Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.7%

Time: 17.1s

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: 55.4% 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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot {a}^{2}\right) \cdot -0.375\\ t_1 := \left(3 \cdot a\right) \cdot c\\ t_2 := {b}^{2} - t\_1\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_1} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{t\_2 - {\left(-b\right)}^{2}}{b + \sqrt{t\_2}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\mathsf{fma}\left(-0.75, \left(a \cdot c\right) \cdot t\_0, \mathsf{fma}\left(-0.2222222222222222, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 6.328125}{a \cdot {c}^{2}}, 0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-3, \frac{t\_0}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right)\right) - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c (pow a 2.0)) -0.375))
        (t_1 (* (* 3.0 a) c))
        (t_2 (- (pow b 2.0) t_1)))
   (if (<= (/ (- (sqrt (- (* b b) t_1)) b) (* 3.0 a)) -283.0)
     (/ (/ (- t_2 (pow (- b) 2.0)) (+ b (sqrt t_2))) (* 3.0 a))
     (/
      1.0
      (*
       b
       (-
        (fma
         -3.0
         (/
          (fma
           -0.75
           (* (* a c) t_0)
           (fma
            -0.2222222222222222
            (/ (* (* (pow a 4.0) (pow c 4.0)) 6.328125) (* a (pow c 2.0)))
            (* 0.5625 (* (pow c 2.0) (pow a 3.0)))))
          (pow b 6.0))
         (fma -3.0 (/ t_0 (pow b 4.0)) (/ (* a 1.5) (pow b 2.0))))
        (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * pow(a, 2.0)) * -0.375;
	double t_1 = (3.0 * a) * c;
	double t_2 = pow(b, 2.0) - t_1;
	double tmp;
	if (((sqrt(((b * b) - t_1)) - b) / (3.0 * a)) <= -283.0) {
		tmp = ((t_2 - pow(-b, 2.0)) / (b + sqrt(t_2))) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (fma(-3.0, (fma(-0.75, ((a * c) * t_0), fma(-0.2222222222222222, (((pow(a, 4.0) * pow(c, 4.0)) * 6.328125) / (a * pow(c, 2.0))), (0.5625 * (pow(c, 2.0) * pow(a, 3.0))))) / pow(b, 6.0)), fma(-3.0, (t_0 / pow(b, 4.0)), ((a * 1.5) / pow(b, 2.0)))) - (2.0 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * (a ^ 2.0)) * -0.375)
	t_1 = Float64(Float64(3.0 * a) * c)
	t_2 = Float64((b ^ 2.0) - t_1)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_1)) - b) / Float64(3.0 * a)) <= -283.0)
		tmp = Float64(Float64(Float64(t_2 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_2))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(fma(-0.75, Float64(Float64(a * c) * t_0), fma(-0.2222222222222222, Float64(Float64(Float64((a ^ 4.0) * (c ^ 4.0)) * 6.328125) / Float64(a * (c ^ 2.0))), Float64(0.5625 * Float64((c ^ 2.0) * (a ^ 3.0))))) / (b ^ 6.0)), fma(-3.0, Float64(t_0 / (b ^ 4.0)), Float64(Float64(a * 1.5) / (b ^ 2.0)))) - Float64(2.0 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[b, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -283.0], N[(N[(N[(t$95$2 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(-0.75 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-0.2222222222222222 * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-3.0 * N[(t$95$0 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 90.4%

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

      2. pow1/3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. pow3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      4. pow2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. pow-pow 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      6. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   4. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow1/3 91.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   6. Simplified 91.0%

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

      1. flip-+ 91.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      2. pow2 91.0%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. add-sqr-sqrt 91.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      4. pow1/3 89.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      5. pow-pow 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      6. metadata-eval 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. *-commutative 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. pow1/3 92.4%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. pow-pow 92.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      10. metadata-eval 92.9%

