Cubic critical, narrow range

Percentage Accurate: 55.8% → 91.8%

Time: 14.4s

Alternatives: 12

Speedup: 23.2×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_1 := \left(b + t_0\right) \cdot \left(b - t_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t_1}^{1.5}}{{\left(-b\right)}^{2} + \left(t_1 + b \cdot \sqrt{t_1}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625 + {\left({\left(a \cdot c\right)}^{2} \cdot -1.125\right)}^{2}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))) (t_1 (* (+ b t_0) (- b t_0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_1 1.5))
       (+ (pow (- b) 2.0) (+ t_1 (* b (sqrt t_1)))))
      (* 3.0 a))
     (+
      (/ (* c -0.5) b)
      (fma
       -0.16666666666666666
       (/
        (+ (* (pow (* a c) 4.0) 5.0625) (pow (* (pow (* a c) 2.0) -1.125) 2.0))
        (* a (pow b 7.0)))
       (fma
        -0.041666666666666664
        (* (/ a (pow b 3.0)) (* (pow c 2.0) 9.0))
        (*
         -0.020833333333333332
         (/ (* (pow a 2.0) (pow c 3.0)) (/ (pow b 5.0) 27.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double t_1 = (b + t_0) * (b - t_0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((pow(-b, 3.0) + pow(t_1, 1.5)) / (pow(-b, 2.0) + (t_1 + (b * sqrt(t_1))))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + fma(-0.16666666666666666, (((pow((a * c), 4.0) * 5.0625) + pow((pow((a * c), 2.0) * -1.125), 2.0)) / (a * pow(b, 7.0))), fma(-0.041666666666666664, ((a / pow(b, 3.0)) * (pow(c, 2.0) * 9.0)), (-0.020833333333333332 * ((pow(a, 2.0) * pow(c, 3.0)) / (pow(b, 5.0) / 27.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	t_1 = Float64(Float64(b + t_0) * Float64(b - t_0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_1 ^ 1.5)) / Float64((Float64(-b) ^ 2.0) + Float64(t_1 + Float64(b * sqrt(t_1))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + fma(-0.16666666666666666, Float64(Float64(Float64((Float64(a * c) ^ 4.0) * 5.0625) + (Float64((Float64(a * c) ^ 2.0) * -1.125) ^ 2.0)) / Float64(a * (b ^ 7.0))), fma(-0.041666666666666664, Float64(Float64(a / (b ^ 3.0)) * Float64((c ^ 2.0) * 9.0)), Float64(-0.020833333333333332 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / Float64((b ^ 5.0) / 27.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$1 + N[(b * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 5.0625), $MachinePrecision] + N[Power[N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * -1.125), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.041666666666666664 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 2.0], $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. flip3-+ 88.3%

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

   7. Applied egg-rr 89.4%

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

      1. cancel-sign-sub 89.4%

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

   9. Simplified 89.4%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in a around 0 53.4%

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

      1. *-commutative 53.4%

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

   4. Simplified 53.4%

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

      1. add-sqr-sqrt 53.5%

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

      2. pow2 53.5%

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

      3. associate-*l* 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around inf 93.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{3}\right)}^{2}}{b} + \left(-0.16666666666666666 \cdot \frac{0.0625 \cdot \left({a}^{4} \cdot \left({c}^{4} \cdot {\left(\sqrt{3}\right)}^{8}\right)\right) + {\left(-0.125 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot {\left(\sqrt{3}\right)}^{4}\right)\right)\right)}^{2}}{a \cdot {b}^{7}} + \left(-0.041666666666666664 \cdot \frac{a \cdot \left({c}^{2} \cdot {\left(\sqrt{3}\right)}^{4}\right)}{{b}^{3}} + -0.020833333333333332 \cdot \frac{{a}^{2} \cdot \left({c}^{3} \cdot {\left(\sqrt{3}\right)}^{6}\right)}{{b}^{5}}\right)\right)} \]

