Cubic critical, narrow range

Percentage Accurate: 55.2% → 92.1%

Time: 20.0s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ t_1 := c \cdot {a}^{2}\\ \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{t_0 - {b}^{2}}{\left(b + \sqrt{t_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{{a}^{3} \cdot \left(-0.75 \cdot \left(c \cdot \left(c \cdot -0.75 + c \cdot 0.375\right)\right) + \left(-0.2222222222222222 \cdot \frac{1.265625 \cdot {c}^{4} + {c}^{4} \cdot 5.0625}{{c}^{2}} + {c}^{2} \cdot 0.5625\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot t_1 + 0.375 \cdot t_1}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))) (t_1 (* c (pow a 2.0))))
   (if (<= b 0.009)
     (/ (- t_0 (pow b 2.0)) (* (+ b (sqrt t_0)) (* a 3.0)))
     (/
      1.0
      (+
       (*
        -3.0
        (/
         (*
          (pow a 3.0)
          (+
           (* -0.75 (* c (+ (* c -0.75) (* c 0.375))))
           (+
            (*
             -0.2222222222222222
             (/
              (+ (* 1.265625 (pow c 4.0)) (* (pow c 4.0) 5.0625))
              (pow c 2.0)))
            (* (pow c 2.0) 0.5625))))
         (pow b 5.0)))
       (+
        (* -3.0 (/ (+ (* -0.75 t_1) (* 0.375 t_1)) (pow b 3.0)))
        (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double t_1 = c * pow(a, 2.0);
	double tmp;
	if (b <= 0.009) {
		tmp = (t_0 - pow(b, 2.0)) / ((b + sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = 1.0 / ((-3.0 * ((pow(a, 3.0) * ((-0.75 * (c * ((c * -0.75) + (c * 0.375)))) + ((-0.2222222222222222 * (((1.265625 * pow(c, 4.0)) + (pow(c, 4.0) * 5.0625)) / pow(c, 2.0))) + (pow(c, 2.0) * 0.5625)))) / pow(b, 5.0))) + ((-3.0 * (((-0.75 * t_1) + (0.375 * t_1)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	t_1 = Float64(c * (a ^ 2.0))
	tmp = 0.0
	if (b <= 0.009)
		tmp = Float64(Float64(t_0 - (b ^ 2.0)) / Float64(Float64(b + sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64((a ^ 3.0) * Float64(Float64(-0.75 * Float64(c * Float64(Float64(c * -0.75) + Float64(c * 0.375)))) + Float64(Float64(-0.2222222222222222 * Float64(Float64(Float64(1.265625 * (c ^ 4.0)) + Float64((c ^ 4.0) * 5.0625)) / (c ^ 2.0))) + Float64((c ^ 2.0) * 0.5625)))) / (b ^ 5.0))) + Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_1) + Float64(0.375 * t_1)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.009], N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-3.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-0.75 * N[(c * N[(N[(c * -0.75), $MachinePrecision] + N[(c * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.2222222222222222 * N[(N[(N[(1.265625 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * 5.0625), $MachinePrecision]), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 2.0], $MachinePrecision] * 0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$1), $MachinePrecision] + N[(0.375 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00899999999999999932

   1. Initial program 91.7%

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

   3. Simplified 91.8%

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

      1. fma-udef 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 91.7%

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

      4. *-commutative 91.7%

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

      5. +-commutative 91.7%

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

      6. fma-udef 92.0%

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

      7. add-cbrt-cube 89.6%

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

      8. pow1/3 86.7%

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

      9. pow3 86.7%

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

      10. sqrt-pow2 86.6%

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

      11. fma-udef 86.6%

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

      12. +-commutative 86.6%

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

      13. *-commutative 86.6%

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

      14. associate-*r* 86.6%

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

      15. *-commutative 86.6%

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

      16. fma-udef 86.6%

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

      17. pow2 86.6%

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

      18. metadata-eval 86.6%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 86.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 90.0%

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

   7. Simplified 90.0%

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

      1. add-cbrt-cube 89.9%

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

      2. pow3 89.9%

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

      3. pow1/3 86.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a}\right)}^{3}} \]

      4. pow-pow 91.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}{3 \cdot a}\right)}^{3}} \]

      5. metadata-eval 91.6%

         \[\leadsto \sqrt[3]{{\left(\frac{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}{3 \cdot a}\right)}^{3}} \]