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

      11. *-commutative 92.9%

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

   8. Applied egg-rr 92.9%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. rem-cbrt-cube 50.7%

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

      2. clear-num 50.7%

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

      3. inv-pow 50.7%

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

      4. *-commutative 50.7%

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

      5. neg-mul-1 50.7%

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

      6. fma-define 50.7%

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

      7. pow2 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. associate-/l* 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in b around inf 93.0%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left(a \cdot \left(c \cdot \left(-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.2222222222222222 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + 0.5625 \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{6}} + \left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right)\right) - 2 \cdot \frac{1}{c}\right)}} \]

   11. Simplified 93.0%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\mathsf{fma}\left(-0.75, \left(c \cdot a\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.375\right), \mathsf{fma}\left(-0.2222222222222222, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 6.328125}{a \cdot {c}^{2}}, 0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right)\right) - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - \left(3 \cdot a\right) \cdot c\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\mathsf{fma}\left(-0.75, \left(a \cdot c\right) \cdot \left(\left(c \cdot {a}^{2}\right) \cdot -0.375\right), \mathsf{fma}\left(-0.2222222222222222, \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 6.328125}{a \cdot {c}^{2}}, 0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right)\right) - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot a\right) \cdot c\\ t_1 := {b}^{2} - t\_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{t\_1 - {\left(-b\right)}^{2}}{b + \sqrt{t\_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 3.0 a) c)) (t_1 (- (pow b 2.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 3.0 a)) -283.0)
     (/ (/ (- t_1 (pow (- b) 2.0)) (+ b (sqrt t_1))) (* 3.0 a))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (*
         a
         (+
          (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
          (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * a) * c;
	double t_1 = pow(b, 2.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -283.0) {
		tmp = ((t_1 - pow(-b, 2.0)) / (b + sqrt(t_1))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / 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 = (3.0d0 * a) * c
    t_1 = (b ** 2.0d0) - t_0
    if (((sqrt(((b * b) - t_0)) - b) / (3.0d0 * a)) <= (-283.0d0)) then
        tmp = ((t_1 - (-b ** 2.0d0)) / (b + sqrt(t_1))) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (3.0 * a) * c;
	double t_1 = Math.pow(b, 2.0) - t_0;
	double tmp;
	if (((Math.sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -283.0) {
		tmp = ((t_1 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_1))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0)))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (3.0 * a) * c
	t_1 = math.pow(b, 2.0) - t_0
	tmp = 0
	if ((math.sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -283.0:
		tmp = ((t_1 - math.pow(-b, 2.0)) / (b + math.sqrt(t_1))) / (3.0 * a)
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0)))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * a) * c)
	t_1 = Float64((b ^ 2.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(3.0 * a)) <= -283.0)
		tmp = Float64(Float64(Float64(t_1 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (3.0 * a) * c;
	t_1 = (b ^ 2.0) - t_0;
	tmp = 0.0;
	if (((sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -283.0)
		tmp = ((t_1 - (-b ^ 2.0)) / (b + sqrt(t_1))) / (3.0 * a);
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + (-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -283.0], N[(N[(N[(t$95$1 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 90.4%

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

      2. pow1/3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. pow3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      4. pow2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. pow-pow 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      6. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   4. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow1/3 91.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   6. Simplified 91.0%

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

      1. flip-+ 91.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      2. pow2 91.0%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. add-sqr-sqrt 91.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      4. pow1/3 89.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      5. pow-pow 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      6. metadata-eval 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. *-commutative 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. pow1/3 92.4%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. pow-pow 92.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      10. metadata-eval 92.9%

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

      11. *-commutative 92.9%

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

   8. Applied egg-rr 92.9%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around 0 92.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around 0 92.8%

