Cubic critical, narrow range

Percentage Accurate: 55.9% → 91.1%

Time: 18.7s

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: 55.9% 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: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - 3 \cdot \left(a \cdot c\right)\\ t_1 := \sqrt[3]{{\left(3 \cdot a\right)}^{1.5}}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* 3.0 (* a c)))) (t_1 (cbrt (pow (* 3.0 a) 1.5))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* t_1 t_1))
     (+
      (* -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 = pow(b, 2.0) - (3.0 * (a * c));
	double t_1 = cbrt(pow((3.0 * a), 1.5));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0105) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (t_1 * t_1);
	} 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;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (3.0 * (a * c));
	double t_1 = Math.cbrt(Math.pow((3.0 * a), 1.5));
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0105) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (t_1 * t_1);
	} 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((b ^ 2.0) - Float64(3.0 * Float64(a * c)))
	t_1 = cbrt((Float64(3.0 * a) ^ 1.5))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0105)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(t_1 * t_1));
	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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[(3.0 * a), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0105], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $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)) < -0.0105000000000000007

   1. Initial program 82.6%

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

      1. log1p-expm1-u 77.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 70.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 70.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1 70.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}^{1}}} \]

      2. log1p-define 77.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)\right)}}^{1}} \]

      3. log1p-expm1-u 82.6%

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

      4. metadata-eval 82.6%

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

      5. pow-pow 82.3%

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

      6. add-sqr-sqrt 82.3%

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

      7. unpow-prod-down 82.3%

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

      8. pow1/2 82.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left({\left(3 \cdot a\right)}^{3}\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      9. metadata-eval 82.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left({\left(3 \cdot a\right)}^{3}\right)}^{\color{blue}{\left(0.3333333333333333 \cdot 1.5\right)}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      10. metadata-eval 82.3%

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

      11. pow-pow 82.3%

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

      12. pow-pow 82.3%

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

      13. metadata-eval 82.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left({\left(3 \cdot a\right)}^{\color{blue}{1}}\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      14. pow-pow 82.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{\left(1 \cdot \frac{3}{2}\right)}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      15. *-commutative 82.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\color{blue}{\left(a \cdot 3\right)}}^{\left(1 \cdot \frac{3}{2}\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      16. metadata-eval 82.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(a \cdot 3\right)}^{\left(1 \cdot \color{blue}{1.5}\right)}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

      17. metadata-eval 82.3%

          \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\left({\left(a \cdot 3\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(3 \cdot a\right)}^{3}}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 82.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(a \cdot 3\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(a \cdot 3\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   7. Step-by-step derivation

      1. unpow1/3 82.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\sqrt[3]{{\left(a \cdot 3\right)}^{1.5}}} \cdot {\left({\left(a \cdot 3\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      2. unpow1/3 82.6%

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

   8. Simplified 82.6%

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

      1. flip-+ 82.2%

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

      2. pow2 82.2%

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

      3. add-sqr-sqrt 83.1%

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

      4. pow2 83.1%

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

      5. associate-*l* 83.1%

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

      6. pow2 83.1%

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

      7. associate-*l* 83.1%

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

   10. Applied egg-rr 83.1%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in a around 0 95.9%

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

   6. Taylor expanded in c around 0 95.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{{\left(3 \cdot a\right)}^{1.5}} \cdot \sqrt[3]{{\left(3 \cdot a\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 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 tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0105) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (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;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0105)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0105], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(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)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   3. Simplified 82.7%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in a around 0 95.9%

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

   6. Taylor expanded in c around 0 95.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0105) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0105)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0105], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   3. Simplified 82.7%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in a around 0 93.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \mathsf{fma}\left(-3, \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/
    -1.0
    (*
     b
     (-
      (/ 2.0 c)
      (fma
       -3.0
       (/ (* -0.375 (* c (pow a 2.0))) (pow b 4.0))
       (/ (* a 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)) <= -0.0105) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - fma(-3.0, ((-0.375 * (c * pow(a, 2.0))) / pow(b, 4.0)), ((a * 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)) <= -0.0105)
		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(2.0 / c) - fma(-3.0, Float64(Float64(-0.375 * Float64(c * (a ^ 2.0))) / (b ^ 4.0)), Float64(Float64(a * 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], -0.0105], 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[(2.0 / c), $MachinePrecision] - N[(-3.0 * N[(N[(-0.375 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   3. Simplified 82.7%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

      1. log1p-expm1-u 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 29.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 29.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 44.9%

