Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.4%

Time: 13.3s

Alternatives: 21

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\left(\frac{\mathsf{fma}\left(\frac{-1.40625}{a}, \left(c \cdot c\right) \cdot {a}^{4}, \mathsf{fma}\left(0.5625 \cdot \left(c \cdot c\right), {a}^{3}, \left(-0.375 \cdot {\left(a \cdot c\right)}^{2}\right) \cdot \left(-0.75 \cdot a\right)\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot -0.375, a, \frac{-0.5}{b \cdot b}\right), a, \frac{0.6666666666666666}{c}\right)\right) \cdot \left(-b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/
      0.3333333333333333
      (*
       (+
        (/
         (fma
          (/ -1.40625 a)
          (* (* c c) (pow a 4.0))
          (fma
           (* 0.5625 (* c c))
           (pow a 3.0)
           (* (* -0.375 (pow (* a c) 2.0)) (* -0.75 a))))
         (pow b 6.0))
        (fma
         (fma (* (/ c (pow b 4.0)) -0.375) a (/ -0.5 (* b b)))
         a
         (/ 0.6666666666666666 c)))
       (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 0.3333333333333333 / (((fma((-1.40625 / a), ((c * c) * pow(a, 4.0)), fma((0.5625 * (c * c)), pow(a, 3.0), ((-0.375 * pow((a * c), 2.0)) * (-0.75 * a)))) / pow(b, 6.0)) + fma(fma(((c / pow(b, 4.0)) * -0.375), a, (-0.5 / (b * b))), a, (0.6666666666666666 / c))) * -b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(0.3333333333333333 / Float64(Float64(Float64(fma(Float64(-1.40625 / a), Float64(Float64(c * c) * (a ^ 4.0)), fma(Float64(0.5625 * Float64(c * c)), (a ^ 3.0), Float64(Float64(-0.375 * (Float64(a * c) ^ 2.0)) * Float64(-0.75 * a)))) / (b ^ 6.0)) + fma(fma(Float64(Float64(c / (b ^ 4.0)) * -0.375), a, Float64(-0.5 / Float64(b * b))), a, Float64(0.6666666666666666 / c))) * Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(N[(N[(-1.40625 / a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5625 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision] + N[(N[(-0.375 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(-0.75 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(-0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(b \cdot \left(-1 \cdot \frac{\frac{-3}{4} \cdot \left(a \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \left({a}^{2} \cdot c\right) + \frac{3}{8} \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(\frac{-2}{9} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + \frac{9}{16} \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{6}} - \left(\frac{-3}{4} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \left(\frac{-1}{2} \cdot \frac{a}{{b}^{2}} + \left(\frac{3}{8} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{2}{3} \cdot \frac{1}{c}\right)\right)\right)\right)\right)}}^{-1} \]

   7. Applied rewrites 96.3%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -0.75, \mathsf{fma}\left(\frac{-0.5}{b}, \frac{a}{b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, 0.375, \frac{0.6666666666666666}{c}\right)\right)\right)\right) \cdot b\right)}}^{-1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{3} \cdot {\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{\frac{405}{64}}{a}, \frac{-2}{9}, \mathsf{fma}\left(\frac{9}{16} \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \frac{-3}{8}\right) \cdot c\right) \cdot a\right) \cdot \frac{-3}{4}\right)\right)}{-{b}^{6}} - \left(a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{4}} + \frac{3}{8} \cdot \frac{c}{{b}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{b}^{2}}\right) + \frac{2}{3} \cdot \frac{1}{c}\right)\right) \cdot b\right)}^{-1} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.3%

       \[\leadsto 0.3333333333333333 \cdot {\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot -0.375, a, \frac{-0.5}{b \cdot b}\right), a, \frac{0.6666666666666666}{c}\right)\right) \cdot b\right)}^{-1} \]

