Quadratic roots, narrow range

Percentage Accurate: 56.1% → 91.5%

Time: 18.2s

Alternatives: 12

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: 56.1% 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: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}\right)}^{3}}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot t_1\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* 4.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.62)
     (/
      (/
       (cbrt (pow (- (pow (fma b b (* a (* c -4.0))) 1.5) (pow b 3.0)) 3.0))
       (+ (pow (- b) 2.0) (+ t_0 (* b t_1))))
      (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
        (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))))
       (/ c b))
      (/ a (/ (pow b 3.0) (* c c)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((4.0 * a) * c);
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.62) {
		tmp = (cbrt(pow((pow(fma(b, b, (a * (c * -4.0))), 1.5) - pow(b, 3.0)), 3.0)) / (pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0)))))) - (c / b)) - (a / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(cbrt((Float64((fma(b, b, Float64(a * Float64(c * -4.0))) ^ 1.5) - (b ^ 3.0)) ^ 3.0)) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * t_1)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0)))))) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[Power[N[Power[N[(N[Power[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $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)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. flip3-+ 86.6%

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

      2. cube-neg 86.6%

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

      3. pow1/2 86.6%

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

      4. pow-pow 87.7%

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

      5. *-commutative 87.7%

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

      6. *-commutative 87.7%

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

      7. metadata-eval 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      8. pow2 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 87.7%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right) \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)\right) \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. Applied egg-rr 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right) \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)\right) \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*l* 87.7%

         \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{\left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right) \cdot \left(\left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right) \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)\right)}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      2. cube-unmult 87.8%

         \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}^{3}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. Simplified 88.1%

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{{\left({\left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}^{1.5} - {b}^{3}\right)}^{3}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. neg-sub0 49.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 49.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 49.2%

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

      4. neg-mul-1 49.2%

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

      5. associate-*l/ 49.2%

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

      6. *-commutative 49.2%

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

      7. associate-/r* 49.2%

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

      8. /-rgt-identity 49.2%

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

      9. metadata-eval 49.2%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in a around 0 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in b around 0 94.2%

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

      1. associate-/l* 94.2%

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

      2. distribute-rgt-out 94.2%

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

      3. metadata-eval 94.2%

         \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   9. Simplified 94.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}\right)}^{3}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{\left(b \cdot b + t_0\right) + b \cdot \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.62)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ (+ (* b b) t_0) (* b (sqrt t_0))))
      (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
        (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))))
       (/ c b))
      (/ a (/ (pow b 3.0) (* c c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.62) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (((b * b) + t_0) + (b * sqrt(t_0)))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0)))))) - (c / b)) - (a / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(Float64(Float64(b * b) + t_0) + Float64(b * sqrt(t_0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0)))))) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $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)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. pow1/2 86.6%

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

      2. pow-to-exp 82.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{2 \cdot a} \]

      3. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      4. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}\right) \cdot 0.5}}{2 \cdot a} \]

   3. Applied egg-rr 82.8%

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

      1. exp-to-pow 86.6%

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

      2. pow1/2 86.6%

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

      3. flip3-+ 86.6%

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

      4. unpow2 86.6%

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

      5. add-sqr-sqrt 86.6%

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

      6. cancel-sign-sub 86.6%

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

   5. Applied egg-rr 86.6%

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

   7. Simplified 88.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b + \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}}}}{2 \cdot a} \]

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. neg-sub0 49.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 49.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 49.2%

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

      4. neg-mul-1 49.2%

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

      5. associate-*l/ 49.2%

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

      6. *-commutative 49.2%

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

      7. associate-/r* 49.2%

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

      8. /-rgt-identity 49.2%

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

      9. metadata-eval 49.2%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in a around 0 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in b around 0 94.2%

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

      1. associate-/l* 94.2%

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

      2. distribute-rgt-out 94.2%

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

      3. metadata-eval 94.2%

         \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}} \]

