Quadratic roots, narrow range

Percentage Accurate: 55.5% → 91.8%

Time: 13.0s

Alternatives: 9

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}\\ t_1 := \sqrt[3]{t_0}\\ \mathbf{if}\;b \leq 0.102:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{3}}{\mathsf{fma}\left(t_1, b + t_1, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma b b (* (* c a) -4.0)) 1.5)) (t_1 (cbrt t_0)))
   (if (<= b 0.102)
     (/ (/ (- t_0 (pow b 3.0)) (fma t_1 (+ b t_1) (* b b))) (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (* 20.0 (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0))))
        (/ (* (* -2.0 (* a a)) (pow c 3.0)) (pow b 5.0)))
       (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(b, b, ((c * a) * -4.0)), 1.5);
	double t_1 = cbrt(t_0);
	double tmp;
	if (b <= 0.102) {
		tmp = ((t_0 - pow(b, 3.0)) / fma(t_1, (b + t_1), (b * b))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (20.0 * (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0)))), (((-2.0 * (a * a)) * pow(c, 3.0)) / pow(b, 5.0))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(c * a) * -4.0)) ^ 1.5
	t_1 = cbrt(t_0)
	tmp = 0.0
	if (b <= 0.102)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 3.0)) / fma(t_1, Float64(b + t_1), Float64(b * b))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(20.0 * Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0)))), Float64(Float64(Float64(-2.0 * Float64(a * a)) * (c ^ 3.0)) / (b ^ 5.0))) - 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[Power[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[b, 0.102], N[(N[(N[(t$95$0 - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(b + t$95$1), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(20.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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 b < 0.101999999999999993

   1. Initial program 85.1%

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

      1. add-cbrt-cube 83.5%

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

      2. pow3 83.3%

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

      3. sqrt-pow2 83.7%

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

      4. *-commutative 83.7%

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

      5. *-commutative 83.7%

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

      6. metadata-eval 83.7%

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

   3. Applied egg-rr 83.7%

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

      1. flip3-+ 83.8%

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

      2. fma-neg 84.4%

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

      3. *-commutative 84.4%

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

      4. *-commutative 84.4%

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

      5. associate-*r* 84.4%

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

   5. Applied egg-rr 84.3%

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

      1. cube-neg 84.3%

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

      2. mul-1-neg 84.3%

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

      3. rem-cube-cbrt 86.0%

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

      4. +-commutative 86.0%

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

      5. mul-1-neg 86.0%

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

      6. unsub-neg 86.0%

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

      7. distribute-rgt-neg-in 86.0%

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

      8. metadata-eval 86.0%

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

   7. Simplified 86.1%

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

   if 0.101999999999999993 < b

   1. Initial program 50.4%

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

      1. neg-sub0 50.4%

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

      2. associate-+l- 50.4%

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

      3. sub0-neg 50.4%

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

      4. neg-mul-1 50.4%

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

      5. associate-*l/ 50.4%

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

      6. *-commutative 50.4%

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

      7. associate-/r* 50.4%

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

      8. /-rgt-identity 50.4%

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

      9. metadata-eval 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in a around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in c around 0 94.8%

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

      1. associate-/l* 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.102:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}}, b + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}}, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}\\ t_1 := \sqrt[3]{t_0}\\ \mathbf{if}\;b \leq 0.104:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{3}}{b \cdot b + t_1 \cdot \left(b + t_1\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma b b (* (* c a) -4.0)) 1.5)) (t_1 (cbrt t_0)))
   (if (<= b 0.104)
     (/ (/ (- t_0 (pow b 3.0)) (+ (* b b) (* t_1 (+ b t_1)))) (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (* 20.0 (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0))))
        (/ (* (* -2.0 (* a a)) (pow c 3.0)) (pow b 5.0)))
       (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(b, b, ((c * a) * -4.0)), 1.5);
	double t_1 = cbrt(t_0);
	double tmp;
	if (b <= 0.104) {
		tmp = ((t_0 - pow(b, 3.0)) / ((b * b) + (t_1 * (b + t_1)))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (20.0 * (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0)))), (((-2.0 * (a * a)) * pow(c, 3.0)) / pow(b, 5.0))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(c * a) * -4.0)) ^ 1.5
	t_1 = cbrt(t_0)
	tmp = 0.0
	if (b <= 0.104)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 3.0)) / Float64(Float64(b * b) + Float64(t_1 * Float64(b + t_1)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(20.0 * Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0)))), Float64(Float64(Float64(-2.0 * Float64(a * a)) * (c ^ 3.0)) / (b ^ 5.0))) - 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[Power[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[b, 0.104], N[(N[(N[(t$95$0 - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(t$95$1 * N[(b + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(20.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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 b < 0.103999999999999995