      6. pow1/2 91.6%

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

      7. *-commutative 91.6%

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

   9. Applied egg-rr 91.6%

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

       1. rem-cbrt-cube 91.8%

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

       2. div-inv 91.6%

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

       3. flip-- 91.8%

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

       4. frac-times 91.9%

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

       5. add-sqr-sqrt 92.8%

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

       6. unpow2 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. *-rgt-identity 92.8%

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

       2. +-commutative 92.8%

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

   13. Simplified 92.8%

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

   if 0.00899999999999999932 < b

   1. Initial program 53.7%

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

   3. Simplified 53.8%

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

      1. fma-udef 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. +-commutative 53.7%

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

      6. fma-udef 53.7%

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

      7. add-cbrt-cube 52.6%

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

      8. pow1/3 50.7%

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

      9. pow3 50.7%

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

      10. sqrt-pow2 50.7%

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

      11. fma-udef 50.7%

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

      12. +-commutative 50.7%

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

      13. *-commutative 50.7%

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

      14. associate-*r* 50.7%

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

      15. *-commutative 50.7%

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

      16. fma-udef 50.7%

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

      17. pow2 50.7%

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

      18. metadata-eval 50.7%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 50.7%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 52.5%

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

   7. Simplified 52.5%

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

      1. clear-num 52.5%

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

      2. inv-pow 52.5%

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

      3. *-commutative 52.5%

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

      4. pow1/3 50.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 53.8%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 53.8%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 53.8%

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

   9. Applied egg-rr 53.8%

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

       1. unpow-1 53.8%

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

       2. associate-/l* 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in b around inf 94.2%

       \[\leadsto \frac{1}{\color{blue}{-3 \cdot \frac{-0.75 \cdot \left(a \cdot \left(c \cdot \left(-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(-0.2222222222222222 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {c}^{2}} + 0.5625 \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}} \]

   13. Taylor expanded in a around 0 94.2%

       \[\leadsto \frac{1}{-3 \cdot \frac{\color{blue}{{a}^{3} \cdot \left(-0.75 \cdot \left(c \cdot \left(-0.75 \cdot c + 0.375 \cdot c\right)\right) + \left(-0.2222222222222222 \cdot \frac{1.265625 \cdot {c}^{4} + 5.0625 \cdot {c}^{4}}{{c}^{2}} + 0.5625 \cdot {c}^{2}\right)\right)}}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{{a}^{3} \cdot \left(-0.75 \cdot \left(c \cdot \left(c \cdot -0.75 + c \cdot 0.375\right)\right) + \left(-0.2222222222222222 \cdot \frac{1.265625 \cdot {c}^{4} + {c}^{4} \cdot 5.0625}{{c}^{2}} + {c}^{2} \cdot 0.5625\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{-0.75 \cdot \left(c \cdot {a}^{2}\right) + 0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]

Alternative 2: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{t_0 - {b}^{2}}{\left(b + \sqrt{t_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))))
   (if (<= b 0.009)
     (/ (- t_0 (pow b 2.0)) (* (+ b (sqrt t_0)) (* a 3.0)))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 6.328125 a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double tmp;
	if (b <= 0.009) {
		tmp = (t_0 - pow(b, 2.0)) / ((b + sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / pow(b, 7.0)) * (6.328125 / a)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.009)
		tmp = Float64(Float64(t_0 - (b ^ 2.0)) / Float64(Float64(b + sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.009], N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00899999999999999932

   1. Initial program 91.7%

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

   3. Simplified 91.8%

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

      1. fma-udef 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 91.7%

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

      4. *-commutative 91.7%

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

      5. +-commutative 91.7%

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

      6. fma-udef 92.0%

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

      7. add-cbrt-cube 89.6%

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

      8. pow1/3 86.7%

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

      9. pow3 86.7%

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

      10. sqrt-pow2 86.6%

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

      11. fma-udef 86.6%

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

      12. +-commutative 86.6%

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

      13. *-commutative 86.6%

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

      14. associate-*r* 86.6%

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

      15. *-commutative 86.6%

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

      16. fma-udef 86.6%

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

      17. pow2 86.6%

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

      18. metadata-eval 86.6%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 86.6%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 90.0%

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

   7. Simplified 90.0%

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

      1. add-cbrt-cube 89.9%

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

      2. pow3 89.9%

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

      3. pow1/3 86.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a}\right)}^{3}} \]

      4. pow-pow 91.6%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}{3 \cdot a}\right)}^{3}} \]

      5. metadata-eval 91.6%

         \[\leadsto \sqrt[3]{{\left(\frac{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}{3 \cdot a}\right)}^{3}} \]