   9. Simplified 94.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}}\right) - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 3: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{\left(b \cdot b + t_0\right) + b \cdot \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.62)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ (+ (* b b) t_0) (* b (sqrt t_0))))
      (* a 2.0))
     (-
      (fma
       -0.25
       (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 20.0 a))
       (- (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))) (/ c b)))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.62) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (((b * b) + t_0) + (b * sqrt(t_0)))) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((pow((a * c), 4.0) / pow(b, 7.0)) * (20.0 / a)), ((-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))) - (c / b))) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(Float64(Float64(b * b) + t_0) + Float64(b * sqrt(t_0)))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(20.0 / a)), Float64(Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(c / b))) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. pow1/2 86.6%

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

      2. pow-to-exp 82.8%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{2 \cdot a} \]

      3. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      4. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}\right) \cdot 0.5}}{2 \cdot a} \]

   3. Applied egg-rr 82.8%

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

      1. exp-to-pow 86.6%

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

      2. pow1/2 86.6%

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

      3. flip3-+ 86.6%

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

      4. unpow2 86.6%

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

      5. add-sqr-sqrt 86.6%

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

      6. cancel-sign-sub 86.6%

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

   5. Applied egg-rr 86.6%

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

   7. Simplified 88.1%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b + \mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}}}}{2 \cdot a} \]

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. pow1/2 49.2%

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

      2. pow-to-exp 46.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{2 \cdot a} \]

      3. *-commutative 46.2%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      4. *-commutative 46.2%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}\right) \cdot 0.5}}{2 \cdot a} \]

   3. Applied egg-rr 46.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) \cdot 0.5}}}{2 \cdot a} \]

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in c around 0 94.2%

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

      1. distribute-rgt-out 94.2%

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

      2. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{c}^{4} \cdot \left({a}^{4} \cdot \color{blue}{20}\right)}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      3. associate-*r* 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      4. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({c}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {a}^{4}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      5. pow-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left({c}^{2} \cdot {c}^{2}\right)} \cdot {a}^{4}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      6. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      7. pow-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot \color{blue}{\left({a}^{2} \cdot {a}^{2}\right)}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      8. unswap-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left(\left({c}^{2} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      9. unpow2 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      10. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      11. unswap-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)} \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      12. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{{\left(c \cdot a\right)}^{2}} \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      13. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      14. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      15. unswap-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      16. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{{\left(c \cdot a\right)}^{2}}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      17. pow-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{\left(c \cdot a\right)}^{\left(2 \cdot 2\right)}} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      18. metadata-eval 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{\color{blue}{4}} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

   9. Simplified 94.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot t_1\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* 4.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.62)
     (/
      (/
       (- (pow (fma b b (* a (* c -4.0))) 1.5) (pow b 3.0))
       (+ (pow (- b) 2.0) (+ t_0 (* b t_1))))
      (* a 2.0))
     (-
      (fma
       -0.25
       (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 20.0 a))
       (- (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))) (/ c b)))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((4.0 * a) * c);
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.62) {
		tmp = ((pow(fma(b, b, (a * (c * -4.0))), 1.5) - pow(b, 3.0)) / (pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	} else {
		tmp = fma(-0.25, ((pow((a * c), 4.0) / pow(b, 7.0)) * (20.0 / a)), ((-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))) - (c / b))) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(Float64((fma(b, b, Float64(a * Float64(c * -4.0))) ^ 1.5) - (b ^ 3.0)) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * t_1)))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(20.0 / a)), Float64(Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(c / b))) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[(N[Power[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. flip3-+ 86.6%

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

      2. cube-neg 86.6%

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

      3. pow1/2 86.6%

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

      4. pow-pow 87.7%

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

      5. *-commutative 87.7%

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

      6. *-commutative 87.7%

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

      7. metadata-eval 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      8. pow2 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 87.7%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. Applied egg-rr 87.7%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. *-lft-identity 87.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      2. +-commutative 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} + \left(-{b}^{3}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      3. unsub-neg 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} - {b}^{3}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      4. fma-neg 88.1%

         \[\leadsto \frac{\frac{{\color{blue}{\left(\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 4\right)\right)\right)}}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      5. associate-*r* 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, -\color{blue}{\left(c \cdot a\right) \cdot 4}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      6. *-commutative 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(c \cdot a\right)}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      7. distribute-lft-neg-in 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(c \cdot a\right)}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      8. metadata-eval 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(c \cdot a\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      9. associate-*r* 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      10. *-commutative 88.1%

