Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.2%

Time: 10.1s

Alternatives: 10

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\frac{\sqrt{\mathsf{fma}\left(-4, c, \frac{b}{a} \cdot b\right) \cdot a} + b}{\left(a \cdot 4\right) \cdot c}}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ -1.0 (/ (+ (sqrt (* (fma -4.0 c (* (/ b a) b)) a)) b) (* (* a 4.0) c)))
  (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-1.0 / ((sqrt((fma(-4.0, c, ((b / a) * b)) * a)) + b) / ((a * 4.0) * c))) / (2.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(-1.0 / Float64(Float64(sqrt(Float64(fma(-4.0, c, Float64(Float64(b / a) * b)) * a)) + b) / Float64(Float64(a * 4.0) * c))) / Float64(2.0 * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.0 / N[(N[(N[Sqrt[N[(N[(-4.0 * c + N[(N[(b / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 56.4

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

5. Applied rewrites 56.4%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 57.2%

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

8. Taylor expanded in c around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{-1}{\frac{\sqrt{\mathsf{fma}\left(-4, c, \frac{b}{a} \cdot b\right) \cdot a} + b}{\left(a \cdot 4\right) \cdot c}}}{2 \cdot a} \]
12. Add Preprocessing

Alternative 2: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 19.5)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (/ 1.0 (/ (fma -0.5 (/ b a) (* (/ c b) 0.5)) c)) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 19.5) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = (1.0 / (fma(-0.5, (b / a), ((c / b) * 0.5)) / c)) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.5, Float64(b / a), Float64(Float64(c / b) * 0.5)) / c)) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 19.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.5 * N[(b / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 79.3%

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 81.0%

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

   if 19.5 < b

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{2 \cdot a} \]

   10. Applied rewrites 87.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 19.5)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/ (/ 1.0 (/ (fma -0.5 (/ b a) (* (/ c b) 0.5)) c)) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 19.5) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = (1.0 / (fma(-0.5, (b / a), ((c / b) * 0.5)) / c)) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.5, Float64(b / a), Float64(Float64(c / b) * 0.5)) / c)) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 19.5], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.5 * N[(b / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 79.3%

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. *-inverses N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. associate-/l/ N/A

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

   6. Applied rewrites 81.0%

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

   if 19.5 < b

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{2 \cdot a} \]

   10. Applied rewrites 87.9%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.2)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (/ 1.0 (/ (fma -0.5 (/ b a) (* (/ c b) 0.5)) c)) (* 2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.2) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (fma(-0.5, (b / a), ((c / b) * 0.5)) / c)) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.5, Float64(b / a), Float64(Float64(c / b) * 0.5)) / c)) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.2], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.5 * N[(b / a), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14.199999999999999

   1. Initial program 79.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

   if 14.199999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 49.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.5

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{2 \cdot a} \]

   10. Applied rewrites 87.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \frac{b}{a}, \frac{c}{b} \cdot 0.5\right)}{c}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{0.5}{b \cdot b} - \frac{0.5}{c \cdot a}\right) \cdot b}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.2)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (/ 1.0 (* (- (/ 0.5 (* b b)) (/ 0.5 (* c a))) b)) (* 2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.2) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (((0.5 / (b * b)) - (0.5 / (c * a))) * b)) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(0.5 / Float64(b * b)) - Float64(0.5 / Float64(c * a))) * b)) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.2], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 14.199999999999999

   1. Initial program 79.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 80.1

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

   4. Applied rewrites 80.1%

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

   if 14.199999999999999 < b

   1. Initial program 48.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7

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

   5. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 49.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 87.4

          \[\leadsto \frac{\frac{1}{b \cdot \left(\frac{0.5}{b \cdot b} - \frac{0.5}{\color{blue}{a \cdot c}}\right)}}{2 \cdot a} \]

   10. Applied rewrites 87.4%

       \[\leadsto \frac{\frac{1}{\color{blue}{b \cdot \left(\frac{0.5}{b \cdot b} - \frac{0.5}{a \cdot c}\right)}}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{0.5}{b \cdot b} - \frac{0.5}{c \cdot a}\right) \cdot b}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 19.5)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (fma -1.0 c (/ (* (* c c) a) (* (- b) b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma(-1.0, c, (((c * c) * a) / (-b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(-1.0, c, Float64(Float64(Float64(c * c) * a) / Float64(Float64(-b) * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 19.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * c + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 79.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 79.6

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

   4. Applied rewrites 79.6%

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

   if 19.5 < b

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.7

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

   10. Applied rewrites 87.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, c, \frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 19.5)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (fma -1.0 c (/ (* (* c c) a) (* (- b) b))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma(-1.0, c, (((c * c) * a) / (-b * b))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(-1.0, c, Float64(Float64(Float64(c * c) * a) / Float64(Float64(-b) * b))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 19.5], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * c + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 79.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 79.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 79.4

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

   4. Applied rewrites 79.5%

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

   if 19.5 < b

   1. Initial program 48.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.7

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

   10. Applied rewrites 87.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, c, \frac{\left(-a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma -1.0 c (/ (* (* c c) a) (* (- b) b))) b))

C

double code(double a, double b, double c) {
	return fma(-1.0, c, (((c * c) * a) / (-b * b))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-1.0, c, Float64(Float64(Float64(c * c) * a) / Float64(Float64(-b) * b))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.0 * c + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 56.4

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

5. Applied rewrites 56.4%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

7. Applied rewrites 57.2%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 80.4

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

10. Applied rewrites 80.4%

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

11. Final simplification 80.4%

    \[\leadsto \frac{\mathsf{fma}\left(-1, c, \frac{\left(c \cdot c\right) \cdot a}{\left(-b\right) \cdot b}\right)}{b} \]
12. Add Preprocessing

Alternative 9: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* (fma (- a) (/ c (* b b)) -1.0) c) b))

C

double code(double a, double b, double c) {
	return (fma(-a, (c / (b * b)), -1.0) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-a), Float64(c / Float64(b * b)), -1.0) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-a) * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot -2\right) \cdot a, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.25}{a}, \left({c}^{4} \cdot {a}^{4}\right) \cdot \frac{20}{{b}^{6}}, -\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)\right)\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 80.3%

   \[\leadsto \frac{\mathsf{fma}\left(-a, \frac{c}{b \cdot b}, -1\right) \cdot c}{b} \]
9. Add Preprocessing

Alternative 10: 64.0% accurate, 3.6× 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(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 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 63.4

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

5. Applied rewrites 63.4%

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

Reproduce

herbie shell --seed 2024263 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))