      \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - \left(3 \cdot a\right) \cdot c\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot a\right) \cdot c\\ t_1 := {b}^{2} - t\_0\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{t\_1 - {\left(-b\right)}^{2}}{b + \sqrt{t\_1}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* 3.0 a) c)) (t_1 (- (pow b 2.0) t_0)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* 3.0 a)) -283.0)
     (/ (/ (- t_1 (pow (- b) 2.0)) (+ b (sqrt t_1))) (* 3.0 a))
     (/
      1.0
      (*
       b
       (-
        (fma
         -3.0
         (/ (* (* c (pow a 2.0)) -0.375) (pow b 4.0))
         (/ (* a 1.5) (pow b 2.0)))
        (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = (3.0 * a) * c;
	double t_1 = pow(b, 2.0) - t_0;
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (3.0 * a)) <= -283.0) {
		tmp = ((t_1 - pow(-b, 2.0)) / (b + sqrt(t_1))) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (fma(-3.0, (((c * pow(a, 2.0)) * -0.375) / pow(b, 4.0)), ((a * 1.5) / pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(3.0 * a) * c)
	t_1 = Float64((b ^ 2.0) - t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(3.0 * a)) <= -283.0)
		tmp = Float64(Float64(Float64(t_1 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(Float64(Float64(c * (a ^ 2.0)) * -0.375) / (b ^ 4.0)), Float64(Float64(a * 1.5) / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -283.0], N[(N[(N[(t$95$1 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 90.4%

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

      2. pow1/3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. pow3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      4. pow2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. pow-pow 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      6. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   4. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow1/3 91.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   6. Simplified 91.0%

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

      1. flip-+ 91.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      2. pow2 91.0%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. add-sqr-sqrt 91.0%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      4. pow1/3 89.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      5. pow-pow 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      6. metadata-eval 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      7. *-commutative 93.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      8. pow1/3 92.4%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      9. pow-pow 92.9%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      10. metadata-eval 92.9%

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

      11. *-commutative 92.9%

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

   8. Applied egg-rr 92.9%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. rem-cbrt-cube 50.7%

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

      2. clear-num 50.7%

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

      3. inv-pow 50.7%

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

      4. *-commutative 50.7%

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

      5. neg-mul-1 50.7%

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

      6. fma-define 50.7%

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

      7. pow2 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. associate-/l* 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in b around inf 90.7%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. fma-define 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\mathsf{fma}\left(-3, \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right)} - 2 \cdot \frac{1}{c}\right)} \]

       2. distribute-rgt-out 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       3. *-commutative 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       4. metadata-eval 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       5. associate-*r/ 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \color{blue}{\frac{1.5 \cdot a}{{b}^{2}}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       6. *-commutative 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{\color{blue}{a \cdot 1.5}}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       7. associate-*r/ 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       8. metadata-eval 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 90.7%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - \left(3 \cdot a\right) \cdot c\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{3 \cdot a}\\ t_1 := \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\\ \mathbf{if}\;\frac{t\_1}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* 3.0 a))) (t_1 (- (sqrt (- (* b b) (* (* 3.0 a) c))) b)))
   (if (<= (/ t_1 (* 3.0 a)) -283.0)
     (/ t_1 (* t_0 (pow t_0 2.0)))
     (/
      1.0
      (*
       b
       (-
        (fma
         -3.0
         (/ (* (* c (pow a 2.0)) -0.375) (pow b 4.0))
         (/ (* a 1.5) (pow b 2.0)))
        (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((3.0 * a));
	double t_1 = sqrt(((b * b) - ((3.0 * a) * c))) - b;
	double tmp;
	if ((t_1 / (3.0 * a)) <= -283.0) {
		tmp = t_1 / (t_0 * pow(t_0, 2.0));
	} else {
		tmp = 1.0 / (b * (fma(-3.0, (((c * pow(a, 2.0)) * -0.375) / pow(b, 4.0)), ((a * 1.5) / pow(b, 2.0))) - (2.0 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(3.0 * a))
	t_1 = Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b)
	tmp = 0.0
	if (Float64(t_1 / Float64(3.0 * a)) <= -283.0)
		tmp = Float64(t_1 / Float64(t_0 * (t_0 ^ 2.0)));
	else
		tmp = Float64(1.0 / Float64(b * Float64(fma(-3.0, Float64(Float64(Float64(c * (a ^ 2.0)) * -0.375) / (b ^ 4.0)), Float64(Float64(a * 1.5) / (b ^ 2.0))) - Float64(2.0 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(3.0 * a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -283.0], N[(t$95$1 / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 92.0%

