Quadratic roots, narrow range

Percentage Accurate: 56.0% → 91.1%

Time: 11.7s

Alternatives: 13

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: 56.0% 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: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (fma
      (fma
       (fma
        (* -0.25 a)
        (* (/ (pow c 4.0) (pow b 6.0)) (/ 20.0 b))
        (/ (* (pow c 3.0) -2.0) (pow b 5.0)))
       a
       (/ (* (- c) c) (pow b 3.0)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = fma(fma(fma((-0.25 * a), ((pow(c, 4.0) / pow(b, 6.0)) * (20.0 / b)), ((pow(c, 3.0) * -2.0) / pow(b, 5.0))), a, ((-c * c) / pow(b, 3.0))), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = fma(fma(fma(Float64(-0.25 * a), Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * Float64(20.0 / b)), Float64(Float64((c ^ 3.0) * -2.0) / (b ^ 5.0))), a, Float64(Float64(Float64(-c) * c) / (b ^ 3.0))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.25 * a), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[((-c) * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-b, b, \left(c \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (fma
      (/
       (fma
        (* -5.0 (* a a))
        (pow c 4.0)
        (* (* (fma (- b) b (* (* c a) -2.0)) (* c c)) (* b b)))
       (pow b 7.0))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = fma((fma((-5.0 * (a * a)), pow(c, 4.0), ((fma(-b, b, ((c * a) * -2.0)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(fma(Float64(-5.0 * Float64(a * a)), (c ^ 4.0), Float64(Float64(fma(Float64(-b), b, Float64(Float64(c * a) * -2.0)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-5.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[((-b) * b + N[(N[(c * a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-2 \cdot a, {c}^{3}, \left(\left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.0%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-b, b, \left(c \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-b, b, \left(c \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(b, b, a \cdot c\right), b \cdot b, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right), b \cdot b, {\left(a \cdot c\right)}^{3} \cdot -5\right)}{{b}^{7}} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (*
      (/
       (fma
        (fma (- (fma b b (* a c))) (* b b) (* (* (* a a) -2.0) (* c c)))
        (* b b)
        (* (pow (* a c) 3.0) -5.0))
       (pow b 7.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (fma(fma(-fma(b, b, (a * c)), (b * b), (((a * a) * -2.0) * (c * c))), (b * b), (pow((a * c), 3.0) * -5.0)) / pow(b, 7.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(fma(Float64(-fma(b, b, Float64(a * c))), Float64(b * b), Float64(Float64(Float64(a * a) * -2.0) * Float64(c * c))), Float64(b * b), Float64((Float64(a * c) ^ 3.0) * -5.0)) / (b ^ 7.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[((-N[(b * b + N[(a * c), $MachinePrecision]), $MachinePrecision]) * N[(b * b), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right) \cdot c} \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot \frac{c}{a}, \frac{{a}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{-5 \cdot \left({a}^{3} \cdot {c}^{3}\right) + {b}^{2} \cdot \left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(-1 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(b, b, a \cdot c\right), b \cdot b, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right), b \cdot b, {\left(a \cdot c\right)}^{3} \cdot -5\right)}{{b}^{7}} \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(b, b, a \cdot c\right), b \cdot b, \left(\left(a \cdot a\right) \cdot -2\right) \cdot \left(c \cdot c\right)\right), b \cdot b, {\left(a \cdot c\right)}^{3} \cdot -5\right)}{{b}^{7}} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c, \left(-a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (*
      (-
       (* (/ (fma (* -2.0 (* a a)) c (* (- a) (* b b))) (pow b 5.0)) c)
       (pow b -1.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (((fma((-2.0 * (a * a)), c, (-a * (b * b))) / pow(b, 5.0)) * c) - pow(b, -1.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(-2.0 * Float64(a * a)), c, Float64(Float64(-a) * Float64(b * b))) / (b ^ 5.0)) * c) - (b ^ -1.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * c + N[((-a) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \cdot c} \]

   8. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{5}}, -2, \frac{-a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

   9. Taylor expanded in b around 0

      \[\leadsto \left(\frac{-2 \cdot \left({a}^{2} \cdot c\right) + -1 \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}} \cdot c - \frac{1}{b}\right) \cdot c \]
   10. Step-by-step derivation

