Quadratic roots, narrow range

Percentage Accurate: 55.8% → 91.8%

Time: 23.2s

Alternatives: 10

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}}\\ t_1 := \sqrt{a \cdot c}\\ t_2 := \left(b + 2 \cdot t_1\right) \cdot \left(b + t_1 \cdot -2\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_2}{\left(-b\right) - \sqrt{t_2}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(\frac{-0.25 \cdot {a}^{3}}{\frac{b}{\mathsf{fma}\left(16, t_0, 4 \cdot t_0\right)}} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 4.0) (pow b 6.0)))
        (t_1 (sqrt (* a c)))
        (t_2 (* (+ b (* 2.0 t_1)) (+ b (* t_1 -2.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -3.5)
     (/ (/ (- (* b b) t_2) (- (- b) (sqrt t_2))) (* a 2.0))
     (fma
      -2.0
      (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
      (-
       (-
        (/ (* -0.25 (pow a 3.0)) (/ b (fma 16.0 t_0 (* 4.0 t_0))))
        (* (/ a (pow b 3.0)) (* c c)))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(c, 4.0) / pow(b, 6.0);
	double t_1 = sqrt((a * c));
	double t_2 = (b + (2.0 * t_1)) * (b + (t_1 * -2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -3.5) {
		tmp = (((b * b) - t_2) / (-b - sqrt(t_2))) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), ((((-0.25 * pow(a, 3.0)) / (b / fma(16.0, t_0, (4.0 * t_0)))) - ((a / pow(b, 3.0)) * (c * c))) - (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((c ^ 4.0) / (b ^ 6.0))
	t_1 = sqrt(Float64(a * c))
	t_2 = Float64(Float64(b + Float64(2.0 * t_1)) * Float64(b + Float64(t_1 * -2.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -3.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_2) / Float64(Float64(-b) - sqrt(t_2))) / Float64(a * 2.0));
	else
		tmp = fma(-2.0, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), Float64(Float64(Float64(Float64(-0.25 * (a ^ 3.0)) / Float64(b / fma(16.0, t_0, Float64(4.0 * t_0)))) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(b + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(-0.25 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[(b / N[(16.0 * t$95$0 + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -3.5

   1. Initial program 88.3%

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

      1. add-sqr-sqrt 88.1%

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

      2. difference-of-squares 88.3%

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

      3. associate-*l* 88.3%

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

      4. sqrt-prod 88.3%

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

      5. metadata-eval 88.3%

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

      6. associate-*l* 88.3%

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

      7. sqrt-prod 88.3%

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

      8. metadata-eval 88.3%

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

   3. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. cancel-sign-sub-inv 88.3%

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

      3. metadata-eval 88.3%

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

   5. Simplified 88.3%

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

      1. flip-+ 88.4%

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

      2. add-sqr-sqrt 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. sqr-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.4%

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

   2. Taylor expanded in a around 0 92.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}\right)\right)} \]

   4. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(\frac{-0.25 \cdot {a}^{3}}{\frac{b}{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) - \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}{\left(-b\right) - \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(\frac{-0.25 \cdot {a}^{3}}{\frac{b}{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}} - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \left(b + 2 \cdot t_0\right) \cdot \left(b + t_0 \cdot -2\right)\\ t_2 := {\left(a \cdot \left(c \cdot -2\right)\right)}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_1}{\left(-b\right) - \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{t_2}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot t_2\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c)))
        (t_1 (* (+ b (* 2.0 t_0)) (+ b (* t_0 -2.0))))
        (t_2 (pow (* a (* c -2.0)) 2.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -3.5)
     (/ (/ (- (* b b) t_1) (- (- b) (sqrt t_1))) (* a 2.0))
     (/
      (fma
       -2.0
       (/ a (/ b c))
       (*
        -0.5
        (+
         (+
          (/ t_2 (pow b 3.0))
          (/ (fma 2.0 (* a (* c t_2)) (* a (* c 0.0))) (pow b 5.0)))
         (* 20.0 (/ (pow (* a c) 4.0) (pow b 7.0))))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0));
	double t_2 = pow((a * (c * -2.0)), 2.0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -3.5) {
		tmp = (((b * b) - t_1) / (-b - sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (a / (b / c)), (-0.5 * (((t_2 / pow(b, 3.0)) + (fma(2.0, (a * (c * t_2)), (a * (c * 0.0))) / pow(b, 5.0))) + (20.0 * (pow((a * c), 4.0) / pow(b, 7.0)))))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = Float64(Float64(b + Float64(2.0 * t_0)) * Float64(b + Float64(t_0 * -2.0)))
	t_2 = Float64(a * Float64(c * -2.0)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -3.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_1) / Float64(Float64(-b) - sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(a / Float64(b / c)), Float64(-0.5 * Float64(Float64(Float64(t_2 / (b ^ 3.0)) + Float64(fma(2.0, Float64(a * Float64(c * t_2)), Float64(a * Float64(c * 0.0))) / (b ^ 5.0))) + Float64(20.0 * Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)))))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(a * N[(c * -2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(a * N[(c * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(20.0 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -3.5