      6. pow1/2 91.6%

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

      7. *-commutative 91.6%

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

   9. Applied egg-rr 91.6%

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

       1. rem-cbrt-cube 91.8%

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

       2. div-inv 91.6%

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

       3. flip-- 91.8%

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

       4. frac-times 91.9%

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

       5. add-sqr-sqrt 92.8%

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

       6. unpow2 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. *-rgt-identity 92.8%

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

       2. +-commutative 92.8%

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

   13. Simplified 92.8%

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

   if 0.00899999999999999932 < b

   1. Initial program 53.7%

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

   2. Taylor expanded in b around inf 93.9%

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

   3. Taylor expanded in c around 0 93.9%

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

      1. distribute-rgt-in 93.9%

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

      2. associate-*r* 93.9%

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

      3. associate-*r* 93.9%

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

      4. distribute-rgt-out 93.9%

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

      5. *-commutative 93.9%

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

      6. times-frac 93.9%

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

   5. Simplified 93.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \]

Alternative 3: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\\ t_1 := c \cdot {a}^{2}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{t_0 - {b}^{2}}{\left(b + \sqrt{t_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot t_1 + 0.375 \cdot t_1}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (pow b 2.0))) (t_1 (* c (pow a 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
     (/ (- t_0 (pow b 2.0)) (* (+ b (sqrt t_0)) (* a 3.0)))
     (/
      1.0
      (+
       (* -3.0 (/ (+ (* -0.75 t_1) (* 0.375 t_1)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), pow(b, 2.0));
	double t_1 = c * pow(a, 2.0);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.25) {
		tmp = (t_0 - pow(b, 2.0)) / ((b + sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_1) + (0.375 * t_1)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), (b ^ 2.0))
	t_1 = Float64(c * (a ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.25)
		tmp = Float64(Float64(t_0 - (b ^ 2.0)) / Float64(Float64(b + sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_1) + Float64(0.375 * t_1)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$1), $MachinePrecision] + N[(0.375 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.2%

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

      1. fma-udef 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*r* 83.1%

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

      4. *-commutative 83.1%

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

      5. +-commutative 83.1%

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

      6. fma-udef 83.3%

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

      7. add-cbrt-cube 81.9%

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

      8. pow1/3 78.8%

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

      9. pow3 78.8%

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

      10. sqrt-pow2 78.8%

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

      11. fma-udef 78.8%

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

      12. +-commutative 78.8%

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

      13. *-commutative 78.8%

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

      14. associate-*r* 78.8%

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

      15. *-commutative 78.8%

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

      16. fma-udef 78.8%

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

      17. pow2 78.8%

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

      18. metadata-eval 78.8%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 78.8%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 82.1%

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

   7. Simplified 82.1%

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

      1. add-cbrt-cube 82.0%

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

      2. pow3 82.0%

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

      3. pow1/3 78.8%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a}\right)}^{3}} \]

      4. pow-pow 83.1%

         \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}{3 \cdot a}\right)}^{3}} \]

      5. metadata-eval 83.1%

         \[\leadsto \sqrt[3]{{\left(\frac{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}{3 \cdot a}\right)}^{3}} \]

      6. pow1/2 83.1%

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

      7. *-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. rem-cbrt-cube 83.2%

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

       2. div-inv 83.1%

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

       3. flip-- 83.4%

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

       4. frac-times 83.5%

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

       5. add-sqr-sqrt 84.9%

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

       6. unpow2 84.9%

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

   11. Applied egg-rr 84.9%

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

       1. *-rgt-identity 84.9%

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

       2. +-commutative 84.9%

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

   13. Simplified 84.9%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. fma-udef 50.2%

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

      2. *-commutative 50.2%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. +-commutative 50.2%

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

      6. fma-udef 50.1%

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

      7. add-cbrt-cube 49.0%

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

      8. pow1/3 47.3%

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

      9. pow3 47.2%

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

      10. sqrt-pow2 47.2%

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

      11. fma-udef 47.2%

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

      12. +-commutative 47.2%

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

      13. *-commutative 47.2%

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

      14. associate-*r* 47.2%

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

      15. *-commutative 47.2%

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

      16. fma-udef 47.2%

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

      17. pow2 47.2%

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

      18. metadata-eval 47.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 47.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 48.8%

