Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.7%

Time: 17.9s

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: 55.5% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -13:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (fma (* a c) -8.0 (pow b 2.0)) (* 4.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -13.0)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 2.0))
     (+
      (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (-
       (-
        (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
        (/ (* a (pow c 2.0)) (pow b 3.0)))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -8.0, pow(b, 2.0)) + (4.0 * (a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -13.0) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(Float64(a * c), -8.0, (b ^ 2.0)) + Float64(4.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -13.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] * -8.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * c), $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], -13.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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)) < -13

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. *-un-lft-identity 86.9%

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

      2. prod-diff 86.8%

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

      3. add-sqr-sqrt 86.8%

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

      4. sqrt-unprod 86.8%

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

      5. swap-sqr 86.8%

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

      6. metadata-eval 86.8%

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

      7. metadata-eval 86.8%

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

      8. swap-sqr 86.8%

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

      9. *-commutative 86.8%

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

      10. *-commutative 86.8%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in b around 0 86.9%

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

      1. flip-+ 87.1%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow2 87.1%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      3. add-sqr-sqrt 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      4. cancel-sign-sub-inv 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      5. *-commutative 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      6. fma-def 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      7. metadata-eval 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      8. cancel-sign-sub-inv 88.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   9. Applied egg-rr 88.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   if -13 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.9%

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

      1. *-commutative 49.9%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in b around inf 93.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      2. unpow-prod-down 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{2} \cdot {-2}^{2}}}{a \cdot {b}^{7}}\right)\right) \]

      3. pow-prod-down 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      4. pow-pow 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      5. metadata-eval 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      6. metadata-eval 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot \color{blue}{4}}{a \cdot {b}^{7}}\right)\right) \]

   7. Applied egg-rr 93.8%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 4}}{a \cdot {b}^{7}}\right)\right) \]

   8. Taylor expanded in c around 0 93.8%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-in 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left(4 \cdot {a}^{4}\right) \cdot {c}^{4} + \left(16 \cdot {a}^{4}\right) \cdot {c}^{4}}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{4 \cdot \left({a}^{4} \cdot {c}^{4}\right)} + \left(16 \cdot {a}^{4}\right) \cdot {c}^{4}}{a \cdot {b}^{7}}\right)\right) \]

      3. associate-*r* 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}}{a \cdot {b}^{7}}\right)\right) \]

      4. distribute-rgt-out 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

      5. times-frac 93.8%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)}\right)\right) \]

   10. Simplified 93.8%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -13:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (fma (* a c) -8.0 (pow b 2.0)) (* 4.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -6.2)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 2.0))
     (-
      (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -8.0, pow(b, 2.0)) + (4.0 * (a * c));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -6.2) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(Float64(a * c), -8.0, (b ^ 2.0)) + Float64(4.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -6.2)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] * -8.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * c), $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], -6.2], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -6.20000000000000018

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. prod-diff 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. sqrt-unprod 86.4%

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

      5. swap-sqr 86.4%

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

      6. metadata-eval 86.4%

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

      7. metadata-eval 86.4%

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

      8. swap-sqr 86.4%

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

      9. *-commutative 86.4%

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

      10. *-commutative 86.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in b around 0 86.5%

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

      1. flip-+ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow2 86.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      3. add-sqr-sqrt 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      4. cancel-sign-sub-inv 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      5. *-commutative 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      6. fma-def 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      7. metadata-eval 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      8. cancel-sign-sub-inv 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   9. Applied egg-rr 88.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   if -6.20000000000000018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around inf 91.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a -8.0) c (fma b b (* a (* 4.0 c))))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -6.2)
     (/ (/ (- (pow b 2.0) t_0) (- (- b) (sqrt t_0))) (* a 2.0))
     (-
      (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * -8.0), c, fma(b, b, (a * (4.0 * c))));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -6.2) {
		tmp = ((pow(b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * -8.0), c, fma(b, b, Float64(a * Float64(4.0 * c))))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -6.2)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -8.0), $MachinePrecision] * c + N[(b * b + N[(a * N[(4.0 * c), $MachinePrecision]), $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], -6.2], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -6.20000000000000018

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. prod-diff 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. sqrt-unprod 86.4%

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

      5. swap-sqr 86.4%

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

      6. metadata-eval 86.4%

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

      7. metadata-eval 86.4%

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

      8. swap-sqr 86.4%

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

      9. *-commutative 86.4%

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

      10. *-commutative 86.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in b around 0 86.5%

