2024235 
(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)))