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

   7. Simplified 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

      3. *-commutative 48.8%

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

      4. pow1/3 47.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 50.2%

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

   9. Applied egg-rr 50.2%

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

       1. unpow-1 50.2%

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

       2. associate-/l* 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in b around inf 93.2%

       \[\leadsto \frac{1}{\color{blue}{-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot \left(c \cdot {a}^{2}\right) + 0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b} \cdot \frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot t_0 + 0.375 \cdot t_0}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))))
   (if (<= b 2.1)
     (/
      1.0
      (*
       (/ 1.0 (- (sqrt (fma a (* c -3.0) (pow b 2.0))) b))
       (/ a 0.3333333333333333)))
     (/
      1.0
      (+
       (* -3.0 (/ (+ (* -0.75 t_0) (* 0.375 t_0)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double tmp;
	if (b <= 2.1) {
		tmp = 1.0 / ((1.0 / (sqrt(fma(a, (c * -3.0), pow(b, 2.0))) - b)) * (a / 0.3333333333333333));
	} else {
		tmp = 1.0 / ((-3.0 * (((-0.75 * t_0) + (0.375 * t_0)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(1.0 / Float64(Float64(1.0 / Float64(sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0))) - b)) * Float64(a / 0.3333333333333333)));
	else
		tmp = Float64(1.0 / Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(0.375 * t_0)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.1], N[(1.0 / N[(N[(1.0 / N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] * N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 82.5%

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

   3. Simplified 82.7%

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

      1. fma-udef 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*r* 82.5%

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

      4. *-commutative 82.5%

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

      5. +-commutative 82.5%

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

      6. fma-udef 82.7%

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

      7. add-cbrt-cube 80.5%

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

      8. pow1/3 79.7%

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

      9. pow3 79.4%

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

      10. sqrt-pow2 79.3%

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

      11. fma-udef 79.4%

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

      12. +-commutative 79.4%

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

      13. *-commutative 79.4%

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

      14. associate-*r* 79.4%

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

      15. *-commutative 79.4%

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

      16. fma-udef 79.5%

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

      17. pow2 79.5%

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

      18. metadata-eval 79.5%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 79.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 80.8%

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

   7. Simplified 80.8%

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

      1. clear-num 80.8%

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

      2. inv-pow 80.8%

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

      3. *-commutative 80.8%

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

      4. pow1/3 79.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 82.6%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 82.6%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 82.6%

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

   9. Applied egg-rr 82.6%

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

       1. unpow-1 82.6%

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

       2. associate-/l* 82.7%

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

   11. Simplified 82.7%

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

       1. *-un-lft-identity 82.7%

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

       2. div-inv 82.8%

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

       3. metadata-eval 82.8%

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

       4. times-frac 82.8%

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

   13. Applied egg-rr 82.8%

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

   if 2.10000000000000009 < b

   1. Initial program 51.5%

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

   3. Simplified 51.5%

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

      1. fma-udef 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*r* 51.5%

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

      4. *-commutative 51.5%

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

      5. +-commutative 51.5%

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

      6. fma-udef 51.4%

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

      7. add-cbrt-cube 50.4%

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

      8. pow1/3 48.3%

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

      9. pow3 48.3%

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

      10. sqrt-pow2 48.3%

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

      11. fma-udef 48.3%

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

      12. +-commutative 48.3%

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

      13. *-commutative 48.3%

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

      14. associate-*r* 48.3%

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

      15. *-commutative 48.3%

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

      16. fma-udef 48.3%

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

      17. pow2 48.3%

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

      18. metadata-eval 48.3%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 48.3%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 50.2%

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

   7. Simplified 50.2%

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

      1. clear-num 50.2%

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

      2. inv-pow 50.2%

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

      3. *-commutative 50.2%

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

      4. pow1/3 48.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 51.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 51.5%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 51.5%

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

   9. Applied egg-rr 51.5%

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

       1. unpow-1 51.5%

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

       2. associate-/l* 51.5%

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

   11. Simplified 51.5%

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

   12. Taylor expanded in b around inf 92.9%

       \[\leadsto \frac{1}{\color{blue}{-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b} \cdot \frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 \cdot \frac{-0.75 \cdot \left(c \cdot {a}^{2}\right) + 0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}\\ \end{array} \]

Alternative 5: 85.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.2%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. fma-udef 50.2%

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

      2. *-commutative 50.2%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. +-commutative 50.2%

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

      6. fma-udef 50.1%

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

      7. add-cbrt-cube 49.0%

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

      8. pow1/3 47.3%

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

      9. pow3 47.2%

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

      10. sqrt-pow2 47.2%

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

      11. fma-udef 47.2%

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

      12. +-commutative 47.2%

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

      13. *-commutative 47.2%

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

      14. associate-*r* 47.2%

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

      15. *-commutative 47.2%

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

      16. fma-udef 47.2%

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

      17. pow2 47.2%

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

      18. metadata-eval 47.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 47.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 48.8%