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

      3. clear-num 44.9%

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

      4. inv-pow 44.9%

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

      5. *-commutative 44.9%

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

      6. neg-mul-1 44.9%

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

      7. fma-define 44.9%

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

      8. pow2 44.9%

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

      9. associate-*l* 44.9%

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

   6. Applied egg-rr 44.9%

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

      1. unpow-1 44.9%

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

      2. sub-neg 44.9%

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

      3. +-commutative 44.9%

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

      4. distribute-lft-neg-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. fma-define 44.9%

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

   8. Simplified 44.9%

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

   9. Taylor expanded in b around inf 93.9%

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

       1. fma-define 93.9%

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

       2. distribute-rgt-out 93.9%

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

       3. *-commutative 93.9%

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

       4. metadata-eval 93.9%

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

       5. associate-*r/ 93.9%

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

       6. *-commutative 93.9%

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

       7. associate-*r/ 93.9%

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

       8. metadata-eval 93.9%

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

   11. Simplified 93.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \mathsf{fma}\left(-3, \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}}, \frac{a \cdot 1.5}{{b}^{2}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
       (* -0.375 (/ a (pow b 3.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0105) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0105)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0105], 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[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $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)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   3. Simplified 82.7%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in c around 0 93.7%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0105)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ -1.0 (* b (- (/ 2.0 c) (/ (* a 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)) <= -0.0105) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 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)) <= -0.0105)
		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(2.0 / c) - Float64(Float64(a * 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], -0.0105], 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[(2.0 / c), $MachinePrecision] - N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   3. Simplified 82.7%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

      1. log1p-expm1-u 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 29.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 29.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 44.9%

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

      3. clear-num 44.9%

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

      4. inv-pow 44.9%

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

      5. *-commutative 44.9%

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

      6. neg-mul-1 44.9%

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

      7. fma-define 44.9%

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

      8. pow2 44.9%

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

      9. associate-*l* 44.9%

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

   6. Applied egg-rr 44.9%

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

      1. unpow-1 44.9%

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

      2. sub-neg 44.9%

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

      3. +-commutative 44.9%

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

      4. distribute-lft-neg-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. fma-define 44.9%

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

   8. Simplified 44.9%

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

   9. Taylor expanded in b around inf 89.2%

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

       1. associate-*r/ 89.2%

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

       2. *-commutative 89.2%

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

       3. associate-*r/ 89.2%

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

       4. metadata-eval 89.2%

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

   11. Simplified 89.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.0105:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= t_0 -0.0105)
     t_0
     (/ -1.0 (* b (- (/ 2.0 c) (/ (* a 1.5) (pow b 2.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.0105) {
		tmp = t_0;
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / pow(b, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.0105d0)) then
        tmp = t_0
    else
        tmp = (-1.0d0) / (b * ((2.0d0 / c) - ((a * 1.5d0) / (b ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.0105) {
		tmp = t_0;
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / Math.pow(b, 2.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.0105:
		tmp = t_0
	else:
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / math.pow(b, 2.0))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.0105)
		tmp = t_0;
	else
		tmp = Float64(-1.0 / Float64(b * Float64(Float64(2.0 / c) - Float64(Float64(a * 1.5) / (b ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.0105)
		tmp = t_0;
	else
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / (b ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0105], t$95$0, N[(-1.0 / N[(b * N[(N[(2.0 / c), $MachinePrecision] - N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0105000000000000007

   1. Initial program 82.6%

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

   if -0.0105000000000000007 < (/.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 44.9%

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

      1. log1p-expm1-u 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 29.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 29.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 29.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 44.9%

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

      3. clear-num 44.9%

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

      4. inv-pow 44.9%

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

      5. *-commutative 44.9%

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

      6. neg-mul-1 44.9%

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

      7. fma-define 44.9%

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

      8. pow2 44.9%

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

      9. associate-*l* 44.9%

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

   6. Applied egg-rr 44.9%

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

      1. unpow-1 44.9%

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

      2. sub-neg 44.9%

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

      3. +-commutative 44.9%

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

      4. distribute-lft-neg-in 44.9%

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

      5. metadata-eval 44.9%

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

      6. fma-define 44.9%

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

   8. Simplified 44.9%

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

   9. Taylor expanded in b around inf 89.2%

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

       1. associate-*r/ 89.2%

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

       2. *-commutative 89.2%

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

       3. associate-*r/ 89.2%

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

       4. metadata-eval 89.2%

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

   11. Simplified 89.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0105:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 112.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 112.2)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/ -1.0 (* b (- (/ 2.0 c) (/ (* a 1.5) (pow b 2.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 112.2) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / pow(b, 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 112.2d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-1.0d0) / (b * ((2.0d0 / c) - ((a * 1.5d0) / (b ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 112.2) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / Math.pow(b, 2.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 112.2:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / math.pow(b, 2.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 112.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-1.0 / Float64(b * Float64(Float64(2.0 / c) - Float64(Float64(a * 1.5) / (b ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 112.2)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -1.0 / (b * ((2.0 / c) - ((a * 1.5) / (b ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 112.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(b * N[(N[(2.0 / c), $MachinePrecision] - N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 112.200000000000003