   1. Initial program 88.3%

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

      1. add-sqr-sqrt 88.1%

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

      2. difference-of-squares 88.3%

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

      3. associate-*l* 88.3%

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

      4. sqrt-prod 88.3%

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

      5. metadata-eval 88.3%

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

      6. associate-*l* 88.3%

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

      7. sqrt-prod 88.3%

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

      8. metadata-eval 88.3%

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

   3. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. cancel-sign-sub-inv 88.3%

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

      3. metadata-eval 88.3%

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

   5. Simplified 88.3%

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

      1. flip-+ 88.4%

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

      2. add-sqr-sqrt 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. sqr-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.4%

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

      1. flip3-- 51.3%

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

      2. clear-num 51.2%

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

      3. pow2 51.2%

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

      4. pow2 51.2%

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

      5. pow-prod-up 50.8%

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

      6. metadata-eval 50.8%

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

      7. distribute-rgt-out 50.8%

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

      8. associate-*l* 50.8%

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

      9. +-commutative 50.8%

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

      10. fma-def 50.8%

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

      11. associate-*l* 50.8%

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

   3. Applied egg-rr 51.1%

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

   4. Taylor expanded in b around inf 92.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + \left(-0.5 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{{b}^{7}} + \left(-0.5 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{{b}^{3}} + -0.5 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{5}}\right)\right)}}{2 \cdot a} \]

   6. Simplified 92.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\left({\left(0 + {\left(a \cdot \left(c \cdot -2\right)\right)}^{2} \cdot -0.5\right)}^{2} + 0\right) + \mathsf{fma}\left(a \cdot 2, c \cdot \mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

   7. Taylor expanded in a around 0 92.5%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{\frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{2 \cdot a} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{a}^{4} \cdot \color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

      2. associate-*l* 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

      3. *-commutative 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right)} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

      4. associate-*l* 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{{c}^{4} \cdot \left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

      5. distribute-rgt-out 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{c}^{4} \cdot \color{blue}{\left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

      6. associate-/l* 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{\frac{{c}^{4}}{\frac{{b}^{7}}{4 \cdot {a}^{4} + 16 \cdot {a}^{4}}}}\right)\right)}{2 \cdot a} \]

      7. distribute-rgt-out 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{c}^{4}}{\frac{{b}^{7}}{\color{blue}{{a}^{4} \cdot \left(4 + 16\right)}}}\right)\right)}{2 \cdot a} \]

      8. metadata-eval 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{4} \cdot \color{blue}{20}}}\right)\right)}{2 \cdot a} \]

   9. Simplified 92.5%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{\frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{4} \cdot 20}}}\right)\right)}{2 \cdot a} \]

   10. Taylor expanded in c around 0 92.5%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{20 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}}}\right)\right)}{2 \cdot a} \]
   11. Step-by-step derivation

       1. metadata-eval 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{{a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       2. pow-sqr 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot {c}^{4}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       3. metadata-eval 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left({a}^{2} \cdot {a}^{2}\right) \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       4. pow-sqr 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left({a}^{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       5. unswap-sqr 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       6. unpow2 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       7. unpow2 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       8. swap-sqr 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       9. unpow2 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       10. unpow2 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       11. swap-sqr 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       12. unpow2 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       13. unpow2 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{{\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       14. pow-sqr 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

