Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.8%

Time: 17.5s

Alternatives: 17

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ t_1 := \frac{c}{{b}^{3}} \cdot -0.5\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {t\_0}^{1.5}}{t\_0 + \mathsf{fma}\left(b, b, b \cdot \sqrt{t\_0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -2 \cdot \left(t\_1 + a \cdot \mathsf{fma}\left(-1, c \cdot \frac{t\_1}{{b}^{2}}, \mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right)\right)\right), \frac{1}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))) (t_1 (* (/ c (pow b 3.0)) -0.5)))
   (if (<= b 1.15)
     (/
      -1.0
      (/
       (* a 2.0)
       (/ (- (pow b 3.0) (pow t_0 1.5)) (+ t_0 (fma b b (* b (sqrt t_0)))))))
     (/
      1.0
      (fma
       -1.0
       (/ b c)
       (*
        a
        (fma
         a
         (*
          -2.0
          (+
           t_1
           (*
            a
            (fma
             -1.0
             (* c (/ t_1 (pow b 2.0)))
             (fma
              -0.125
              (* b (/ (* (/ (pow c 4.0) (pow b 6.0)) 20.0) (pow c 2.0)))
              (/ (pow c 2.0) (pow b 5.0)))))))
         (/ 1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double t_1 = (c / pow(b, 3.0)) * -0.5;
	double tmp;
	if (b <= 1.15) {
		tmp = -1.0 / ((a * 2.0) / ((pow(b, 3.0) - pow(t_0, 1.5)) / (t_0 + fma(b, b, (b * sqrt(t_0))))));
	} else {
		tmp = 1.0 / fma(-1.0, (b / c), (a * fma(a, (-2.0 * (t_1 + (a * fma(-1.0, (c * (t_1 / pow(b, 2.0))), fma(-0.125, (b * (((pow(c, 4.0) / pow(b, 6.0)) * 20.0) / pow(c, 2.0))), (pow(c, 2.0) / pow(b, 5.0))))))), (1.0 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	t_1 = Float64(Float64(c / (b ^ 3.0)) * -0.5)
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(-1.0 / Float64(Float64(a * 2.0) / Float64(Float64((b ^ 3.0) - (t_0 ^ 1.5)) / Float64(t_0 + fma(b, b, Float64(b * sqrt(t_0)))))));
	else
		tmp = Float64(1.0 / fma(-1.0, Float64(b / c), Float64(a * fma(a, Float64(-2.0 * Float64(t_1 + Float64(a * fma(-1.0, Float64(c * Float64(t_1 / (b ^ 2.0))), fma(-0.125, Float64(b * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0) / (c ^ 2.0))), Float64((c ^ 2.0) / (b ^ 5.0))))))), Float64(1.0 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(-1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[(N[Power[b, 3.0], $MachinePrecision] - N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision] + N[(a * N[(a * N[(-2.0 * N[(t$95$1 + N[(a * N[(-1.0 * N[(c * N[(t$95$1 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(b * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. flip3-- 86.0%

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

       2. pow1/2 86.0%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{{\color{blue}{\left({\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{0.5}\right)}}^{3} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       3. pow-pow 86.5%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{\color{blue}{{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{\left(0.5 \cdot 3\right)}} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       4. metadata-eval 86.5%

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

       5. add-sqr-sqrt 86.5%

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

       6. fma-define 86.5%

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

   14. Applied egg-rr 86.5%

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

       1. unpow2 86.5%

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

       2. fmm-undef 86.8%

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

       3. distribute-rgt-neg-in 86.8%

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

       4. distribute-rgt-neg-in 86.8%

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

       5. metadata-eval 86.8%

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

       6. unpow2 86.8%

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

       7. fmm-undef 86.9%

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

       8. distribute-rgt-neg-in 86.9%

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

       9. distribute-rgt-neg-in 86.9%

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

       10. metadata-eval 86.9%

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

   16. Simplified 86.9%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 95.0%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(-0.125 \cdot \frac{b \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -2 \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