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

      2. pow3 91.9%

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

   4. Applied egg-rr 91.9%

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

      1. rem-cbrt-cube 92.2%

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

      2. add-cube-cbrt 92.5%

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

      3. pow2 92.5%

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

      4. *-commutative 92.5%

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

      5. *-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. rem-cbrt-cube 50.7%

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

      2. clear-num 50.7%

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

      3. inv-pow 50.7%

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

      4. *-commutative 50.7%

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

      5. neg-mul-1 50.7%

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

      6. fma-define 50.7%

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

      7. pow2 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. associate-/l* 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in b around inf 90.7%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. fma-define 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\mathsf{fma}\left(-3, \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right)} - 2 \cdot \frac{1}{c}\right)} \]

       2. distribute-rgt-out 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       3. *-commutative 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\color{blue}{\left(c \cdot {a}^{2}\right)} \cdot \left(-0.75 + 0.375\right)}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       4. metadata-eval 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot \color{blue}{-0.375}}{{b}^{4}}, 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       5. associate-*r/ 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \color{blue}{\frac{1.5 \cdot a}{{b}^{2}}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       6. *-commutative 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{\color{blue}{a \cdot 1.5}}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)} \]

       7. associate-*r/ 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       8. metadata-eval 90.7%

          \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 90.7%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{\sqrt[3]{3 \cdot a} \cdot {\left(\sqrt[3]{3 \cdot a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\mathsf{fma}\left(-3, \frac{\left(c \cdot {a}^{2}\right) \cdot -0.375}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right) - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{3 \cdot a}\\ t_1 := \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\\ \mathbf{if}\;\frac{t\_1}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot {t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b \cdot \left(\frac{2}{a \cdot c} + \left(\frac{\left(a \cdot c\right) \cdot -1.125}{{b}^{4}} - \frac{1.5}{{b}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (cbrt (* 3.0 a))) (t_1 (- (sqrt (- (* b b) (* (* 3.0 a) c))) b)))
   (if (<= (/ t_1 (* 3.0 a)) -283.0)
     (/ t_1 (* t_0 (pow t_0 2.0)))
     (/
      -1.0
      (*
       a
       (*
        b
        (+
         (/ 2.0 (* a c))
         (- (/ (* (* a c) -1.125) (pow b 4.0)) (/ 1.5 (pow b 2.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = cbrt((3.0 * a));
	double t_1 = sqrt(((b * b) - ((3.0 * a) * c))) - b;
	double tmp;
	if ((t_1 / (3.0 * a)) <= -283.0) {
		tmp = t_1 / (t_0 * pow(t_0, 2.0));
	} else {
		tmp = -1.0 / (a * (b * ((2.0 / (a * c)) + ((((a * c) * -1.125) / pow(b, 4.0)) - (1.5 / pow(b, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.cbrt((3.0 * a));
	double t_1 = Math.sqrt(((b * b) - ((3.0 * a) * c))) - b;
	double tmp;
	if ((t_1 / (3.0 * a)) <= -283.0) {
		tmp = t_1 / (t_0 * Math.pow(t_0, 2.0));
	} else {
		tmp = -1.0 / (a * (b * ((2.0 / (a * c)) + ((((a * c) * -1.125) / Math.pow(b, 4.0)) - (1.5 / Math.pow(b, 2.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = cbrt(Float64(3.0 * a))
	t_1 = Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b)
	tmp = 0.0
	if (Float64(t_1 / Float64(3.0 * a)) <= -283.0)
		tmp = Float64(t_1 / Float64(t_0 * (t_0 ^ 2.0)));
	else
		tmp = Float64(-1.0 / Float64(a * Float64(b * Float64(Float64(2.0 / Float64(a * c)) + Float64(Float64(Float64(Float64(a * c) * -1.125) / (b ^ 4.0)) - Float64(1.5 / (b ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(3.0 * a), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -283.0], N[(t$95$1 / N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a * N[(b * N[(N[(2.0 / N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * c), $MachinePrecision] * -1.125), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(1.5 / N[Power[b, 2.0], $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 92.0%