          \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right)} \cdot a\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. Simplified 88.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}^{1.5} - {b}^{3}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. pow1/2 49.2%

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

      2. pow-to-exp 46.2%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot 0.5}}}{2 \cdot a} \]

      3. *-commutative 46.2%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}\right) \cdot 0.5}}{2 \cdot a} \]

      4. *-commutative 46.2%

         \[\leadsto \frac{\left(-b\right) + e^{\log \left(b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}\right) \cdot 0.5}}{2 \cdot a} \]

   3. Applied egg-rr 46.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{e^{\log \left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) \cdot 0.5}}}{2 \cdot a} \]

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in c around 0 94.2%

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

      1. distribute-rgt-out 94.2%

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

      2. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{c}^{4} \cdot \left({a}^{4} \cdot \color{blue}{20}\right)}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      3. associate-*r* 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 20}}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      4. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({c}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot {a}^{4}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      5. pow-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left({c}^{2} \cdot {c}^{2}\right)} \cdot {a}^{4}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      6. metadata-eval 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot {a}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      7. pow-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left({c}^{2} \cdot {c}^{2}\right) \cdot \color{blue}{\left({a}^{2} \cdot {a}^{2}\right)}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      8. unswap-sqr 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left(\left({c}^{2} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      9. unpow2 94.2%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      10. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      11. unswap-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)} \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      12. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left(\color{blue}{{\left(c \cdot a\right)}^{2}} \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      13. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      14. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      15. unswap-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      16. unpow2 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\left({\left(c \cdot a\right)}^{2} \cdot \color{blue}{{\left(c \cdot a\right)}^{2}}\right) \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      17. pow-sqr 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{\left(c \cdot a\right)}^{\left(2 \cdot 2\right)}} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

      18. metadata-eval 94.2%

          \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{\color{blue}{4}} \cdot 20}{a \cdot {b}^{7}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

   9. Simplified 94.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot t_1\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* 4.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.62)
     (/
      (/
       (- (pow (fma b b (* a (* c -4.0))) 1.5) (pow b 3.0))
       (+ (pow (- b) 2.0) (+ t_0 (* b t_1))))
      (* a 2.0))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))) (/ c b))
      (/ a (/ (pow b 3.0) (* c c)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((4.0 * a) * c);
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.62) {
		tmp = ((pow(fma(b, b, (a * (c * -4.0))), 1.5) - pow(b, 3.0)) / (pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0))))) - (c / b)) - (a / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(Float64((fma(b, b, Float64(a * Float64(c * -4.0))) ^ 1.5) - (b ^ 3.0)) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * t_1)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0))))) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[(N[Power[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $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)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. flip3-+ 86.6%

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

      2. cube-neg 86.6%

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

      3. pow1/2 86.6%

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

      4. pow-pow 87.7%

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

      5. *-commutative 87.7%

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

      6. *-commutative 87.7%

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

      7. metadata-eval 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      8. pow2 87.7%

         \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. Applied egg-rr 87.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 87.7%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. Applied egg-rr 87.7%

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. *-lft-identity 87.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      2. +-commutative 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} + \left(-{b}^{3}\right)}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      3. unsub-neg 87.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} - {b}^{3}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      4. fma-neg 88.1%

         \[\leadsto \frac{\frac{{\color{blue}{\left(\mathsf{fma}\left(b, b, -c \cdot \left(a \cdot 4\right)\right)\right)}}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      5. associate-*r* 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, -\color{blue}{\left(c \cdot a\right) \cdot 4}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      6. *-commutative 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(c \cdot a\right)}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      7. distribute-lft-neg-in 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(c \cdot a\right)}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      8. metadata-eval 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(c \cdot a\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      9. associate-*r* 88.1%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