   9. Simplified 94.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625 + {\left({\left(a \cdot c\right)}^{2} \cdot -1.125\right)}^{2}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right) + b \cdot \sqrt{\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625 + {\left({\left(a \cdot c\right)}^{2} \cdot -1.125\right)}^{2}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\right)\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_1 := \left(b + t_0\right) \cdot \left(b - t_0\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t_1}^{1.5}}{{\left(-b\right)}^{2} + \left(t_1 + b \cdot \sqrt{t_1}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))) (t_1 (* (+ b t_0) (- b t_0))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_1 1.5))
       (+ (pow (- b) 2.0) (+ t_1 (* b (sqrt t_1)))))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double t_1 = (b + t_0) * (b - t_0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((pow(-b, 3.0) + pow(t_1, 1.5)) / (pow(-b, 2.0) + (t_1 + (b * sqrt(t_1))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a * (3.0d0 * c)))
    t_1 = (b + t_0) * (b - t_0)
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-14.0d0)) then
        tmp = (((-b ** 3.0d0) + (t_1 ** 1.5d0)) / ((-b ** 2.0d0) + (t_1 + (b * sqrt(t_1))))) / (3.0d0 * a)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (3.0 * c)));
	double t_1 = (b + t_0) * (b - t_0);
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((Math.pow(-b, 3.0) + Math.pow(t_1, 1.5)) / (Math.pow(-b, 2.0) + (t_1 + (b * Math.sqrt(t_1))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (3.0 * c)))
	t_1 = (b + t_0) * (b - t_0)
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0:
		tmp = ((math.pow(-b, 3.0) + math.pow(t_1, 1.5)) / (math.pow(-b, 2.0) + (t_1 + (b * math.sqrt(t_1))))) / (3.0 * a)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	t_1 = Float64(Float64(b + t_0) * Float64(b - t_0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_1 ^ 1.5)) / Float64((Float64(-b) ^ 2.0) + Float64(t_1 + Float64(b * sqrt(t_1))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * (3.0 * c)));
	t_1 = (b + t_0) * (b - t_0);
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0)
		tmp = (((-b ^ 3.0) + (t_1 ^ 1.5)) / ((-b ^ 2.0) + (t_1 + (b * sqrt(t_1))))) / (3.0 * a);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$1 + N[(b * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. flip3-+ 88.3%

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

   7. Applied egg-rr 89.4%

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

      1. cancel-sign-sub 89.4%

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

   9. Simplified 89.4%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in b around inf 94.3%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 94.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. times-frac 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

   5. Simplified 94.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right) + b \cdot \sqrt{\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_1 := b + t_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{b}^{2} + t_1 \cdot \left(t_0 - b\right)}{\left(-b\right) - \sqrt{t_1 \cdot \left(b - t_0\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))) (t_1 (+ b t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/
      (/ (+ (pow b 2.0) (* t_1 (- t_0 b))) (- (- b) (sqrt (* t_1 (- b t_0)))))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double t_1 = b + t_0;
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((a * (3.0d0 * c)))
    t_1 = b + t_0
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-14.0d0)) then
        tmp = (((b ** 2.0d0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (3.0d0 * a)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (3.0 * c)));
	double t_1 = b + t_0;
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((Math.pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - Math.sqrt((t_1 * (b - t_0))))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (3.0 * c)))
	t_1 = b + t_0
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0:
		tmp = ((math.pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - math.sqrt((t_1 * (b - t_0))))) / (3.0 * a)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	t_1 = Float64(b + t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) + Float64(t_1 * Float64(t_0 - b))) / Float64(Float64(-b) - sqrt(Float64(t_1 * Float64(b - t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * (3.0 * c)));
	t_1 = b + t_0;
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0)
		tmp = (((b ^ 2.0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (3.0 * a);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(t$95$1 * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. flip-+ 88.1%

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

      2. pow2 88.1%

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

      3. add-sqr-sqrt 89.4%

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

      4. associate-*l* 89.4%

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

      5. associate-*l* 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

      2. sqr-neg 89.4%

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

      3. unpow2 89.4%

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

   9. Simplified 89.4%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in b around inf 94.3%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 94.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. times-frac 94.3%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