   11. Applied rewrites 93.4%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c, \left(-a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c - \frac{1}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c, \left(-a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \left(c \cdot c\right) \cdot \frac{c}{{b}^{4}}, -\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/
      (fma
       (* (* -2.0 a) a)
       (* (* c c) (/ c (pow b 4.0)))
       (- (fma (/ (* c c) b) (/ a b) c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = fma(((-2.0 * a) * a), ((c * c) * (c / pow(b, 4.0))), -fma(((c * c) / b), (a / b), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(-2.0 * a) * a), Float64(Float64(c * c) * Float64(c / (b ^ 4.0))), Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision])), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \frac{{c}^{3}}{{b}^{4}}, -\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)\right)}{b}} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \left(c \cdot c\right) \cdot \frac{c}{{b}^{4}}, -\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-2 \cdot a\right) \cdot a, \left(c \cdot c\right) \cdot \frac{c}{{b}^{4}}, -\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c \cdot c, \left(-\mathsf{fma}\left(b, b, c \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (*
      (/
       (fma (* -2.0 (* a a)) (* c c) (* (- (fma b b (* c a))) (* b b)))
       (pow b 5.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (fma((-2.0 * (a * a)), (c * c), (-fma(b, b, (c * a)) * (b * b))) / pow(b, 5.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(-2.0 * Float64(a * a)), Float64(c * c), Float64(Float64(-fma(b, b, Float64(c * a))) * Float64(b * b))) / (b ^ 5.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.232], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[((-N[(b * b + N[(c * a), $MachinePrecision]), $MachinePrecision]) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 83.5%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \cdot c} \]

   8. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{5}}, -2, \frac{-a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{-2 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(-1 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)}{{b}^{5}} \cdot c \]
   10. Step-by-step derivation

   11. Applied rewrites 93.1%

       \[\leadsto \frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c \cdot c, \left(-\mathsf{fma}\left(b, b, c \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c \cdot c, \left(-\mathsf{fma}\left(b, b, c \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - t\_0\right)}{b + \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.005)
     (/ (/ (- (- (* b b) t_0)) (+ b (sqrt t_0))) (* 2.0 a))
     (/ (- (fma (/ (* c c) b) (/ a b) c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.005) {
		tmp = (-((b * b) - t_0) / (b + sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.005)
		tmp = Float64(Float64(Float64(-Float64(Float64(b * b) - t_0)) / Float64(b + sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.005], N[(N[((-N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]) / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0050000000000000001

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 79.7%

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

   if -0.0050000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 91.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{\frac{-\left(b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 c (/ (* b b) a)) a)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.005)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (/ (- (fma (/ (* c c) b) (/ a b) c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, c, ((b * b) / a)) * a;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.005) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.005)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[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], -0.005], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0050000000000000001

   1. Initial program 78.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 79.7%

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

   if -0.0050000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 44.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 91.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-4, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.232:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.232)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (- (fma (/ (* c c) b) (/ a b) c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.232) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.232)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.232], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.232000000000000012

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 82.9

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

   4. Applied rewrites 82.9%

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

   if -0.232000000000000012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 88.2%

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

Alternative 10: 81.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 82.1%

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

Alternative 11: 80.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 91.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot \frac{c}{a}, \frac{{a}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}\right), c, \frac{-a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

6. Taylor expanded in b around -inf

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

8. Applied rewrites 82.0%

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

Alternative 12: 63.9% 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 55.6%

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

3. Taylor expanded in a around 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. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 64.3

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

5. Applied rewrites 64.3%

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

Alternative 13: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. fp-cancel-sub-sign-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. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 55.5

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

5. Applied rewrites 55.5%

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

7. Applied rewrites 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. lower-+.f64 N/A

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

9. Applied rewrites 55.2%

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

10. Taylor expanded in a around 0

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

    1. div-add N/A

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

    2. associate-*r/ N/A

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

    3. associate-*r/ N/A

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

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

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

    5. metadata-eval N/A

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

    6. +-inverses 3.2

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 3.2%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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)))