   1. Initial program 78.2%

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

      1. sqr-neg 78.2%

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

      2. sqr-neg 78.2%

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

      3. associate-*l* 78.2%

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

   3. Simplified 78.2%

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

   if 112.200000000000003 < b

   1. Initial program 42.6%

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

      1. log1p-expm1-u 28.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-undefine 27.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   4. Applied egg-rr 27.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
   5. Step-by-step derivation

      1. log1p-define 28.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

      2. log1p-expm1-u 42.6%

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

      3. clear-num 42.6%

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

      4. inv-pow 42.6%

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

      5. *-commutative 42.6%

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

      6. neg-mul-1 42.6%

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

      7. fma-define 42.6%

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

      8. pow2 42.6%

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

      9. associate-*l* 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. unpow-1 42.5%

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

      2. sub-neg 42.5%

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

      3. +-commutative 42.5%

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

      4. distribute-lft-neg-in 42.5%

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

      5. metadata-eval 42.5%

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

      6. fma-define 42.6%

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

   8. Simplified 42.6%

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

   9. Taylor expanded in b around inf 90.5%

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

       1. associate-*r/ 90.5%

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

       2. *-commutative 90.5%

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

       3. associate-*r/ 90.5%

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

       4. metadata-eval 90.5%

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

   11. Simplified 90.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 112.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{b \cdot \left(\frac{2}{c} - \frac{a \cdot 1.5}{{b}^{2}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((2.0d0 / c) - ((a * 1.5d0) / (b ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. log1p-expm1-u 43.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 41.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 41.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 43.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-expm1-u 55.5%

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

   3. clear-num 55.5%

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

   4. inv-pow 55.5%

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

   5. *-commutative 55.5%

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

   6. neg-mul-1 55.5%

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

   7. fma-define 55.5%

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

   8. pow2 55.5%

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

   9. associate-*l* 55.5%

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

6. Applied egg-rr 55.5%

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

   1. unpow-1 55.5%

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

   2. sub-neg 55.5%

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

   3. +-commutative 55.5%

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

   4. distribute-lft-neg-in 55.5%

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

   5. metadata-eval 55.5%

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

   6. fma-define 55.5%

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

8. Simplified 55.5%

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

9. Taylor expanded in b around inf 81.0%

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

    1. associate-*r/ 81.0%

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

    2. *-commutative 81.0%

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

    3. associate-*r/ 81.0%

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

    4. metadata-eval 81.0%

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

11. Simplified 81.0%

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

12. Final simplification 81.0%

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

Alternative 10: 81.0% accurate, 1.0× speedup

Math

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

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(a * Float64(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[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

3. Simplified 55.4%

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

5. Taylor expanded in c around 0 80.4%

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

   1. associate-/l* 80.4%

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

   2. associate-*r/ 80.4%

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

   3. metadata-eval 80.4%

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

7. Simplified 80.4%

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

Alternative 11: 63.9% 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 55.5%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around inf 64.5%

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

Alternative 12: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. log1p-expm1-u 43.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-undefine 41.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

4. Applied egg-rr 41.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\log \left(1 + \mathsf{expm1}\left(3 \cdot a\right)\right)}} \]
5. Step-by-step derivation

   1. log1p-define 43.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(3 \cdot a\right)\right)}} \]

   2. log1p-expm1-u 55.5%

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

   3. clear-num 55.5%

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

   4. inv-pow 55.5%

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

   5. *-commutative 55.5%

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

   6. neg-mul-1 55.5%

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

   7. fma-define 55.5%

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

   8. pow2 55.5%

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

   9. associate-*l* 55.5%

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

6. Applied egg-rr 55.5%

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

   1. unpow-1 55.5%

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

   2. sub-neg 55.5%

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

   3. +-commutative 55.5%

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

   4. distribute-lft-neg-in 55.5%

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

   5. metadata-eval 55.5%

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

   6. fma-define 55.5%

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

8. Simplified 55.5%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))