   15. Simplified 95.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-1, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -2 \cdot \left(a \cdot \mathsf{fma}\left(-1, c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.5}{{b}^{2}}, \mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right)\right) + \frac{c}{{b}^{3}} \cdot -0.5\right), \frac{1}{b}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + \mathsf{fma}\left(b, b, b \cdot \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, -2 \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5 + a \cdot \mathsf{fma}\left(-1, c \cdot \frac{\frac{c}{{b}^{3}} \cdot -0.5}{{b}^{2}}, \mathsf{fma}\left(-0.125, b \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{c}^{2}}, \frac{{c}^{2}}{{b}^{5}}\right)\right)\right), \frac{1}{b}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ t_1 := \frac{{c}^{4}}{{b}^{6}}\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {t\_0}^{1.5}}{t\_0 + \mathsf{fma}\left(b, b, b \cdot \sqrt{t\_0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot t\_1 + t\_1 \cdot 16\right)}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))) (t_1 (/ (pow c 4.0) (pow b 6.0))))
   (if (<= b 1.15)
     (/
      -1.0
      (/
       (* a 2.0)
       (/ (- (pow b 3.0) (pow t_0 1.5)) (+ t_0 (fma b b (* b (sqrt t_0)))))))
     (-
      (*
       a
       (-
        (*
         a
         (+
          (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
          (* -0.25 (/ (* a (+ (* 4.0 t_1) (* t_1 16.0))) b))))
        (/ (pow c 2.0) (pow b 3.0))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double t_1 = pow(c, 4.0) / pow(b, 6.0);
	double tmp;
	if (b <= 1.15) {
		tmp = -1.0 / ((a * 2.0) / ((pow(b, 3.0) - pow(t_0, 1.5)) / (t_0 + fma(b, b, (b * sqrt(t_0))))));
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * ((a * ((4.0 * t_1) + (t_1 * 16.0))) / b)))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	t_1 = Float64((c ^ 4.0) / (b ^ 6.0))
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(-1.0 / Float64(Float64(a * 2.0) / Float64(Float64((b ^ 3.0) - (t_0 ^ 1.5)) / Float64(t_0 + fma(b, b, Float64(b * sqrt(t_0)))))));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-0.25 * Float64(Float64(a * Float64(Float64(4.0 * t_1) + Float64(t_1 * 16.0))) / b)))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(-1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[(N[Power[b, 3.0], $MachinePrecision] - N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(a * N[(N[(4.0 * t$95$1), $MachinePrecision] + N[(t$95$1 * 16.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. flip3-- 86.0%

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

       2. pow1/2 86.0%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{{\color{blue}{\left({\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{0.5}\right)}}^{3} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       3. pow-pow 86.5%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{\color{blue}{{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{\left(0.5 \cdot 3\right)}} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       4. metadata-eval 86.5%