      10. *-commutative 88.1%

          \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot -4\right)} \cdot a\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. Simplified 88.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)\right)}^{1.5} - {b}^{3}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. neg-sub0 49.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 49.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 49.2%

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

      4. neg-mul-1 49.2%

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

      5. associate-*l/ 49.2%

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

      6. *-commutative 49.2%

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

      7. associate-/r* 49.2%

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

      8. /-rgt-identity 49.2%

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

      9. metadata-eval 49.2%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in b around inf 91.9%

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

      1. +-commutative 91.9%

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

      2. mul-1-neg 91.9%

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

      3. unsub-neg 91.9%

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

      4. +-commutative 91.9%

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

      5. mul-1-neg 91.9%

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

      6. unsub-neg 91.9%

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

      7. associate-*r/ 91.9%

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

      8. associate-/l* 91.9%

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

      9. *-commutative 91.9%

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

      10. unpow2 91.9%

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

      11. associate-*l* 91.9%

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

      12. *-commutative 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 6: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot a\right) \cdot c\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 4.0 a) c)) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.62)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))) (/ c b))
      (/ a (/ (pow b 3.0) (* c c)))))))

C

double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.62) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0))))) - (c / b)) - (a / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 2.0d0)) <= (-0.62d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = (((-2.0d0) / ((b ** 5.0d0) / (a * (a * (c ** 3.0d0))))) - (c / b)) - (a / ((b ** 3.0d0) / (c * c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.62) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (Math.pow(b, 5.0) / (a * (a * Math.pow(c, 3.0))))) - (c / b)) - (a / (Math.pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (4.0 * a) * c
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -0.62:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = ((-2.0 / (math.pow(b, 5.0) / (a * (a * math.pow(c, 3.0))))) - (c / b)) - (a / (math.pow(b, 3.0) / (c * c)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(4.0 * a) * c)
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.62)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0))))) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (4.0 * a) * c;
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -0.62)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	else
		tmp = ((-2.0 / ((b ^ 5.0) / (a * (a * (c ^ 3.0))))) - (c / b)) - (a / ((b ^ 3.0) / (c * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.62], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $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)) < -0.619999999999999996

   1. Initial program 86.6%

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

      1. flip-+ 86.3%

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

      2. pow2 86.3%

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

      3. add-sqr-sqrt 88.0%

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

      4. *-commutative 88.0%

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

      5. *-commutative 88.0%

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

      6. *-commutative 88.0%

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

      7. *-commutative 88.0%

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

   3. Applied egg-rr 88.0%

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

   if -0.619999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.2%

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

      1. neg-sub0 49.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 49.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 49.2%

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

      4. neg-mul-1 49.2%

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

      5. associate-*l/ 49.2%

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

      6. *-commutative 49.2%

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

      7. associate-/r* 49.2%

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

      8. /-rgt-identity 49.2%

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

      9. metadata-eval 49.2%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in b around inf 91.9%