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

      1. flip-+ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow2 86.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      3. add-sqr-sqrt 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      4. cancel-sign-sub-inv 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      5. *-commutative 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      6. fma-def 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      7. metadata-eval 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]

      8. cancel-sign-sub-inv 88.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   9. Applied egg-rr 88.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left(\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   11. Simplified 87.7%

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

   if -6.20000000000000018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around inf 91.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -6.2)
   (/
    1.0
    (/
     a
     (/ (fma -1.0 b (sqrt (fma (* a -8.0) c (fma b b (* a (* 4.0 c)))))) 2.0)))
   (-
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -6.2) {
		tmp = 1.0 / (a / (fma(-1.0, b, sqrt(fma((a * -8.0), c, fma(b, b, (a * (4.0 * c)))))) / 2.0));
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	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)) <= -6.2)
		tmp = Float64(1.0 / Float64(a / Float64(fma(-1.0, b, sqrt(fma(Float64(a * -8.0), c, fma(b, b, Float64(a * Float64(4.0 * c)))))) / 2.0)));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	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], -6.2], N[(1.0 / N[(a / N[(N[(-1.0 * b + N[Sqrt[N[(N[(a * -8.0), $MachinePrecision] * c + N[(b * b + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $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)) < -6.20000000000000018

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. prod-diff 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. sqrt-unprod 86.4%

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

      5. swap-sqr 86.4%

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

      6. metadata-eval 86.4%

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

      7. metadata-eval 86.4%

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

      8. swap-sqr 86.4%

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

      9. *-commutative 86.4%

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

      10. *-commutative 86.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in b around 0 86.5%

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

      1. clear-num 86.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}} \]

      2. inv-pow 86.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}\right)}^{-1}} \]

      3. neg-mul-1 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

      4. fma-def 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}\right)}}\right)}^{-1} \]

      5. cancel-sign-sub-inv 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}\right)}\right)}^{-1} \]

      6. *-commutative 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

      7. fma-def 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

      8. metadata-eval 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

   9. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]

   11. Simplified 86.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)}\right)}{2}}}} \]

   if -6.20000000000000018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in b around inf 91.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ t_1 := \left(a \cdot c\right) \cdot -8 - t_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{t_0 \cdot t_0}{{b}^{3}} + \left(0.0625 \cdot \frac{{t_1}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t_1}{b}\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)) (t_1 (- (* (* a c) -8.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -6.2)
     (/
      1.0
      (/
       a
       (/
        (fma -1.0 b (sqrt (fma (* a -8.0) c (fma b b (* a (* 4.0 c))))))
        2.0)))
     (/
      (+
       (* -0.125 (/ (* t_0 t_0) (pow b 3.0)))
       (+ (* 0.0625 (/ (pow t_1 3.0) (pow b 5.0))) (* 0.5 (/ t_1 b))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double t_1 = ((a * c) * -8.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -6.2) {
		tmp = 1.0 / (a / (fma(-1.0, b, sqrt(fma((a * -8.0), c, fma(b, b, (a * (4.0 * c)))))) / 2.0));
	} else {
		tmp = ((-0.125 * ((t_0 * t_0) / pow(b, 3.0))) + ((0.0625 * (pow(t_1, 3.0) / pow(b, 5.0))) + (0.5 * (t_1 / b)))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	t_1 = Float64(Float64(Float64(a * c) * -8.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -6.2)
		tmp = Float64(1.0 / Float64(a / Float64(fma(-1.0, b, sqrt(fma(Float64(a * -8.0), c, fma(b, b, Float64(a * Float64(4.0 * c)))))) / 2.0)));
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(t_0 * t_0) / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_1 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_1 / b)))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * c), $MachinePrecision] * -8.0), $MachinePrecision] - t$95$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], -6.2], N[(1.0 / N[(a / N[(N[(-1.0 * b + N[Sqrt[N[(N[(a * -8.0), $MachinePrecision] * c + N[(b * b + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$1, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 / b), $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)) < -6.20000000000000018

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. prod-diff 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. sqrt-unprod 86.4%

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

      5. swap-sqr 86.4%

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

      6. metadata-eval 86.4%

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

      7. metadata-eval 86.4%

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

      8. swap-sqr 86.4%

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

      9. *-commutative 86.4%

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

      10. *-commutative 86.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in b around 0 86.5%