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

       5. add-sqr-sqrt 86.5%

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

       6. fma-define 86.5%

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

   14. Applied egg-rr 86.5%

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

       1. unpow2 86.5%

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

       2. fmm-undef 86.8%

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

       3. distribute-rgt-neg-in 86.8%

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

       4. distribute-rgt-neg-in 86.8%

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

       5. metadata-eval 86.8%

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

       6. unpow2 86.8%

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

       7. fmm-undef 86.9%

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

       8. distribute-rgt-neg-in 86.9%

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

       9. distribute-rgt-neg-in 86.9%

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

       10. metadata-eval 86.9%

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

   16. Simplified 86.9%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in a around 0 94.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + \mathsf{fma}\left(b, b, b \cdot \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{{c}^{4}}{{b}^{6}} \cdot 16\right)}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {t\_0}^{1.5}}{t\_0 + \mathsf{fma}\left(b, b, b \cdot \sqrt{t\_0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) - \left(c + \left(a \cdot \left({c}^{2} \cdot {b}^{-2}\right) - -0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= b 1.5)
     (/
      -1.0
      (/
       (* a 2.0)
       (/ (- (pow b 3.0) (pow t_0 1.5)) (+ t_0 (fma b b (* b (sqrt t_0)))))))
     (/
      (-
       (* -2.0 (* (* a a) (* (pow c 3.0) (pow b -4.0))))
       (+
        c
        (-
         (* a (* (pow c 2.0) (pow b -2.0)))
         (* -0.25 (* (/ (pow (* a c) 4.0) a) (* 20.0 (pow b -6.0)))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (b <= 1.5) {
		tmp = -1.0 / ((a * 2.0) / ((pow(b, 3.0) - pow(t_0, 1.5)) / (t_0 + fma(b, b, (b * sqrt(t_0))))));
	} else {
		tmp = ((-2.0 * ((a * a) * (pow(c, 3.0) * pow(b, -4.0)))) - (c + ((a * (pow(c, 2.0) * pow(b, -2.0))) - (-0.25 * ((pow((a * c), 4.0) / a) * (20.0 * pow(b, -6.0))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (b <= 1.5)
		tmp = Float64(-1.0 / Float64(Float64(a * 2.0) / Float64(Float64((b ^ 3.0) - (t_0 ^ 1.5)) / Float64(t_0 + fma(b, b, Float64(b * sqrt(t_0)))))));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * Float64((c ^ 3.0) * (b ^ -4.0)))) - Float64(c + Float64(Float64(a * Float64((c ^ 2.0) * (b ^ -2.0))) - Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 * (b ^ -6.0))))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.5], N[(-1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[(N[Power[b, 3.0], $MachinePrecision] - N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * b + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c + N[(N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. flip3-- 86.0%

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

       2. pow1/2 86.0%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{{\color{blue}{\left({\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{0.5}\right)}}^{3} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       3. pow-pow 86.5%

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\frac{\color{blue}{{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right)}^{\left(0.5 \cdot 3\right)}} - {b}^{3}}{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} + \left(b \cdot b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} \cdot b\right)}}} \]

       4. metadata-eval 86.5%

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

       5. add-sqr-sqrt 86.5%

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

       6. fma-define 86.5%

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

   14. Applied egg-rr 86.5%

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

       1. unpow2 86.5%

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

       2. fmm-undef 86.8%

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

       3. distribute-rgt-neg-in 86.8%

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

       4. distribute-rgt-neg-in 86.8%

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

       5. metadata-eval 86.8%

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

       6. unpow2 86.8%

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

       7. fmm-undef 86.9%

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

       8. distribute-rgt-neg-in 86.9%

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

       9. distribute-rgt-neg-in 86.9%

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

       10. metadata-eval 86.9%

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

   16. Simplified 86.9%

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

   if 1.5 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in b around inf 94.9%

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

   7. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, \left(-a\right) \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   9. Applied egg-rr 94.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot \left({a}^{2} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}}{b} \]
   10. Step-by-step derivation

       1. unpow2 94.9%

          \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   11. Applied egg-rr 94.9%