       15. metadata-eval 92.5%

           \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{{\left(a \cdot c\right)}^{\color{blue}{4}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   12. Simplified 92.5%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left({\left(a \cdot \left(c \cdot -2\right)\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{20 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}}\right)\right)}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}{\left(-b\right) - \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, -0.5 \cdot \left(\left(\frac{{\left(a \cdot \left(c \cdot -2\right)\right)}^{2}}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot {\left(a \cdot \left(c \cdot -2\right)\right)}^{2}\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + 20 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}}\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternative 3: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \left(b + 2 \cdot t_0\right) \cdot \left(b + t_0 \cdot -2\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t_1}{\left(-b\right) - \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot -10\right)\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))) (t_1 (* (+ b (* 2.0 t_0)) (+ b (* t_0 -2.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -3.5)
     (/ (/ (- (* b b) t_1) (- (- b) (sqrt t_1))) (* a 2.0))
     (/
      (fma
       -4.0
       (/ (pow (* a c) 3.0) (pow b 5.0))
       (fma
        -2.0
        (+ (/ a (/ b c)) (/ a (/ (/ (pow b 3.0) (* c c)) a)))
        (* (/ (pow (* a c) 4.0) (pow b 7.0)) -10.0)))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -3.5) {
		tmp = (((b * b) - t_1) / (-b - sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = fma(-4.0, (pow((a * c), 3.0) / pow(b, 5.0)), fma(-2.0, ((a / (b / c)) + (a / ((pow(b, 3.0) / (c * c)) / a))), ((pow((a * c), 4.0) / pow(b, 7.0)) * -10.0))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = Float64(Float64(b + Float64(2.0 * t_0)) * Float64(b + Float64(t_0 * -2.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -3.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_1) / Float64(Float64(-b) - sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-4.0, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), fma(-2.0, Float64(Float64(a / Float64(b / c)) + Float64(a / Float64(Float64((b ^ 3.0) / Float64(c * c)) / a))), Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * -10.0))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -3.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -3.5

   1. Initial program 88.3%

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

      1. add-sqr-sqrt 88.1%

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

      2. difference-of-squares 88.3%

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

      3. associate-*l* 88.3%

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

      4. sqrt-prod 88.3%

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

      5. metadata-eval 88.3%

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

      6. associate-*l* 88.3%

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

      7. sqrt-prod 88.3%

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

      8. metadata-eval 88.3%

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

   3. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. cancel-sign-sub-inv 88.3%

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

      3. metadata-eval 88.3%

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

   5. Simplified 88.3%

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

      1. flip-+ 88.4%

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

      2. add-sqr-sqrt 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. sqr-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. *-commutative 89.2%

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

      4. *-commutative 89.2%

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

      5. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -3.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.4%

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

   2. Taylor expanded in b around inf 92.4%

      \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

   4. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \frac{-0.5 \cdot \mathsf{fma}\left(16, {a}^{4} \cdot {c}^{4}, 4 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

   5. Taylor expanded in a around 0 92.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{-0.5 \cdot \frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{2 \cdot a} \]

   7. Simplified 92.4%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot -10}\right)\right)}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -3.5:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}{\left(-b\right) - \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot -10\right)\right)}{a \cdot 2}\\ \end{array} \]