   5. Simplified 94.3%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{b}^{2} + \left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(\sqrt{a \cdot \left(3 \cdot c\right)} - b\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 4: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_1 := b + t_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{b}^{2} + t_1 \cdot \left(t_0 - b\right)}{\left(-b\right) - \sqrt{t_1 \cdot \left(b - t_0\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))) (t_1 (+ b t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/
      (/ (+ (pow b 2.0) (* t_1 (- t_0 b))) (- (- b) (sqrt (* t_1 (- b t_0)))))
      (* 3.0 a))
     (+
      (/ (* c -0.5) b)
      (fma
       -0.041666666666666664
       (* (/ a (pow b 3.0)) (* (pow c 2.0) 9.0))
       (*
        -0.020833333333333332
        (/ (* (pow a 2.0) (pow c 3.0)) (/ (pow b 5.0) 27.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double t_1 = b + t_0;
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = ((pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + fma(-0.041666666666666664, ((a / pow(b, 3.0)) * (pow(c, 2.0) * 9.0)), (-0.020833333333333332 * ((pow(a, 2.0) * pow(c, 3.0)) / (pow(b, 5.0) / 27.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	t_1 = Float64(b + t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) + Float64(t_1 * Float64(t_0 - b))) / Float64(Float64(-b) - sqrt(Float64(t_1 * Float64(b - t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + fma(-0.041666666666666664, Float64(Float64(a / (b ^ 3.0)) * Float64((c ^ 2.0) * 9.0)), Float64(-0.020833333333333332 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / Float64((b ^ 5.0) / 27.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(t$95$1 * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(-0.041666666666666664 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 2.0], $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. flip-+ 88.1%

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

      2. pow2 88.1%

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

      3. add-sqr-sqrt 89.4%

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

      4. associate-*l* 89.4%

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

      5. associate-*l* 89.4%

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

   7. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

      2. sqr-neg 89.4%

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

      3. unpow2 89.4%

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

   9. Simplified 89.4%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in a around 0 53.4%

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

      1. *-commutative 53.4%

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

   4. Simplified 53.4%

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

      1. add-sqr-sqrt 53.5%

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

      2. pow2 53.5%

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

      3. associate-*l* 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{3}\right)}^{2}}{b} + \left(-0.041666666666666664 \cdot \frac{a \cdot \left({c}^{2} \cdot {\left(\sqrt{3}\right)}^{4}\right)}{{b}^{3}} + -0.020833333333333332 \cdot \frac{{a}^{2} \cdot \left({c}^{3} \cdot {\left(\sqrt{3}\right)}^{6}\right)}{{b}^{5}}\right)} \]

   9. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\frac{{b}^{2} + \left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(\sqrt{a \cdot \left(3 \cdot c\right)} - b\right)}{\left(-b\right) - \sqrt{\left(b + \sqrt{a \cdot \left(3 \cdot c\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\\ \end{array} \]

Alternative 5: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{b + t_0}, \sqrt{b - t_0}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/ (fma (sqrt (+ b t_0)) (sqrt (- b t_0)) (- b)) (* 3.0 a))
     (+
      (/ (* c -0.5) b)
      (fma
       -0.041666666666666664
       (* (/ a (pow b 3.0)) (* (pow c 2.0) 9.0))
       (*
        -0.020833333333333332
        (/ (* (pow a 2.0) (pow c 3.0)) (/ (pow b 5.0) 27.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = fma(sqrt((b + t_0)), sqrt((b - t_0)), -b) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) / b) + fma(-0.041666666666666664, ((a / pow(b, 3.0)) * (pow(c, 2.0) * 9.0)), (-0.020833333333333332 * ((pow(a, 2.0) * pow(c, 3.0)) / (pow(b, 5.0) / 27.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(fma(sqrt(Float64(b + t_0)), sqrt(Float64(b - t_0)), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + fma(-0.041666666666666664, Float64(Float64(a / (b ^ 3.0)) * Float64((c ^ 2.0) * 9.0)), Float64(-0.020833333333333332 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / Float64((b ^ 5.0) / 27.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[Sqrt[N[(b + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(b - t$95$0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(-0.041666666666666664 * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 2.0], $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. sqrt-prod 87.8%