       \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 2}{\frac{{b}^{3} - {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + \mathsf{fma}\left(b, b, b \cdot \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) - \left(c + \left(a \cdot \left({c}^{2} \cdot {b}^{-2}\right) - -0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) - \left(c + \left(a \cdot \left({c}^{2} \cdot {b}^{-2}\right) - -0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 1.2)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (-
       (* -2.0 (* (* a a) (* (pow c 3.0) (pow b -4.0))))
       (+
        c
        (-
         (* a (* (pow c 2.0) (pow b -2.0)))
         (* -0.25 (* (/ (pow (* a c) 4.0) a) (* 20.0 (pow b -6.0)))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.2) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((-2.0 * ((a * a) * (pow(c, 3.0) * pow(b, -4.0)))) - (c + ((a * (pow(c, 2.0) * pow(b, -2.0))) - (-0.25 * ((pow((a * c), 4.0) / a) * (20.0 * pow(b, -6.0))))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 1.2d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = (((-2.0d0) * ((a * a) * ((c ** 3.0d0) * (b ** (-4.0d0))))) - (c + ((a * ((c ** 2.0d0) * (b ** (-2.0d0)))) - ((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 * (b ** (-6.0d0)))))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.2) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((-2.0 * ((a * a) * (Math.pow(c, 3.0) * Math.pow(b, -4.0)))) - (c + ((a * (Math.pow(c, 2.0) * Math.pow(b, -2.0))) - (-0.25 * ((Math.pow((a * c), 4.0) / a) * (20.0 * Math.pow(b, -6.0))))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 1.2:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = ((-2.0 * ((a * a) * (math.pow(c, 3.0) * math.pow(b, -4.0)))) - (c + ((a * (math.pow(c, 2.0) * math.pow(b, -2.0))) - (-0.25 * ((math.pow((a * c), 4.0) / a) * (20.0 * math.pow(b, -6.0))))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 1.2)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * Float64((c ^ 3.0) * (b ^ -4.0)))) - Float64(c + Float64(Float64(a * Float64((c ^ 2.0) * (b ^ -2.0))) - Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 * (b ^ -6.0))))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 1.2)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	else
		tmp = ((-2.0 * ((a * a) * ((c ^ 3.0) * (b ^ -4.0)))) - (c + ((a * ((c ^ 2.0) * (b ^ -2.0))) - (-0.25 * ((((a * c) ^ 4.0) / a) * (20.0 * (b ^ -6.0))))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.2], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c + N[(N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.19999999999999996

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 81.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 81.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. add-sqr-sqrt 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. Applied egg-rr 86.1%

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

   if 1.19999999999999996 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in b around inf 94.9%

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

   7. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, \left(-a\right) \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   9. Applied egg-rr 94.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot \left({a}^{2} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}}{b} \]
   10. Step-by-step derivation

       1. unpow2 94.9%

          \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   11. Applied egg-rr 94.9%

       \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) - \left(c + \left(a \cdot \left({c}^{2} \cdot {b}^{-2}\right) - -0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 1.15)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (/ (pow c 2.0) (pow b 2.0))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.15) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - (pow(c, 2.0) / pow(b, 2.0)))) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 1.15d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = ((a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c ** 2.0d0) / (b ** 2.0d0)))) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.15) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((a * ((a * ((-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - (Math.pow(c, 2.0) / Math.pow(b, 2.0)))) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 1.15:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = ((a * ((a * ((-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(c, 3.0) / math.pow(b, 4.0))))) - (math.pow(c, 2.0) / math.pow(b, 2.0)))) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 4.0))))) - Float64((c ^ 2.0) / (b ^ 2.0)))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 1.15)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	else
		tmp = ((a * ((a * ((-5.0 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-2.0 * ((c ^ 3.0) / (b ^ 4.0))))) - ((c ^ 2.0) / (b ^ 2.0)))) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(a * N[(N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 81.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 81.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. add-sqr-sqrt 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. Applied egg-rr 86.1%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in b around inf 94.9%

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

   7. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, \left(-a\right) \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   9. Applied egg-rr 94.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot \left({a}^{2} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}}{b} \]
   10. Step-by-step derivation

       1. unpow2 94.9%

          \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   11. Applied egg-rr 94.9%

       \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   12. Taylor expanded in a around 0 94.9%

       \[\leadsto \frac{\color{blue}{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 1.15)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (*
       c
       (+
        -1.0
        (*
         c
         (-
          (*
           c
           (+
            (* -5.0 (/ (* c (pow a 3.0)) (pow b 6.0)))
            (* -2.0 (/ (pow a 2.0) (pow b 4.0)))))
          (/ a (pow b 2.0))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.15) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 6.0))) + (-2.0 * (pow(a, 2.0) / pow(b, 4.0))))) - (a / pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 1.15d0) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = (c * ((-1.0d0) + (c * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (b ** 4.0d0))))) - (a / (b ** 2.0d0)))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.15) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))) - (a / Math.pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 1.15:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(a, 2.0) / math.pow(b, 4.0))))) - (a / math.pow(b, 2.0)))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 4.0))))) - Float64(a / (b ^ 2.0)))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 1.15)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	else
		tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 6.0))) + (-2.0 * ((a ^ 2.0) / (b ^ 4.0))))) - (a / (b ^ 2.0)))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(-1.0 + N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 81.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 81.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. add-sqr-sqrt 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. Applied egg-rr 86.1%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in b around inf 94.9%