   1. Initial program 85.1%

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

      1. add-cbrt-cube 83.5%

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

      2. pow3 83.3%

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

      3. sqrt-pow2 83.7%

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

      4. *-commutative 83.7%

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

      5. *-commutative 83.7%

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

      6. metadata-eval 83.7%

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

   3. Applied egg-rr 83.7%

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

      1. flip3-+ 83.8%

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

      2. fma-neg 84.4%

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

      3. *-commutative 84.4%

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

      4. *-commutative 84.4%

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

      5. associate-*r* 84.4%

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

   5. Applied egg-rr 84.3%

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

      1. cube-neg 84.3%

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

      2. rem-cube-cbrt 86.0%

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

      3. distribute-rgt-neg-in 86.0%

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

      4. metadata-eval 86.0%

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

      5. sqr-neg 86.0%

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

      6. distribute-rgt-out-- 86.0%

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

   7. Simplified 86.0%

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

   if 0.103999999999999995 < b

   1. Initial program 50.4%

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

      1. neg-sub0 50.4%

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

      2. associate-+l- 50.4%

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

      3. sub0-neg 50.4%

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

      4. neg-mul-1 50.4%

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

      5. associate-*l/ 50.4%

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

      6. *-commutative 50.4%

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

      7. associate-/r* 50.4%

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

      8. /-rgt-identity 50.4%

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

      9. metadata-eval 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in a around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in c around 0 94.8%

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

      1. associate-/l* 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.104:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5} - {b}^{3}}{b \cdot b + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}} \cdot \left(b + \sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)\right)}^{1.5}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 3: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{b \cdot b}\\ \mathbf{if}\;b \leq 0.104:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b}, t_0, \sqrt[3]{b} \cdot t_0\right) + \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* b b))))
   (if (<= b 0.104)
     (/
      (+
       (fma (- (cbrt b)) t_0 (* (cbrt b) t_0))
       (- (sqrt (fma a (* c -4.0) (* b b))) b))
      (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (* 20.0 (/ (pow c 4.0) (/ (pow b 7.0) (pow a 3.0))))
        (/ (* (* -2.0 (* a a)) (pow c 3.0)) (pow b 5.0)))
       (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((b * b));
	double tmp;
	if (b <= 0.104) {
		tmp = (fma(-cbrt(b), t_0, (cbrt(b) * t_0)) + (sqrt(fma(a, (c * -4.0), (b * b))) - b)) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (20.0 * (pow(c, 4.0) / (pow(b, 7.0) / pow(a, 3.0)))), (((-2.0 * (a * a)) * pow(c, 3.0)) / pow(b, 5.0))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(b * b))
	tmp = 0.0
	if (b <= 0.104)
		tmp = Float64(Float64(fma(Float64(-cbrt(b)), t_0, Float64(cbrt(b) * t_0)) + Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(20.0 * Float64((c ^ 4.0) / Float64((b ^ 7.0) / (a ^ 3.0)))), Float64(Float64(Float64(-2.0 * Float64(a * a)) * (c ^ 3.0)) / (b ^ 5.0))) - 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[Power[N[(b * b), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[b, 0.104], N[(N[(N[((-N[Power[b, 1/3], $MachinePrecision]) * t$95$0 + N[(N[Power[b, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(20.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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 b < 0.103999999999999995