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

      1. +-commutative 91.9%

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

      2. mul-1-neg 91.9%

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

      3. unsub-neg 91.9%

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

      4. +-commutative 91.9%

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

      5. mul-1-neg 91.9%

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

      6. unsub-neg 91.9%

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

      7. associate-*r/ 91.9%

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

      8. associate-/l* 91.9%

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

      9. *-commutative 91.9%

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

      10. unpow2 91.9%

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

      11. associate-*l* 91.9%

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

      12. *-commutative 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.62:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(\left(4 \cdot a\right) \cdot c - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 7: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot a\right) \cdot c\\ t_1 := \sqrt{b \cdot b - t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - b \cdot b\right)}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 4.0 a) c)) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.005)
     (/ (/ (+ (pow (- b) 2.0) (- t_0 (* b b))) (- (- b) t_1)) (* a 2.0))
     (- (/ (- a) (/ (pow b 3.0) (* c c))) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double t_1 = sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.005) {
		tmp = ((pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-a / (pow(b, 3.0) / (c * c))) - (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) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    t_1 = sqrt(((b * b) - t_0))
    if (((t_1 - b) / (a * 2.0d0)) <= (-0.005d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = (-a / ((b ** 3.0d0) / (c * c))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (4.0 * a) * c;
	double t_1 = Math.sqrt(((b * b) - t_0));
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.005) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-a / (Math.pow(b, 3.0) / (c * c))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (4.0 * a) * c
	t_1 = math.sqrt(((b * b) - t_0))
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -0.005:
		tmp = ((math.pow(-b, 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (-a / (math.pow(b, 3.0) / (c * c))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(4.0 * a) * c)
	t_1 = sqrt(Float64(Float64(b * b) - t_0))
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.005)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - Float64(b * b))) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-a) / Float64((b ^ 3.0) / Float64(c * c))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (4.0 * a) * c;
	t_1 = sqrt(((b * b) - t_0));
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -0.005)
		tmp = (((-b ^ 2.0) + (t_0 - (b * b))) / (-b - t_1)) / (a * 2.0);
	else
		tmp = (-a / ((b ^ 3.0) / (c * c))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-a) / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.1%

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

      1. flip-+ 81.1%

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

      2. pow2 81.1%

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

      3. add-sqr-sqrt 82.8%

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

      4. *-commutative 82.8%

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

      5. *-commutative 82.8%

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

      6. *-commutative 82.8%

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

      7. *-commutative 82.8%

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

   3. Applied egg-rr 82.8%

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

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

   1. Initial program 44.3%

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

      1. neg-sub0 44.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 44.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 44.3%

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

      4. neg-mul-1 44.3%

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

      5. associate-*l/ 44.3%

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

      6. *-commutative 44.3%

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

      7. associate-/r* 44.3%

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

      8. /-rgt-identity 44.3%

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

      9. metadata-eval 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in b around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. mul-1-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. associate-*r/ 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. associate-/l* 89.8%

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

      8. unpow2 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(\left(4 \cdot a\right) \cdot c - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \]

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.005)
   (* (- (sqrt (fma b b (* -4.0 (* a c)))) b) (/ 0.5 a))
   (- (/ (- a) (/ (pow b 3.0) (* c c))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.005) {
		tmp = (sqrt(fma(b, b, (-4.0 * (a * c)))) - b) * (0.5 / a);
	} else {
		tmp = (-a / (pow(b, 3.0) / (c * c))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.005)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(a * c)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-a) / Float64((b ^ 3.0) / Float64(c * c))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-a) / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.1%

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

      1. /-rgt-identity 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-/l* 81.1%

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

      4. associate-*r/ 81.1%

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

      5. +-commutative 81.1%

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

      6. unsub-neg 81.1%

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

      7. fma-neg 81.2%

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

      8. associate-*l* 81.2%

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

      9. *-commutative 81.2%

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

      10. distribute-rgt-neg-in 81.2%

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

      11. metadata-eval 81.2%

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

      12. associate-/r* 81.2%

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

      13. metadata-eval 81.2%

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

      14. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

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

   1. Initial program 44.3%

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

      1. neg-sub0 44.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 44.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 44.3%

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

      4. neg-mul-1 44.3%

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

      5. associate-*l/ 44.3%

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

      6. *-commutative 44.3%

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

      7. associate-/r* 44.3%

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

      8. /-rgt-identity 44.3%

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

      9. metadata-eval 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in b around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. mul-1-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. associate-*r/ 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. associate-/l* 89.8%

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

      8. unpow2 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \]

Alternative 9: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.005)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* -4.0 (* a c)))) b))
   (- (/ (- a) (/ (pow b 3.0) (* c c))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.005) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (-4.0 * (a * c)))) - b);
	} else {
		tmp = (-a / (pow(b, 3.0) / (c * c))) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)) <= (-0.005d0)) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + ((-4.0d0) * (a * c)))) - b)
    else
        tmp = (-a / ((b ** 3.0d0) / (c * c))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.005) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (-4.0 * (a * c)))) - b);
	} else {
		tmp = (-a / (Math.pow(b, 3.0) / (c * c))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.005:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (-4.0 * (a * c)))) - b)
	else:
		tmp = (-a / (math.pow(b, 3.0) / (c * c))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.005)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(a * c)))) - b));
	else
		tmp = Float64(Float64(Float64(-a) / Float64((b ^ 3.0) / Float64(c * c))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.005)
		tmp = (0.5 / a) * (sqrt(((b * b) + (-4.0 * (a * c)))) - b);
	else
		tmp = (-a / ((b ^ 3.0) / (c * c))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-a) / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.1%