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

   7. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, \left(-a\right) \cdot \frac{{c}^{2}}{{b}^{2}}\right) - c\right)}{b}} \]

   9. Applied egg-rr 94.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot \left({a}^{2} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}}{b} \]
   10. Step-by-step derivation

       1. unpow2 94.9%

          \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   11. Applied egg-rr 94.9%

       \[\leadsto \frac{-2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left({c}^{3} \cdot {b}^{-4}\right)\right) + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \left(20 \cdot {b}^{-6}\right)\right) - a \cdot \left({c}^{2} \cdot {b}^{-2}\right)\right) - c\right)}{b} \]

   12. Taylor expanded in c around 0 94.8%

       \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right) - 1\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ t_1 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\frac{t\_1 - {b}^{2}}{b + \sqrt{t\_1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))) (t_1 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 1.4)
     (/ 1.0 (/ (* a 2.0) (/ (- t_1 (pow b 2.0)) (+ b (sqrt t_1)))))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double t_1 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.4) {
		tmp = 1.0 / ((a * 2.0) / ((t_1 - pow(b, 2.0)) / (b + sqrt(t_1))));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c / (b ** 3.0d0)
    t_1 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 1.4d0) then
        tmp = 1.0d0 / ((a * 2.0d0) / ((t_1 - (b ** 2.0d0)) / (b + sqrt(t_1))))
    else
        tmp = (-1.0d0) / ((b / c) + (a * (((-2.0d0) * (a * (t_0 - (t_0 * 0.5d0)))) + ((-1.0d0) / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / Math.pow(b, 3.0);
	double t_1 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.4) {
		tmp = 1.0 / ((a * 2.0) / ((t_1 - Math.pow(b, 2.0)) / (b + Math.sqrt(t_1))));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c / math.pow(b, 3.0)
	t_1 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 1.4:
		tmp = 1.0 / ((a * 2.0) / ((t_1 - math.pow(b, 2.0)) / (b + math.sqrt(t_1))))
	else:
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	t_1 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(Float64(t_1 - (b ^ 2.0)) / Float64(b + sqrt(t_1)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c / (b ^ 3.0);
	t_1 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 1.4)
		tmp = 1.0 / ((a * 2.0) / ((t_1 - (b ^ 2.0)) / (b + sqrt(t_1))));
	else
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.4], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. flip-- 85.4%

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

       2. add-sqr-sqrt 86.1%

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

       3. unpow2 86.1%

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

   14. Applied egg-rr 86.1%

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

   if 1.3999999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {b}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ t_1 := {b}^{2} - c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq 1.65:\\ \;\;\;\;\frac{\frac{t\_1 - {\left(-b\right)}^{2}}{b + \sqrt{t\_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))) (t_1 (- (pow b 2.0) (* c (* a 4.0)))))
   (if (<= b 1.65)
     (/ (/ (- t_1 (pow (- b) 2.0)) (+ b (sqrt t_1))) (* a 2.0))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double t_1 = pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.65) {
		tmp = ((t_1 - pow(-b, 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c / (b ** 3.0d0)
    t_1 = (b ** 2.0d0) - (c * (a * 4.0d0))
    if (b <= 1.65d0) then
        tmp = ((t_1 - (-b ** 2.0d0)) / (b + sqrt(t_1))) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) + (a * (((-2.0d0) * (a * (t_0 - (t_0 * 0.5d0)))) + ((-1.0d0) / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c / Math.pow(b, 3.0);
	double t_1 = Math.pow(b, 2.0) - (c * (a * 4.0));
	double tmp;
	if (b <= 1.65) {
		tmp = ((t_1 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c / math.pow(b, 3.0)
	t_1 = math.pow(b, 2.0) - (c * (a * 4.0))
	tmp = 0
	if b <= 1.65:
		tmp = ((t_1 - math.pow(-b, 2.0)) / (b + math.sqrt(t_1))) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	t_1 = Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))
	tmp = 0.0
	if (b <= 1.65)
		tmp = Float64(Float64(Float64(t_1 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c / (b ^ 3.0);
	t_1 = (b ^ 2.0) - (c * (a * 4.0));
	tmp = 0.0;
	if (b <= 1.65)
		tmp = ((t_1 - (-b ^ 2.0)) / (b + sqrt(t_1))) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.65], N[(N[(N[(t$95$1 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 81.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