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot c}\\ t_1 := \left(b + 2 \cdot t_0\right) \cdot \left(b + t_0 \cdot -2\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\frac{b \cdot b - t_1}{\left(-b\right) - \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a c))) (t_1 (* (+ b (* 2.0 t_0)) (+ b (* t_0 -2.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.012)
     (/ (/ (- (* b b) t_1) (- (- b) (sqrt t_1))) (* a 2.0))
     (-
      (- (/ (* -2.0 (* a a)) (/ (pow b 5.0) (pow c 3.0))) (/ c b))
      (* (/ a (pow b 3.0)) (* c c))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * c));
	double t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (((b * b) - t_1) / (-b - sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (a * a)) / (pow(b, 5.0) / pow(c, 3.0))) - (c / b)) - ((a / pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a * c))
    t_1 = (b + (2.0d0 * t_0)) * (b + (t_0 * (-2.0d0)))
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-0.012d0)) then
        tmp = (((b * b) - t_1) / (-b - sqrt(t_1))) / (a * 2.0d0)
    else
        tmp = ((((-2.0d0) * (a * a)) / ((b ** 5.0d0) / (c ** 3.0d0))) - (c / b)) - ((a / (b ** 3.0d0)) * (c * c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * c));
	double t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0));
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (((b * b) - t_1) / (-b - Math.sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (a * a)) / (Math.pow(b, 5.0) / Math.pow(c, 3.0))) - (c / b)) - ((a / Math.pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * c))
	t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0))
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012:
		tmp = (((b * b) - t_1) / (-b - math.sqrt(t_1))) / (a * 2.0)
	else:
		tmp = (((-2.0 * (a * a)) / (math.pow(b, 5.0) / math.pow(c, 3.0))) - (c / b)) - ((a / math.pow(b, 3.0)) * (c * c))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * c))
	t_1 = Float64(Float64(b + Float64(2.0 * t_0)) * Float64(b + Float64(t_0 * -2.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.012)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_1) / Float64(Float64(-b) - sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) / Float64((b ^ 5.0) / (c ^ 3.0))) - Float64(c / b)) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * c));
	t_1 = (b + (2.0 * t_0)) * (b + (t_0 * -2.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012)
		tmp = (((b * b) - t_1) / (-b - sqrt(t_1))) / (a * 2.0);
	else
		tmp = (((-2.0 * (a * a)) / ((b ^ 5.0) / (c ^ 3.0))) - (c / b)) - ((a / (b ^ 3.0)) * (c * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b + N[(t$95$0 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.012], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.012

   1. Initial program 81.3%

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

      1. add-sqr-sqrt 81.3%

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

      2. difference-of-squares 81.4%

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

      3. associate-*l* 81.4%

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

      4. sqrt-prod 81.4%

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

      5. metadata-eval 81.4%

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

      6. associate-*l* 81.4%

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

      7. sqrt-prod 81.4%

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

      8. metadata-eval 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. *-commutative 81.4%

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

      2. cancel-sign-sub-inv 81.4%

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

      3. metadata-eval 81.4%

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

   5. Simplified 81.4%

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

      1. flip-+ 81.3%

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

      2. add-sqr-sqrt 82.7%

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

      3. *-commutative 82.7%

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

      4. *-commutative 82.7%

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

      5. *-commutative 82.7%

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

   7. Applied egg-rr 82.7%

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

      1. sqr-neg 82.7%

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

      2. *-commutative 82.7%

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

      3. *-commutative 82.7%

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

      4. *-commutative 82.7%

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

      5. *-commutative 82.7%

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

   9. Simplified 82.7%

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

   if -0.012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.5%

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

   2. Taylor expanded in b around inf 93.6%

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

      1. associate-+r+ 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

      4. mul-1-neg 93.6%

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

      5. unsub-neg 93.6%

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

      6. associate-/l* 93.6%

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

      7. associate-*r/ 93.6%

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

      8. unpow2 93.6%

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

      9. associate-/l* 93.6%

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

      10. associate-/r/ 93.6%

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

      11. unpow2 93.6%

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

   4. Simplified 93.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}{\left(-b\right) - \sqrt{\left(b + 2 \cdot \sqrt{a \cdot c}\right) \cdot \left(b + \sqrt{a \cdot c} \cdot -2\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 5: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.012)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (* -2.0 (* a a)) (/ (pow b 5.0) (pow c 3.0))) (/ c b))
    (* (/ a (pow b 3.0)) (* c c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (a * a)) / (pow(b, 5.0) / pow(c, 3.0))) - (c / b)) - ((a / pow(b, 3.0)) * (c * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.012)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) / Float64((b ^ 5.0) / (c ^ 3.0))) - Float64(c / b)) - Float64(Float64(a / (b ^ 3.0)) * Float64(c * c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.012], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.012

   1. Initial program 81.3%

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

   3. Simplified 81.5%

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

   if -0.012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.5%

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

   2. Taylor expanded in b around inf 93.6%

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

      1. associate-+r+ 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

      4. mul-1-neg 93.6%

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

      5. unsub-neg 93.6%

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

      6. associate-/l* 93.6%

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

      7. associate-*r/ 93.6%

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

      8. unpow2 93.6%

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

      9. associate-/l* 93.6%

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

      10. associate-/r/ 93.6%

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

      11. unpow2 93.6%

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

   4. Simplified 93.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left(a \cdot a\right)}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 6: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.012)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (* (* c c) (/ (- a) (pow b 3.0))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * c) * (-a / pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.012)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * c) * Float64(Float64(-a) / (b ^ 3.0))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.012], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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 4 a) c)))) (*.f64 2 a)) < -0.012