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

      1. /-rgt-identity 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-/l* 81.1%

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

      4. associate-*r/ 81.1%

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

      5. +-commutative 81.1%

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

      6. unsub-neg 81.1%

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

      7. fma-neg 81.2%

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

      8. associate-*l* 81.2%

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

      9. *-commutative 81.2%

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

      10. distribute-rgt-neg-in 81.2%

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

      11. metadata-eval 81.2%

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

      12. associate-/r* 81.2%

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

      13. metadata-eval 81.2%

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

      14. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

      1. fma-udef 81.1%

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

      2. *-commutative 81.1%

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

   5. Applied egg-rr 81.1%

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

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

   1. Initial program 44.3%

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

      1. neg-sub0 44.3%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 44.3%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 44.3%

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

      4. neg-mul-1 44.3%

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

      5. associate-*l/ 44.3%

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

      6. *-commutative 44.3%

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

      7. associate-/r* 44.3%

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

      8. /-rgt-identity 44.3%

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

      9. metadata-eval 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in b around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. mul-1-neg 89.8%

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

      3. unsub-neg 89.8%

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

      4. associate-*r/ 89.8%

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

      5. neg-mul-1 89.8%

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

      6. *-commutative 89.8%

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

      7. associate-/l* 89.8%

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

      8. unpow2 89.8%

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

   6. Simplified 89.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.005:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\frac{{b}^{3}}{c \cdot c}} - \frac{c}{b}\\ \end{array} \]

Alternative 10: 81.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. neg-sub0 56.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 56.2%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 56.2%

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

   4. neg-mul-1 56.2%

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

   5. associate-*l/ 56.2%

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

   6. *-commutative 56.2%

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

   7. associate-/r* 56.2%

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

   8. /-rgt-identity 56.2%

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

   9. metadata-eval 56.2%

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

3. Simplified 56.3%

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

4. Taylor expanded in b around inf 80.3%

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

   1. +-commutative 80.3%

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

   2. mul-1-neg 80.3%

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

   3. unsub-neg 80.3%

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

   4. associate-*r/ 80.3%

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

   5. neg-mul-1 80.3%

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

   6. *-commutative 80.3%

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

   7. associate-/l* 80.3%

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

   8. unpow2 80.3%

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

6. Simplified 80.3%

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

7. Final simplification 80.3%

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

Alternative 11: 63.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. neg-sub0 56.2%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 56.2%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 56.2%

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

   4. neg-mul-1 56.2%

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

   5. associate-*l/ 56.2%

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

   6. *-commutative 56.2%

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

   7. associate-/r* 56.2%

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

   8. /-rgt-identity 56.2%

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

   9. metadata-eval 56.2%

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

3. Simplified 56.3%

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

4. Taylor expanded in b around inf 63.7%

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

   1. associate-*r/ 63.7%

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

   2. neg-mul-1 63.7%

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

6. Simplified 63.7%

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

7. Final simplification 63.7%

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

Alternative 12: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. log1p-expm1-u 48.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right)} \]

   2. neg-mul-1 48.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right) \]

   3. fma-def 48.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\right)\right) \]

   4. *-commutative 48.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{2 \cdot a}\right)\right) \]

   5. *-commutative 48.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{2 \cdot a}\right)\right) \]

   6. *-commutative 48.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{\color{blue}{a \cdot 2}}\right)\right) \]

3. Applied egg-rr 48.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}\right)\right)} \]

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

6. Simplified 3.2%

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

7. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023238 
(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)))