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. +-commutative 85.1%

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

      3. unsub-neg 85.1%

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

      4. fma-neg 85.2%

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

      5. associate-*l* 85.2%

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

      6. *-commutative 85.2%

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

      7. distribute-rgt-neg-in 85.2%

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

      8. metadata-eval 85.2%

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

   3. Simplified 85.2%

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

      1. add-sqr-sqrt 84.1%

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

      2. add-cube-cbrt 81.2%

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

      3. prod-diff 81.4%

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

   5. Applied egg-rr 82.5%

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

      1. +-commutative 82.5%

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

      2. *-commutative 82.5%

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

      3. fma-udef 82.5%

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

   7. Simplified 82.6%

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

      1. *-un-lft-identity 82.6%

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

      2. distribute-rgt-neg-out 82.6%

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

      3. *-commutative 82.6%

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

      4. cbrt-unprod 84.1%

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

      5. add-cbrt-cube 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. *-lft-identity 85.6%

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

       2. unsub-neg 85.6%

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

   11. Simplified 85.6%

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

   if 0.103999999999999995 < b

   1. Initial program 50.4%

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

      1. neg-sub0 50.4%

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

      2. associate-+l- 50.4%

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

      3. sub0-neg 50.4%

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

      4. neg-mul-1 50.4%

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

      5. associate-*l/ 50.4%

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

      6. *-commutative 50.4%

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

      7. associate-/r* 50.4%

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

      8. /-rgt-identity 50.4%

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

      9. metadata-eval 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in a around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in c around 0 94.8%

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

      1. associate-/l* 94.8%

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

   9. Simplified 94.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.104:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b}, \sqrt[3]{b \cdot b}, \sqrt[3]{b} \cdot \sqrt[3]{b \cdot b}\right) + \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, 20 \cdot \frac{{c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{b \cdot b}\\ \mathbf{if}\;b \leq 21.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b}, t_0, \sqrt[3]{b} \cdot t_0\right) + \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* b b))))
   (if (<= b 21.5)
     (/
      (+
       (fma (- (cbrt b)) t_0 (* (cbrt b) t_0))
       (- (sqrt (fma a (* c -4.0) (* b b))) b))
      (* a 2.0))
     (-
      (- (/ (* (* -2.0 (* a a)) (pow c 3.0)) (pow b 5.0)) (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((b * b));
	double tmp;
	if (b <= 21.5) {
		tmp = (fma(-cbrt(b), t_0, (cbrt(b) * t_0)) + (sqrt(fma(a, (c * -4.0), (b * b))) - b)) / (a * 2.0);
	} else {
		tmp = ((((-2.0 * (a * a)) * pow(c, 3.0)) / pow(b, 5.0)) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(b * b))
	tmp = 0.0
	if (b <= 21.5)
		tmp = Float64(Float64(fma(Float64(-cbrt(b)), t_0, Float64(cbrt(b) * t_0)) + Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) * (c ^ 3.0)) / (b ^ 5.0)) - 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[Power[N[(b * b), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[b, 21.5], N[(N[(N[((-N[Power[b, 1/3], $MachinePrecision]) * t$95$0 + N[(N[Power[b, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $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 b < 21.5