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

      1. clear-num 86.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}}} \]

      2. inv-pow 86.5%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\left(-b\right) + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}\right)}^{-1}} \]

      3. neg-mul-1 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{-1 \cdot b} + \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}}\right)}^{-1} \]

      4. fma-def 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}\right)}}\right)}^{-1} \]

      5. cancel-sign-sub-inv 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}\right)}\right)}^{-1} \]

      6. *-commutative 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

      7. fma-def 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

      8. metadata-eval 86.5%

         \[\leadsto {\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)}\right)}\right)}^{-1} \]

   9. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]

   11. Simplified 86.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(c \cdot 4\right)\right)\right)}\right)}{2}}}} \]

   if -6.20000000000000018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

      1. *-un-lft-identity 49.5%

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

      2. prod-diff 49.5%

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

      3. add-sqr-sqrt 49.5%

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

      4. sqrt-unprod 49.5%

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

      5. swap-sqr 49.5%

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

      6. metadata-eval 49.5%

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

      7. metadata-eval 49.5%

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

      8. swap-sqr 49.5%

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

      9. *-commutative 49.5%

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

      10. *-commutative 49.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.9%

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

      13. associate-*r* 1.9%

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

      14. *-commutative 1.9%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in b around inf 91.3%

      \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. unpow2 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      2. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 - -4\right)\right)} \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      3. metadata-eval 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right) \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      4. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 - -4\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      5. metadata-eval 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

   9. Applied egg-rr 91.3%

      \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4}{b}\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ t_1 := \left(a \cdot c\right) \cdot -8 - t_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{t_0 \cdot t_0}{{b}^{3}} + \left(0.0625 \cdot \frac{{t_1}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t_1}{b}\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)) (t_1 (- (* (* a c) -8.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -6.2)
     (/ (- (sqrt (fma (* a -8.0) c (fma b b (* a (* 4.0 c))))) b) (* a 2.0))
     (/
      (+
       (* -0.125 (/ (* t_0 t_0) (pow b 3.0)))
       (+ (* 0.0625 (/ (pow t_1 3.0) (pow b 5.0))) (* 0.5 (/ t_1 b))))
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double t_1 = ((a * c) * -8.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -6.2) {
		tmp = (sqrt(fma((a * -8.0), c, fma(b, b, (a * (4.0 * c))))) - b) / (a * 2.0);
	} else {
		tmp = ((-0.125 * ((t_0 * t_0) / pow(b, 3.0))) + ((0.0625 * (pow(t_1, 3.0) / pow(b, 5.0))) + (0.5 * (t_1 / b)))) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	t_1 = Float64(Float64(Float64(a * c) * -8.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -6.2)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -8.0), c, fma(b, b, Float64(a * Float64(4.0 * c))))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64(Float64(t_0 * t_0) / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_1 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_1 / b)))) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * c), $MachinePrecision] * -8.0), $MachinePrecision] - t$95$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], -6.2], N[(N[(N[Sqrt[N[(N[(a * -8.0), $MachinePrecision] * c + N[(b * b + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$1, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 / b), $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)) < -6.20000000000000018

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. *-un-lft-identity 86.5%

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

      2. prod-diff 86.4%

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

      3. add-sqr-sqrt 86.4%

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

      4. sqrt-unprod 86.4%

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

      5. swap-sqr 86.4%

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

      6. metadata-eval 86.4%

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

      7. metadata-eval 86.4%

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

      8. swap-sqr 86.4%

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

      9. *-commutative 86.4%

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

      10. *-commutative 86.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.5%

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

      13. associate-*r* 1.5%

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

      14. *-commutative 1.5%

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

   6. Applied egg-rr 86.5%

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

   7. Taylor expanded in b around 0 86.5%

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

      1. +-commutative 86.5%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)} + \left(-b\right)}}{a \cdot 2} \]

      2. *-un-lft-identity 86.5%

         \[\leadsto \frac{\color{blue}{1 \cdot \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-def 86.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1, \sqrt{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) - -4 \cdot \left(a \cdot c\right)}, -b\right)}}{a \cdot 2} \]

      4. cancel-sign-sub-inv 86.5%

         \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}}, -b\right)}{a \cdot 2} \]

      5. *-commutative 86.5%

         \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\left(\color{blue}{\left(a \cdot c\right) \cdot -8} + {b}^{2}\right) + \left(--4\right) \cdot \left(a \cdot c\right)}, -b\right)}{a \cdot 2} \]