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

      2. pow3 91.9%

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

   4. Applied egg-rr 91.9%

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

      1. rem-cbrt-cube 92.2%

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

      2. add-cube-cbrt 92.5%

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

      3. pow2 92.5%

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

      4. *-commutative 92.5%

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

      5. *-commutative 92.5%

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

   6. Applied egg-rr 92.5%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. rem-cbrt-cube 50.7%

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

      2. clear-num 50.7%

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

      3. inv-pow 50.7%

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

      4. *-commutative 50.7%

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

      5. neg-mul-1 50.7%

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

      6. fma-define 50.7%

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

      7. pow2 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. associate-/l* 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in b around inf 90.5%

      \[\leadsto \frac{1}{a \cdot \color{blue}{\left(b \cdot \left(\left(-1 \cdot \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{1}{{b}^{2}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\color{blue}{\left(1.5 \cdot \frac{1}{{b}^{2}} + -1 \cdot \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)} - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       2. mul-1-neg 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(1.5 \cdot \frac{1}{{b}^{2}} + \color{blue}{\left(-\frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       3. unsub-neg 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\color{blue}{\left(1.5 \cdot \frac{1}{{b}^{2}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)} - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       4. associate-*r/ 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\color{blue}{\frac{1.5 \cdot 1}{{b}^{2}}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       5. metadata-eval 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{\color{blue}{1.5}}{{b}^{2}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       6. distribute-rgt-out 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-2.25 + 1.125\right)}}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       7. *-commutative 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\color{blue}{\left(c \cdot a\right)} \cdot \left(-2.25 + 1.125\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       8. metadata-eval 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot \color{blue}{-1.125}}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       9. associate-*r/ 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \color{blue}{\frac{2 \cdot 1}{a \cdot c}}\right)\right)} \]

       10. metadata-eval 90.5%

           \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{\color{blue}{2}}{a \cdot c}\right)\right)} \]

       11. *-commutative 90.5%

           \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{2}{\color{blue}{c \cdot a}}\right)\right)} \]

   11. Simplified 90.5%

       \[\leadsto \frac{1}{a \cdot \color{blue}{\left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{2}{c \cdot a}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{\sqrt[3]{3 \cdot a} \cdot {\left(\sqrt[3]{3 \cdot a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b \cdot \left(\frac{2}{a \cdot c} + \left(\frac{\left(a \cdot c\right) \cdot -1.125}{{b}^{4}} - \frac{1.5}{{b}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b \cdot \left(\frac{2}{a \cdot c} + \left(\frac{\left(a \cdot c\right) \cdot -1.125}{{b}^{4}} - \frac{1.5}{{b}^{2}}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -283.0)
   (/ (- (sqrt (fma b b (* (* 3.0 a) (- c)))) b) (* 3.0 a))
   (/
    -1.0
    (*
     a
     (*
      b
      (+
       (/ 2.0 (* a c))
       (- (/ (* (* a c) -1.125) (pow b 4.0)) (/ 1.5 (pow b 2.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -283.0) {
		tmp = (sqrt(fma(b, b, ((3.0 * a) * -c))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (a * (b * ((2.0 / (a * c)) + ((((a * c) * -1.125) / pow(b, 4.0)) - (1.5 / pow(b, 2.0))))));
	}
	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)) <= -283.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(3.0 * a) * Float64(-c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-1.0 / Float64(a * Float64(b * Float64(Float64(2.0 / Float64(a * c)) + Float64(Float64(Float64(Float64(a * c) * -1.125) / (b ^ 4.0)) - Float64(1.5 / (b ^ 2.0)))))));
	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], -283.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(3.0 * a), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a * N[(b * N[(N[(2.0 / N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * c), $MachinePrecision] * -1.125), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(1.5 / N[Power[b, 2.0], $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -283

   1. Initial program 92.2%

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

      1. add-cbrt-cube 90.4%

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

      2. pow1/3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. pow3 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      4. pow2 88.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. pow-pow 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      6. metadata-eval 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   4. Applied egg-rr 88.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow1/3 91.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   6. Simplified 91.0%

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

      1. pow1/3 88.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. pow-pow 92.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. metadata-eval 92.2%