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

   7. Simplified 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

      3. *-commutative 48.8%

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

      4. pow1/3 47.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 50.2%

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

   9. Applied egg-rr 50.2%

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

       1. unpow-1 50.2%

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

       2. associate-/l* 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in a around 0 87.7%

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

4. Final simplification 86.9%

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

Alternative 6: 85.5% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.25)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.25], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.25

   1. Initial program 83.1%

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

   3. Simplified 83.3%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. fma-udef 50.2%

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

      2. *-commutative 50.2%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. +-commutative 50.2%

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

      6. fma-udef 50.1%

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

      7. add-cbrt-cube 49.0%

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

      8. pow1/3 47.3%

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

      9. pow3 47.2%

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

      10. sqrt-pow2 47.2%

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

      11. fma-udef 47.2%

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

      12. +-commutative 47.2%

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

      13. *-commutative 47.2%

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

      14. associate-*r* 47.2%

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

      15. *-commutative 47.2%

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

      16. fma-udef 47.2%

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

      17. pow2 47.2%

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

      18. metadata-eval 47.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 47.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 48.8%

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

   7. Simplified 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

      3. *-commutative 48.8%

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

      4. pow1/3 47.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 50.2%

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

   9. Applied egg-rr 50.2%

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

       1. unpow-1 50.2%

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

       2. associate-/l* 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in a around 0 87.7%

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

4. Final simplification 86.9%

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

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.25:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.25d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.25) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.25:
		tmp = t_0
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.25)
		tmp = t_0;
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.25

   1. Initial program 83.1%

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

   if -0.25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. fma-udef 50.2%

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

      2. *-commutative 50.2%

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

      3. associate-*r* 50.2%

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

      4. *-commutative 50.2%

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

      5. +-commutative 50.2%

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

      6. fma-udef 50.1%

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

      7. add-cbrt-cube 49.0%

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

      8. pow1/3 47.3%

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

      9. pow3 47.2%

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

      10. sqrt-pow2 47.2%

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

      11. fma-udef 47.2%

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

      12. +-commutative 47.2%

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

      13. *-commutative 47.2%

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

      14. associate-*r* 47.2%

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

      15. *-commutative 47.2%

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

      16. fma-udef 47.2%

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

      17. pow2 47.2%

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

      18. metadata-eval 47.2%

          \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. Applied egg-rr 47.2%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow1/3 48.8%

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

   7. Simplified 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

      3. *-commutative 48.8%

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

      4. pow1/3 47.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

      5. pow-pow 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

      6. metadata-eval 50.2%

         \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

      7. pow1/2 50.2%

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

   9. Applied egg-rr 50.2%

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

       1. unpow-1 50.2%

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

       2. associate-/l* 50.2%

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

   11. Simplified 50.2%

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

   12. Taylor expanded in a around 0 87.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.25:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 8: 81.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Simplified 56.0%

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

   1. fma-udef 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*r* 56.0%

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

   4. *-commutative 56.0%

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

   5. +-commutative 56.0%

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

   6. fma-udef 56.0%

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

   7. add-cbrt-cube 54.7%

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

   8. pow1/3 52.8%

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

   9. pow3 52.8%

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

   10. sqrt-pow2 52.8%

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

   11. fma-udef 52.8%

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

   12. +-commutative 52.8%

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

   13. *-commutative 52.8%

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

   14. associate-*r* 52.8%

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

   15. *-commutative 52.8%

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

   16. fma-udef 52.8%

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

   17. pow2 52.8%

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

   18. metadata-eval 52.8%

       \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

5. Applied egg-rr 52.8%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow1/3 54.7%

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

7. Simplified 54.7%

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

   1. clear-num 54.7%

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

   2. inv-pow 54.7%

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

   3. *-commutative 54.7%

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

   4. pow1/3 52.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}\right)}^{-1} \]

   5. pow-pow 56.0%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b}\right)}^{-1} \]

   6. metadata-eval 56.0%

      \[\leadsto {\left(\frac{a \cdot 3}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}} - b}\right)}^{-1} \]

   7. pow1/2 56.0%

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

9. Applied egg-rr 56.0%

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

    1. unpow-1 56.0%

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

    2. associate-/l* 56.0%

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

11. Simplified 56.0%

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

12. Taylor expanded in a around 0 82.5%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]

13. Final simplification 82.5%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 9: 64.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in b around inf 64.1%

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

3. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))