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

      3. fma-def 88.6%

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

      4. associate-*l* 88.6%

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

      5. associate-*l* 88.6%

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

   7. Applied egg-rr 88.6%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in a around 0 53.4%

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

      1. *-commutative 53.4%

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

   4. Simplified 53.4%

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

      1. add-sqr-sqrt 53.5%

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

      2. pow2 53.5%

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

      3. associate-*l* 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{3}\right)}^{2}}{b} + \left(-0.041666666666666664 \cdot \frac{a \cdot \left({c}^{2} \cdot {\left(\sqrt{3}\right)}^{4}\right)}{{b}^{3}} + -0.020833333333333332 \cdot \frac{{a}^{2} \cdot \left({c}^{3} \cdot {\left(\sqrt{3}\right)}^{6}\right)}{{b}^{5}}\right)} \]

   9. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{b + \sqrt{a \cdot \left(3 \cdot c\right)}}, \sqrt{b - \sqrt{a \cdot \left(3 \cdot c\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + \mathsf{fma}\left(-0.041666666666666664, \frac{a}{{b}^{3}} \cdot \left({c}^{2} \cdot 9\right), -0.020833333333333332 \cdot \frac{{a}^{2} \cdot {c}^{3}}{\frac{{b}^{5}}{27}}\right)\\ \end{array} \]

Alternative 6: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{b + t_0}, \sqrt{b - t_0}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -14.0)
     (/ (fma (sqrt (+ b t_0)) (sqrt (- b t_0)) (- b)) (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -14.0) {
		tmp = fma(sqrt((b + t_0)), sqrt((b - t_0)), -b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -14.0)
		tmp = Float64(fma(sqrt(Float64(b + t_0)), sqrt(Float64(b - t_0)), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -14.0], N[(N[(N[Sqrt[N[(b + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(b - t$95$0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -14

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.9%

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

      2. difference-of-squares 88.1%

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

      3. associate-*l* 88.1%

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

      4. associate-*l* 88.2%

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

   3. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

   5. Simplified 88.2%

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

      1. +-commutative 88.2%

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

      2. sqrt-prod 87.8%

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

      3. fma-def 88.6%

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

      4. associate-*l* 88.6%

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

      5. associate-*l* 88.6%

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

   7. Applied egg-rr 88.6%

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

   if -14 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 53.5%

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

   2. Taylor expanded in b around inf 92.2%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -14:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{b + \sqrt{a \cdot \left(3 \cdot c\right)}}, \sqrt{b - \sqrt{a \cdot \left(3 \cdot c\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.024:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.024)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.024) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.024)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.024], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.024

   1. Initial program 78.6%

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

      1. +-commutative 78.6%

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

      2. sqr-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. div-sub 77.2%

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

      5. --rgt-identity 77.2%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 78.6%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 78.8%

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

   if -0.024 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 48.8%

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

   2. Taylor expanded in b around inf 89.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.024:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 8: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00015)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00015) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00015)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00015], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.49999999999999987e-4

   1. Initial program 73.5%

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

      1. +-commutative 73.5%

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

      2. sqr-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. div-sub 72.3%

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

      5. --rgt-identity 72.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 73.5%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 73.8%

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

   if -1.49999999999999987e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.2%

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

   2. Taylor expanded in b around inf 76.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.00015:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.00015) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.00015) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.00015d0)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.00015) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.00015:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.00015)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.00015)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.00015], t$95$0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.49999999999999987e-4

   1. Initial program 73.5%

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

   if -1.49999999999999987e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.2%

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

   2. Taylor expanded in b around inf 76.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.00015)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00015) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-0.00015d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00015) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00015:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.00015)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.00015)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00015], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.49999999999999987e-4

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0 73.5%

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

      1. *-commutative 73.5%

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

   4. Simplified 73.5%

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

   if -1.49999999999999987e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 42.2%

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

   2. Taylor expanded in b around inf 76.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 64.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (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 = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in b around inf 64.3%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

3. Final simplification 64.3%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]

Alternative 12: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. add-sqr-sqrt 56.0%

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

   2. difference-of-squares 56.2%

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

   3. associate-*l* 56.2%

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

   4. associate-*l* 56.3%

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

3. Applied egg-rr 56.3%

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

   1. *-commutative 56.3%

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

   2. *-commutative 56.3%

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

5. Simplified 56.3%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023315 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))