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

      4. pow2 92.2%

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

      5. cancel-sign-sub-inv 92.2%

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

      6. fma-define 92.4%

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

   8. Applied egg-rr 92.4%

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

   if -283 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.7%

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

      1. add-cbrt-cube 50.7%

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

      2. pow3 50.7%

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

   4. Applied egg-rr 50.7%

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

      1. rem-cbrt-cube 50.7%

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

      2. clear-num 50.7%

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

      3. inv-pow 50.7%

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

      4. *-commutative 50.7%

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

      5. neg-mul-1 50.7%

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

      6. fma-define 50.7%

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

      7. pow2 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   6. Applied egg-rr 50.7%

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

      1. unpow-1 50.7%

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

      2. associate-/l* 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in b around inf 90.5%

      \[\leadsto \frac{1}{a \cdot \color{blue}{\left(b \cdot \left(\left(-1 \cdot \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{1}{{b}^{2}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\color{blue}{\left(1.5 \cdot \frac{1}{{b}^{2}} + -1 \cdot \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)} - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       2. mul-1-neg 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(1.5 \cdot \frac{1}{{b}^{2}} + \color{blue}{\left(-\frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       3. unsub-neg 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\color{blue}{\left(1.5 \cdot \frac{1}{{b}^{2}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right)} - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       4. associate-*r/ 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\color{blue}{\frac{1.5 \cdot 1}{{b}^{2}}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       5. metadata-eval 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{\color{blue}{1.5}}{{b}^{2}} - \frac{-2.25 \cdot \left(a \cdot c\right) + 1.125 \cdot \left(a \cdot c\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       6. distribute-rgt-out 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-2.25 + 1.125\right)}}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       7. *-commutative 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\color{blue}{\left(c \cdot a\right)} \cdot \left(-2.25 + 1.125\right)}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       8. metadata-eval 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot \color{blue}{-1.125}}{{b}^{4}}\right) - 2 \cdot \frac{1}{a \cdot c}\right)\right)} \]

       9. associate-*r/ 90.5%

          \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \color{blue}{\frac{2 \cdot 1}{a \cdot c}}\right)\right)} \]

       10. metadata-eval 90.5%

           \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{\color{blue}{2}}{a \cdot c}\right)\right)} \]

       11. *-commutative 90.5%

           \[\leadsto \frac{1}{a \cdot \left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{2}{\color{blue}{c \cdot a}}\right)\right)} \]

   11. Simplified 90.5%

       \[\leadsto \frac{1}{a \cdot \color{blue}{\left(b \cdot \left(\left(\frac{1.5}{{b}^{2}} - \frac{\left(c \cdot a\right) \cdot -1.125}{{b}^{4}}\right) - \frac{2}{c \cdot a}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -283:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b \cdot \left(\frac{2}{a \cdot c} + \left(\frac{\left(a \cdot c\right) \cdot -1.125}{{b}^{4}} - \frac{1.5}{{b}^{2}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 36:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 36.0)
   (/ (- (sqrt (fma b b (* (* 3.0 a) (- c)))) b) (* 3.0 a))
   (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 36.0) {
		tmp = (sqrt(fma(b, b, ((3.0 * a) * -c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 36.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(3.0 * a) * Float64(-c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 36.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(3.0 * a), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 36

   1. Initial program 78.7%

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

      1. add-cbrt-cube 77.8%

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

      2. pow1/3 76.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. pow3 76.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      4. pow2 76.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      5. pow-pow 76.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      6. metadata-eval 76.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   4. Applied egg-rr 76.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow1/3 78.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   6. Simplified 78.0%

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

      1. pow1/3 76.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. pow-pow 78.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      3. metadata-eval 78.7%

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

      4. pow2 78.7%

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

      5. cancel-sign-sub-inv 78.7%

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

      6. fma-define 78.8%

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

   8. Applied egg-rr 78.8%

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

   if 36 < b

   1. Initial program 43.3%

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

      1. add-cbrt-cube 43.3%

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

      2. pow3 43.3%

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

   4. Applied egg-rr 43.3%

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

      1. rem-cbrt-cube 43.3%

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

      2. clear-num 43.3%

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

      3. inv-pow 43.3%

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

      4. *-commutative 43.3%

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

      5. neg-mul-1 43.3%

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

      6. fma-define 43.3%

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

      7. pow2 43.3%

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

      8. *-commutative 43.3%

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

      9. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. unpow-1 43.3%

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

      2. associate-/l* 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in b around inf 89.6%