      2. pow2 81.7%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      3. pow-pow 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      4. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      5. pow-pow 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      6. metadata-eval 85.1%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      7. add-sqr-sqrt 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      8. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      9. *-commutative 85.8%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. Applied egg-rr 86.1%

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

   if 1.6499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - c \cdot \left(a \cdot 4\right)\right) - {\left(-b\right)}^{2}}{b + \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ t_1 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\frac{t\_1 - {b}^{2}}{b + \sqrt{t\_1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))) (t_1 (fma b b (* c (* a -4.0)))))
   (if (<= b 1.15)
     (/ 1.0 (/ (* a 2.0) (/ (- t_1 (pow b 2.0)) (+ b (sqrt t_1)))))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double t_1 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (b <= 1.15) {
		tmp = 1.0 / ((a * 2.0) / ((t_1 - pow(b, 2.0)) / (b + sqrt(t_1))));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	t_1 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(Float64(t_1 - (b ^ 2.0)) / Float64(b + sqrt(t_1)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. flip-- 85.4%

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

       2. add-sqr-sqrt 86.1%

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

       3. unpow2 86.1%

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

   14. Applied egg-rr 86.1%

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

       1. unpow2 86.1%

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

       2. fmm-undef 86.0%

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

       3. distribute-rgt-neg-in 86.0%

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

       4. distribute-rgt-neg-in 86.0%

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

       5. metadata-eval 86.0%

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

       6. +-commutative 86.0%

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

       7. unpow2 86.0%

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

       8. fmm-undef 85.8%

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

       9. distribute-rgt-neg-in 85.8%

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

       10. distribute-rgt-neg-in 85.8%

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

       11. metadata-eval 85.8%

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

   16. Simplified 85.8%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \mathbf{if}\;b \leq 1.35:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} - b\right)}^{3}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (if (<= b 1.35)
     (/
      (cbrt (pow (- (sqrt (- (pow b 2.0) (* c (* a 4.0)))) b) 3.0))
      (* a 2.0))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double tmp;
	if (b <= 1.35) {
		tmp = cbrt(pow((sqrt((pow(b, 2.0) - (c * (a * 4.0)))) - b), 3.0)) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c / Math.pow(b, 3.0);
	double tmp;
	if (b <= 1.35) {
		tmp = Math.cbrt(Math.pow((Math.sqrt((Math.pow(b, 2.0) - (c * (a * 4.0)))) - b), 3.0)) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	tmp = 0.0
	if (b <= 1.35)
		tmp = Float64(cbrt((Float64(sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))) - b) ^ 3.0)) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.35], N[(N[Power[N[Power[N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3500000000000001

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. add-cbrt-cube 83.1%

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

      2. pow3 83.1%

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

   10. Applied egg-rr 85.7%

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

   if 1.3500000000000001 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} - b\right)}^{3}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (if (<= b 1.25)
     (cbrt
      (pow (/ (- (sqrt (- (pow b 2.0) (* c (* a 4.0)))) b) (* a 2.0)) 3.0))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double tmp;
	if (b <= 1.25) {
		tmp = cbrt(pow(((sqrt((pow(b, 2.0) - (c * (a * 4.0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c / Math.pow(b, 3.0);
	double tmp;
	if (b <= 1.25) {
		tmp = Math.cbrt(Math.pow(((Math.sqrt((Math.pow(b, 2.0) - (c * (a * 4.0)))) - b) / (a * 2.0)), 3.0));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	tmp = 0.0
	if (b <= 1.25)
		tmp = cbrt((Float64(Float64(sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) ^ 3.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.25], N[Power[N[Power[N[(N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. add-cbrt-cube 83.1%