      6. fma-def 86.5%

         \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right)} + \left(--4\right) \cdot \left(a \cdot c\right)}, -b\right)}{a \cdot 2} \]

      7. metadata-eval 86.5%

         \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + \color{blue}{4} \cdot \left(a \cdot c\right)}, -b\right)}{a \cdot 2} \]

   9. Applied egg-rr 86.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1, \sqrt{\mathsf{fma}\left(a \cdot c, -8, {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}, -b\right)}}{a \cdot 2} \]

   11. Simplified 86.5%

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

   if -6.20000000000000018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

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

      1. *-un-lft-identity 49.5%

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

      2. prod-diff 49.5%

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

      3. add-sqr-sqrt 49.5%

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

      4. sqrt-unprod 49.5%

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

      5. swap-sqr 49.5%

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

      6. metadata-eval 49.5%

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

      7. metadata-eval 49.5%

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

      8. swap-sqr 49.5%

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

      9. *-commutative 49.5%

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

      10. *-commutative 49.5%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 1.9%

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

      13. associate-*r* 1.9%

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

      14. *-commutative 1.9%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in b around inf 91.3%

      \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. unpow2 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      2. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 - -4\right)\right)} \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      3. metadata-eval 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right) \cdot \left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      4. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-8 - -4\right)\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

      5. metadata-eval 91.3%

         \[\leadsto \frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-4}\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]

   9. Applied egg-rr 91.3%

      \[\leadsto \frac{-0.125 \cdot \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -6.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -8, c, \mathsf{fma}\left(b, b, a \cdot \left(4 \cdot c\right)\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \left(\left(a \cdot c\right) \cdot -4\right)}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4}{b}\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -3.0) {
		tmp = 0.5 * ((sqrt(fma(a, (c * -4.0), pow(b, 2.0))) / a) - (b / a));
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	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)) <= -3.0)
		tmp = Float64(0.5 * Float64(Float64(sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))) / a) - Float64(b / a)));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	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], -3.0], N[(0.5 * N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $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

   1. Initial program 84.7%

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

   3. Simplified 84.6%

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

      1. div-sub 84.7%

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

      2. sub-neg 84.7%

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

      3. *-un-lft-identity 84.7%

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

      4. *-commutative 84.7%

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

      5. times-frac 84.7%

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

      6. metadata-eval 84.7%

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

      7. pow2 84.7%

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

      8. *-un-lft-identity 84.7%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\frac{\color{blue}{1 \cdot b}}{a \cdot 2}\right) \]

      9. *-commutative 84.7%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\frac{1 \cdot b}{\color{blue}{2 \cdot a}}\right) \]

      10. times-frac 84.7%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} + \left(-\color{blue}{\frac{1}{2} \cdot \frac{b}{a}}\right) \]

      11. metadata-eval 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. sub-neg 84.7%

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

      2. distribute-lft-out-- 84.7%

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

   8. Simplified 84.7%

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

   if -3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. unsub-neg 86.6%

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

      3. mul-1-neg 86.6%

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

      4. distribute-neg-frac 86.6%

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

      5. associate-/l* 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 86.3%

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

Alternative 8: 84.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -3.0) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	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) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-3.0d0)) then
        tmp = t_0
    else
        tmp = (-c / b) - (a / ((b ** 3.0d0) / (c ** 2.0d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 84.7%

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

   if -3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 48.9%

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

      1. *-commutative 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. mul-1-neg 86.6%

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

      2. unsub-neg 86.6%

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

      3. mul-1-neg 86.6%

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

      4. distribute-neg-frac 86.6%

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

      5. associate-/l* 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 86.3%

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

Alternative 9: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -2e-6) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -2e-6) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-2d-6)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -2e-6) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -2e-6:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-6)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-6)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -2e-6], t$95$0, N[((-c) / b), $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)) < -1.99999999999999991e-6

   1. Initial program 71.7%

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

   if -1.99999999999999991e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 32.3%

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

      1. *-commutative 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in b around inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. distribute-neg-frac 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.4% 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 53.5%

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

   1. *-commutative 53.5%

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

3. Simplified 53.5%

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

5. Taylor expanded in b around inf 65.8%

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

   1. mul-1-neg 65.8%

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

   2. distribute-neg-frac 65.8%

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

7. Simplified 65.8%

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

8. Final simplification 65.8%

   \[\leadsto \frac{-c}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))