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

       1. associate-*r/ 89.6%

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

       2. *-commutative 89.6%

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

       3. associate-*r/ 89.6%

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

       4. metadata-eval 89.6%

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

   11. Simplified 89.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 36:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 36:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 36.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 36.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 36.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 36.0], 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[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 36

   1. Initial program 78.7%

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

   3. Simplified 78.8%

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

   if 36 < b

   1. Initial program 43.3%

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

      1. add-cbrt-cube 43.3%

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

      2. pow3 43.3%

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

   4. Applied egg-rr 43.3%

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

      1. rem-cbrt-cube 43.3%

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

      2. clear-num 43.3%

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

      3. inv-pow 43.3%

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

      4. *-commutative 43.3%

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

      5. neg-mul-1 43.3%

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

      6. fma-define 43.3%

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

      7. pow2 43.3%

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

      8. *-commutative 43.3%

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

      9. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. unpow-1 43.3%

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

      2. associate-/l* 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in b around inf 89.6%

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

       1. associate-*r/ 89.6%

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

       2. *-commutative 89.6%

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

       3. associate-*r/ 89.6%

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

       4. metadata-eval 89.6%

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

   11. Simplified 89.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 36.0)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 36.0) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 36.0d0) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 36.0) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 36.0:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 36.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 36.0)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 36.0], 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], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 36

   1. Initial program 78.7%

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

   if 36 < b

   1. Initial program 43.3%

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

      1. add-cbrt-cube 43.3%

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

      2. pow3 43.3%

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

   4. Applied egg-rr 43.3%

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

      1. rem-cbrt-cube 43.3%

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

      2. clear-num 43.3%

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

      3. inv-pow 43.3%

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

      4. *-commutative 43.3%

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

      5. neg-mul-1 43.3%

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

      6. fma-define 43.3%

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

      7. pow2 43.3%

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

      8. *-commutative 43.3%

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

      9. *-commutative 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. unpow-1 43.3%

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

      2. associate-/l* 43.3%

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

   8. Simplified 43.3%

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

   9. Taylor expanded in b around inf 89.6%

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

       1. associate-*r/ 89.6%

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

       2. *-commutative 89.6%

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

       3. associate-*r/ 89.6%

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

       4. metadata-eval 89.6%

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

   11. Simplified 89.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 36:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c)))))

C

double code(double a, double b, double c) {
	return 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
}

Python

def code(a, b, c):
	return 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

   1. add-cbrt-cube 53.1%

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

   2. pow3 53.1%

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

4. Applied egg-rr 53.1%

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

   1. rem-cbrt-cube 53.1%

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

   2. clear-num 53.1%

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

   3. inv-pow 53.1%

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

   4. *-commutative 53.1%

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

   5. neg-mul-1 53.1%

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

   6. fma-define 53.1%

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

   7. pow2 53.1%

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

   8. *-commutative 53.1%

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

   9. *-commutative 53.1%

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

6. Applied egg-rr 53.1%

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

   1. unpow-1 53.1%

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

   2. associate-/l* 53.1%

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

8. Simplified 53.1%

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

9. Taylor expanded in b around inf 82.6%

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

    1. associate-*r/ 82.6%

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

    2. *-commutative 82.6%

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

    3. associate-*r/ 82.6%

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

    4. metadata-eval 82.6%

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

11. Simplified 82.6%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
12. Add Preprocessing

Alternative 11: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (/ (* a c) (pow b 3.0))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / pow(b, 3.0))) - (0.5 / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) - (0.5 / b));
end

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

5. Taylor expanded in c around 0 82.0%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 82.0%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   2. metadata-eval 82.0%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{\color{blue}{0.5}}{b}\right) \]

7. Simplified 82.0%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)} \]
8. Add Preprocessing

Alternative 12: 64.3% 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 53.1%

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

3. Simplified 53.2%

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

5. Taylor expanded in b around inf 66.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(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)))