   1. Initial program 81.3%

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

   3. Simplified 81.5%

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

   if -0.012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.5%

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

   2. Taylor expanded in b around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. mul-1-neg 89.6%

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

      4. distribute-neg-frac 89.6%

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

      5. associate-/l* 89.6%

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

      6. associate-/r/ 89.6%

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

      7. unpow2 89.6%

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

   4. Simplified 89.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.012)
   (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
   (- (* (* c c) (/ (- a) (pow b 3.0))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * c) * (-a / pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-0.012d0)) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = ((c * c) * (-a / (b ** 3.0d0))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * c) * (-a / Math.pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = ((c * c) * (-a / math.pow(b, 3.0))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.012)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * c) * Float64(Float64(-a) / (b ^ 3.0))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.012)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = ((c * c) * (-a / (b ^ 3.0))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.012], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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 4 a) c)))) (*.f64 2 a)) < -0.012

   1. Initial program 81.3%

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

   3. Simplified 81.5%

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

      1. *-commutative 81.5%

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

      2. metadata-eval 81.5%

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

      3. distribute-lft-neg-in 81.5%

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

      4. distribute-rgt-neg-in 81.5%

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

      5. *-commutative 81.5%

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

      6. fma-neg 81.3%

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

      7. associate-*l* 81.3%

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

   5. Applied egg-rr 81.3%

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

   if -0.012 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.5%

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

   2. Taylor expanded in b around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. mul-1-neg 89.6%

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

      4. distribute-neg-frac 89.6%

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

      5. associate-/l* 89.6%

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

      6. associate-/r/ 89.6%

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

      7. unpow2 89.6%

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

   4. Simplified 89.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.012:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 8: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (* (* c c) (/ (- a) (pow b 3.0))) (/ c b)))

C

double code(double a, double b, double c) {
	return ((c * c) * (-a / pow(b, 3.0))) - (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 * c) * (-a / (b ** 3.0d0))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return ((c * c) * (-a / Math.pow(b, 3.0))) - (c / b);
}

Python

def code(a, b, c):
	return ((c * c) * (-a / math.pow(b, 3.0))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * c) * Float64(Float64(-a) / (b ^ 3.0))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * c) * (-a / (b ^ 3.0))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * c), $MachinePrecision] * N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

2. Taylor expanded in b around inf 80.1%

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

   1. mul-1-neg 80.1%

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

   2. unsub-neg 80.1%

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

   3. mul-1-neg 80.1%

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

   4. distribute-neg-frac 80.1%

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

   5. associate-/l* 80.1%

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

   6. associate-/r/ 80.1%

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

   7. unpow2 80.1%

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

4. Simplified 80.1%

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

5. Final simplification 80.1%

   \[\leadsto \left(c \cdot c\right) \cdot \frac{-a}{{b}^{3}} - \frac{c}{b} \]

Alternative 9: 64.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

2. Taylor expanded in b around inf 63.6%

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

   1. mul-1-neg 63.6%

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

   2. distribute-neg-frac 63.6%

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

4. Simplified 63.6%

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

5. Final simplification 63.6%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-sqr-sqrt 56.1%

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

   2. difference-of-squares 56.2%

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

   3. associate-*l* 56.2%

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

   4. sqrt-prod 56.2%

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

   5. metadata-eval 56.2%

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

   6. associate-*l* 56.2%

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

   7. sqrt-prod 56.2%

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

   8. metadata-eval 56.2%

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

3. Applied egg-rr 56.2%

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

   1. *-commutative 56.2%

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

   2. cancel-sign-sub-inv 56.2%

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

   3. metadata-eval 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

   2. distribute-rgt-out 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

   3. *-commutative 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{\color{blue}{c \cdot a}} \cdot \left(-2 + 2\right)\right)}{a} \]

   4. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{c \cdot a} \cdot \color{blue}{0}\right)}{a} \]

   5. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   6. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023287 
(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)))