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

      2. pow3 83.1%

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

   10. Applied egg-rr 85.7%

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

   if 1.25 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{3}}\\ \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(t\_0 - t\_0 \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 3.0))))
   (if (<= b 1.15)
     (/ 1.0 (/ 1.0 (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))))
     (/
      -1.0
      (+ (/ b c) (* a (+ (* -2.0 (* a (- t_0 (* t_0 0.5)))) (/ -1.0 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 3.0);
	double tmp;
	if (b <= 1.15) {
		tmp = 1.0 / (1.0 / ((sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0)));
	} else {
		tmp = -1.0 / ((b / c) + (a * ((-2.0 * (a * (t_0 - (t_0 * 0.5)))) + (-1.0 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 3.0))
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0))));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) + Float64(a * Float64(Float64(-2.0 * Float64(a * Float64(t_0 - Float64(t_0 * 0.5)))) + Float64(-1.0 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15], N[(1.0 / N[(1.0 / N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(a * N[(t$95$0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. add-cbrt-cube 84.8%

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

      2. pow1/3 81.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 81.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 81.7%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 81.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 81.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 83.1%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

      3. fmm-undef 83.3%

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

      4. cbrt-prod 83.8%

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

      5. add-sqr-sqrt 83.4%

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

      6. add-cbrt-cube 85.6%

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

   10. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   12. Simplified 85.6%

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

       1. clear-num 85.6%

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

       2. inv-pow 85.6%

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

   14. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

       2. associate-/r* 85.6%

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

       3. associate-/l/ 85.6%

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

       4. unpow2 85.6%

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

       5. fmm-undef 85.7%

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

       6. distribute-rgt-neg-in 85.7%

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

       7. distribute-rgt-neg-in 85.7%

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

       8. metadata-eval 85.7%

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

   16. Simplified 85.7%

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

   if 1.1499999999999999 < b

   1. Initial program 48.6%

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

      1. *-commutative 48.6%

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

   3. Simplified 48.6%

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

      1. add-cbrt-cube 48.1%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 45.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 45.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 45.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 46.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.9%

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

      1. clear-num 46.9%

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

      2. inv-pow 46.9%

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

      3. fmm-undef 46.6%

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

      4. cbrt-prod 47.8%

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

      5. add-sqr-sqrt 47.4%

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

      6. add-cbrt-cube 48.6%

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

   10. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

   12. Simplified 48.6%

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

   13. Taylor expanded in a around 0 92.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(\frac{c}{{b}^{3}} - \frac{c}{{b}^{3}} \cdot 0.5\right)\right) + \frac{-1}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.4)
   (/ 1.0 (/ 1.0 (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4) {
		tmp = 1.0 / (1.0 / ((sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0)));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.4)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.4], N[(1.0 / N[(1.0 / N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.39999999999999991

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

   3. Simplified 84.2%

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

      1. add-cbrt-cube 83.4%

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

      2. pow1/3 80.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 80.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 80.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 80.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 80.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 80.8%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 79.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 79.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 81.8%