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. +-commutative 81.4%

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

      3. unsub-neg 81.4%

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

      4. fma-neg 81.4%

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

      5. associate-*l* 81.4%

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

      6. *-commutative 81.4%

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

      7. distribute-rgt-neg-in 81.4%

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

      8. metadata-eval 81.4%

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

   3. Simplified 81.4%

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

      1. add-sqr-sqrt 80.3%

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

      2. add-cube-cbrt 77.3%

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

      3. prod-diff 77.2%

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

   5. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. fma-udef 78.8%

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

   7. Simplified 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. distribute-rgt-neg-out 79.0%

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

      3. *-commutative 79.0%

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

      4. cbrt-unprod 80.0%

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

      5. add-cbrt-cube 81.5%

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

   9. Applied egg-rr 81.5%

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

       1. *-lft-identity 81.5%

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

       2. unsub-neg 81.5%

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

   11. Simplified 81.5%

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

   if 21.5 < b

   1. Initial program 46.2%

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

      1. neg-sub0 46.2%

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

      2. associate-+l- 46.2%

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

      3. sub0-neg 46.2%

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

      4. neg-mul-1 46.2%

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

      5. associate-*l/ 46.3%

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

      6. *-commutative 46.3%

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

      7. associate-/r* 46.3%

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

      8. /-rgt-identity 46.3%

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

      9. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

      4. +-commutative 94.2%

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

      5. mul-1-neg 94.2%

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

      6. unsub-neg 94.2%

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

      7. associate-*r/ 94.2%

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

      8. *-commutative 94.2%

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

      9. associate-*r* 94.2%

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

      10. unpow2 94.2%

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

      11. associate-/l* 94.2%

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

      12. unpow2 94.2%

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

   6. Simplified 94.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 21.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\sqrt[3]{b}, \sqrt[3]{b \cdot b}, \sqrt[3]{b} \cdot \sqrt[3]{b \cdot b}\right) + \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 5: 88.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 20.0)
   (* (- b (sqrt (fma a (* c -4.0) (* b b)))) (/ -0.5 a))
   (-
    (- (/ (* (* -2.0 (* a a)) (pow c 3.0)) (pow b 5.0)) (/ c b))
    (/ (* c c) (/ (pow b 3.0) a)))))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 20.0], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $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 b < 20

   1. Initial program 81.4%

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

      1. neg-sub0 81.4%

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

      2. associate-+l- 81.4%

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

      3. sub0-neg 81.4%

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

      4. neg-mul-1 81.4%

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

      5. associate-*l/ 81.4%

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

      6. *-commutative 81.4%

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

      7. associate-/r* 81.4%

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

      8. /-rgt-identity 81.4%

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

      9. metadata-eval 81.4%

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

   3. Simplified 81.5%

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

   if 20 < b

   1. Initial program 46.2%

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

      1. neg-sub0 46.2%

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

      2. associate-+l- 46.2%

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

      3. sub0-neg 46.2%

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

      4. neg-mul-1 46.2%

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

      5. associate-*l/ 46.3%

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

      6. *-commutative 46.3%

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

      7. associate-/r* 46.3%

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

      8. /-rgt-identity 46.3%

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

      9. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

      4. +-commutative 94.2%

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

      5. mul-1-neg 94.2%

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

      6. unsub-neg 94.2%

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

      7. associate-*r/ 94.2%

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

      8. *-commutative 94.2%

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

      9. associate-*r* 94.2%

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

      10. unpow2 94.2%

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

      11. associate-/l* 94.2%

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

      12. unpow2 94.2%

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

   6. Simplified 94.2%

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

4. Final simplification 91.4%

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

Alternative 6: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 20:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-c\right)}{\frac{b \cdot \left(b \cdot b\right)}{a}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 20.0)
   (* (- b (sqrt (fma a (* c -4.0) (* b b)))) (/ -0.5 a))
   (- (/ (* c (- c)) (/ (* b (* b b)) a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 20.0) {
		tmp = (b - sqrt(fma(a, (c * -4.0), (b * b)))) * (-0.5 / a);
	} else {
		tmp = ((c * -c) / ((b * (b * b)) / a)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 20.0)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(c * Float64(-c)) / Float64(Float64(b * Float64(b * b)) / a)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 20.0], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * (-c)), $MachinePrecision] / N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 20