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

      1. clear-num 81.8%

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

      2. inv-pow 81.8%

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

      3. fmm-undef 81.9%

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

      4. cbrt-prod 82.6%

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

      5. add-sqr-sqrt 82.2%

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

      6. add-cbrt-cube 84.2%

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

   10. Applied egg-rr 84.2%

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

       1. unpow-1 84.2%

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

   12. Simplified 84.2%

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

       1. clear-num 84.2%

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

       2. inv-pow 84.2%

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

   14. Applied egg-rr 84.2%

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

       1. unpow-1 84.2%

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

       2. associate-/r* 84.2%

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

       3. associate-/l/ 84.2%

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

       4. unpow2 84.2%

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

       5. fmm-undef 84.3%

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

       6. distribute-rgt-neg-in 84.3%

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

       7. distribute-rgt-neg-in 84.3%

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

       8. metadata-eval 84.3%

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

   16. Simplified 84.3%

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

   if 3.39999999999999991 < b

   1. Initial program 47.6%

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

      1. *-commutative 47.6%

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

   3. Simplified 47.6%

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

      1. add-cbrt-cube 47.2%

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

      2. pow1/3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 45.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.0%

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

      1. clear-num 46.0%

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

      2. inv-pow 46.0%

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

      3. fmm-undef 45.7%

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

      4. cbrt-prod 46.8%

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

      5. add-sqr-sqrt 46.5%

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

      6. add-cbrt-cube 47.6%

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

   10. Applied egg-rr 47.6%

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

       1. unpow-1 47.6%

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

   12. Simplified 47.6%

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

   13. Taylor expanded in a around 0 88.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

   3. Simplified 84.3%

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

   if 3.5 < b

   1. Initial program 47.6%

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

      1. *-commutative 47.6%

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

   3. Simplified 47.6%

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

      1. add-cbrt-cube 47.2%

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

      2. pow1/3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 45.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.0%

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

      1. clear-num 46.0%

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

      2. inv-pow 46.0%

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

      3. fmm-undef 45.7%

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

      4. cbrt-prod 46.8%

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

      5. add-sqr-sqrt 46.5%

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

      6. add-cbrt-cube 47.6%

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

   10. Applied egg-rr 47.6%

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

       1. unpow-1 47.6%

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

   12. Simplified 47.6%

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

   13. Taylor expanded in a around 0 88.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.4)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / 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 <= 3.4d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.4:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.4)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.4)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.4], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.39999999999999991

   1. Initial program 84.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 3.39999999999999991 < b

   1. Initial program 47.6%

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

      1. *-commutative 47.6%

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

   3. Simplified 47.6%

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

      1. add-cbrt-cube 47.2%

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

      2. pow1/3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 44.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 44.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 44.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 45.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 45.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   8. Applied egg-rr 46.0%

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

      1. clear-num 46.0%

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

      2. inv-pow 46.0%

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

      3. fmm-undef 45.7%

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

      4. cbrt-prod 46.8%

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

      5. add-sqr-sqrt 46.5%

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

      6. add-cbrt-cube 47.6%

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

   10. Applied egg-rr 47.6%

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

       1. unpow-1 47.6%

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

   12. Simplified 47.6%

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

   13. Taylor expanded in a around 0 88.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / 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 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

   1. add-cbrt-cube 53.8%

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

   2. pow1/3 51.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 51.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 51.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 51.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 51.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 51.5%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. +-commutative 51.5%

      \[\leadsto \frac{\color{blue}{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   2. add-cube-cbrt 51.5%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

   3. fma-define 51.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{{\left({b}^{6}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

8. Applied egg-rr 52.6%

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

   1. clear-num 52.6%

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

   2. inv-pow 52.6%

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

   3. fmm-undef 52.4%

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

   4. cbrt-prod 53.4%

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

   5. add-sqr-sqrt 53.1%

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

   6. add-cbrt-cube 54.3%

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

10. Applied egg-rr 54.3%

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

    1. unpow-1 54.3%

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

12. Simplified 54.3%

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

13. Taylor expanded in a around 0 82.6%

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

14. Final simplification 82.6%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
15. Add Preprocessing

Alternative 17: 64.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. *-commutative 54.4%

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

3. Simplified 54.4%

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

5. Taylor expanded in b around inf 65.4%

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

   1. associate-*r/ 65.4%

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

   2. mul-1-neg 65.4%

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

7. Simplified 65.4%

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

8. Final simplification 65.4%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))