   1. Initial program 81.4%

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

      1. neg-sub0 81.4%

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

      2. associate-+l- 81.4%

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

      3. sub0-neg 81.4%

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

      4. neg-mul-1 81.4%

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

      5. associate-*l/ 81.4%

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

      6. *-commutative 81.4%

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

      7. associate-/r* 81.4%

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

      8. /-rgt-identity 81.4%

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

      9. metadata-eval 81.4%

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

   3. Simplified 81.5%

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

   if 20 < b

   1. Initial program 46.2%

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

      1. neg-sub0 46.2%

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

      2. associate-+l- 46.2%

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

      3. sub0-neg 46.2%

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

      4. neg-mul-1 46.2%

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

      5. associate-*l/ 46.3%

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

      6. *-commutative 46.3%

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

      7. associate-/r* 46.3%

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

      8. /-rgt-identity 46.3%

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

      9. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. associate-*r/ 89.9%

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

      5. neg-mul-1 89.9%

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

      6. associate-/l* 89.9%

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

      7. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. unpow3 89.9%

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

   8. Applied egg-rr 89.9%

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

4. Final simplification 88.0%

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

Alternative 7: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 20:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-c\right)}{\frac{b \cdot \left(b \cdot b\right)}{a}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 20.0)
   (* (/ -0.5 a) (- b (sqrt (+ (* b b) (* a (* c -4.0))))))
   (- (/ (* c (- c)) (/ (* b (* b b)) a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 20.0) {
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = ((c * -c) / ((b * (b * b)) / a)) - (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 (b <= 20.0d0) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((b * b) + (a * (c * (-4.0d0))))))
    else
        tmp = ((c * -c) / ((b * (b * b)) / a)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 20.0) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((b * b) + (a * (c * -4.0)))));
	} else {
		tmp = ((c * -c) / ((b * (b * b)) / a)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 20.0:
		tmp = (-0.5 / a) * (b - math.sqrt(((b * b) + (a * (c * -4.0)))))
	else:
		tmp = ((c * -c) / ((b * (b * b)) / a)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 20.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))));
	else
		tmp = Float64(Float64(Float64(c * Float64(-c)) / Float64(Float64(b * Float64(b * b)) / a)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 20.0)
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (a * (c * -4.0)))));
	else
		tmp = ((c * -c) / ((b * (b * b)) / a)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 20.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * (-c)), $MachinePrecision] / N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 20

   1. Initial program 81.4%

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

      1. neg-sub0 81.4%

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

      2. associate-+l- 81.4%

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

      3. sub0-neg 81.4%

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

      4. neg-mul-1 81.4%

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

      5. associate-*l/ 81.4%

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

      6. *-commutative 81.4%

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

      7. associate-/r* 81.4%

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

      8. /-rgt-identity 81.4%

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

      9. metadata-eval 81.4%

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

   3. Simplified 81.5%

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

      1. fma-udef 81.4%

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

   5. Applied egg-rr 81.4%

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

   if 20 < b

   1. Initial program 46.2%

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

      1. neg-sub0 46.2%

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

      2. associate-+l- 46.2%

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

      3. sub0-neg 46.2%

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

      4. neg-mul-1 46.2%

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

      5. associate-*l/ 46.3%

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

      6. *-commutative 46.3%

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

      7. associate-/r* 46.3%

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

      8. /-rgt-identity 46.3%

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

      9. metadata-eval 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in b around inf 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. associate-*r/ 89.9%

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

      5. neg-mul-1 89.9%

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

      6. associate-/l* 89.9%

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

      7. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. unpow3 89.9%

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

   8. Applied egg-rr 89.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 20:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-c\right)}{\frac{b \cdot \left(b \cdot b\right)}{a}} - \frac{c}{b}\\ \end{array} \]

Alternative 8: 81.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. neg-sub0 53.9%

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

   2. associate-+l- 53.9%

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

   3. sub0-neg 53.9%

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

   4. neg-mul-1 53.9%

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

   5. associate-*l/ 53.9%

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

   6. *-commutative 53.9%

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

   7. associate-/r* 53.9%

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

   8. /-rgt-identity 53.9%

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

   9. metadata-eval 53.9%

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

3. Simplified 54.0%

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

4. Taylor expanded in b around inf 83.1%

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

   1. +-commutative 83.1%

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

   2. mul-1-neg 83.1%

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

   3. unsub-neg 83.1%

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

   4. associate-*r/ 83.1%

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

   5. neg-mul-1 83.1%

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

   6. associate-/l* 83.1%

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

   7. unpow2 83.1%

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

6. Simplified 83.1%

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

   1. unpow3 83.1%

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

8. Applied egg-rr 83.1%

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

9. Final simplification 83.1%

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

Alternative 9: 64.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. neg-sub0 53.9%

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

   2. associate-+l- 53.9%

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

   3. sub0-neg 53.9%

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

   4. neg-mul-1 53.9%

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

   5. associate-*l/ 53.9%

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

   6. *-commutative 53.9%

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

   7. associate-/r* 53.9%

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

   8. /-rgt-identity 53.9%

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

   9. metadata-eval 53.9%

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

3. Simplified 54.0%

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

4. Taylor expanded in b around inf 66.0%

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

   1. associate-*r/ 66.0%

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

   2. neg-mul-1 66.0%

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

6. Simplified 66.0%

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

7. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2023178 
(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)))