   12. Applied rewrites 96.4%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(\frac{\mathsf{fma}\left(\frac{-1.40625}{a}, {a}^{4} \cdot \left(c \cdot c\right), \mathsf{fma}\left(\left(c \cdot c\right) \cdot 0.5625, {a}^{3}, \left(-0.75 \cdot a\right) \cdot \left({\left(c \cdot a\right)}^{2} \cdot -0.375\right)\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot -0.375, a, \frac{-0.5}{b \cdot b}\right), a, \frac{0.6666666666666666}{c}\right)\right) \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\left(\frac{\mathsf{fma}\left(\frac{-1.40625}{a}, \left(c \cdot c\right) \cdot {a}^{4}, \mathsf{fma}\left(0.5625 \cdot \left(c \cdot c\right), {a}^{3}, \left(-0.375 \cdot {\left(a \cdot c\right)}^{2}\right) \cdot \left(-0.75 \cdot a\right)\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot -0.375, a, \frac{-0.5}{b \cdot b}\right), a, \frac{0.6666666666666666}{c}\right)\right) \cdot \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -1.0546875, {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (fma
      (/
       (fma
        (* (* a a) -1.0546875)
        (pow c 4.0)
        (*
         (fma (* -0.375 (* b b)) (* c c) (* -0.5625 (* (pow c 3.0) a)))
         (* b b)))
       (pow b 7.0))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma((fma(((a * a) * -1.0546875), pow(c, 4.0), (fma((-0.375 * (b * b)), (c * c), (-0.5625 * (pow(c, 3.0) * a))) * (b * b))) / pow(b, 7.0)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = fma(Float64(fma(Float64(Float64(a * a) * -1.0546875), (c ^ 4.0), Float64(fma(Float64(-0.375 * Float64(b * b)), Float64(c * c), Float64(-0.5625 * Float64((c ^ 3.0) * a))) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -1.0546875), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -1.0546875, {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, -0.5625 \cdot \left({c}^{3} \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \left(\frac{0.5}{b} - \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a\right) \cdot a\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (pow
       (fma
        -0.6666666666666666
        (/ b c)
        (* (- (/ 0.5 b) (* (* (/ c (pow b 3.0)) -0.375) a)) a))
       -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(fma(-0.6666666666666666, (b / c), (((0.5 / b) - (((c / pow(b, 3.0)) * -0.375) * a)) * a)), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((fma(-0.6666666666666666, Float64(b / c), Float64(Float64(Float64(0.5 / b) - Float64(Float64(Float64(c / (b ^ 3.0)) * -0.375) * a)) * a)) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(N[(0.5 / b), $MachinePrecision] - N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(b \cdot \left(-1 \cdot \frac{\frac{-3}{4} \cdot \left(a \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \left({a}^{2} \cdot c\right) + \frac{3}{8} \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(\frac{-2}{9} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + \frac{9}{16} \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{6}} - \left(\frac{-3}{4} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \left(\frac{-1}{2} \cdot \frac{a}{{b}^{2}} + \left(\frac{3}{8} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{2}{3} \cdot \frac{1}{c}\right)\right)\right)\right)\right)}}^{-1} \]

   7. Applied rewrites 96.3%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -0.75, \mathsf{fma}\left(\frac{-0.5}{b}, \frac{a}{b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, 0.375, \frac{0.6666666666666666}{c}\right)\right)\right)\right) \cdot b\right)}}^{-1} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{3} \cdot {\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{\frac{405}{64}}{a}, \frac{-2}{9}, \mathsf{fma}\left(\frac{9}{16} \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \frac{-3}{8}\right) \cdot c\right) \cdot a\right) \cdot \frac{-3}{4}\right)\right)}{-{b}^{6}} - \left(a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{4}} + \frac{3}{8} \cdot \frac{c}{{b}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{b}^{2}}\right) + \frac{2}{3} \cdot \frac{1}{c}\right)\right) \cdot b\right)}^{-1} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.3%

       \[\leadsto 0.3333333333333333 \cdot {\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{4}} \cdot -0.375, a, \frac{-0.5}{b \cdot b}\right), a, \frac{0.6666666666666666}{c}\right)\right) \cdot b\right)}^{-1} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\frac{-2}{3} \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}}^{-1} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}}^{-1} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \color{blue}{\frac{b}{c}}, a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}^{-1} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, \color{blue}{a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)}\right)\right)}^{-1} \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + -1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)}\right)\right)}^{-1} \]

       5. mul-1-neg N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{1}{2} \cdot \frac{1}{b} + \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)}\right)\right)\right)}^{-1} \]

       6. unsub-neg N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} - a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)}\right)\right)}^{-1} \]

       7. lower--.f64 N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} - a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)}\right)\right)}^{-1} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b}} - a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)\right)}^{-1} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{\color{blue}{\frac{1}{2}}}{b} - a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)\right)}^{-1} \]

       10. lower-/.f64 N/A

           \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\color{blue}{\frac{\frac{1}{2}}{b}} - a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)\right)}^{-1} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{\frac{1}{2}}{b} - \color{blue}{a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}\right)\right)\right)}^{-1} \]

       12. distribute-rgt-out N/A

           \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{\frac{1}{2}}{b} - a \cdot \color{blue}{\left(\frac{c}{{b}^{3}} \cdot \left(\frac{-3}{4} + \frac{3}{8}\right)\right)}\right)\right)\right)}^{-1} \]

       13. metadata-eval N/A

           \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{\frac{1}{2}}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot \color{blue}{\frac{-3}{8}}\right)\right)\right)\right)}^{-1} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\frac{-2}{3}, \frac{b}{c}, a \cdot \left(\frac{\frac{1}{2}}{b} - a \cdot \color{blue}{\left(\frac{c}{{b}^{3}} \cdot \frac{-3}{8}\right)}\right)\right)\right)}^{-1} \]

   13. Applied rewrites 94.1%

       \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \left(\frac{0.5}{b} - a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.375\right)\right)\right)\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \left(\frac{0.5}{b} - \left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a\right) \cdot a\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.375, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (pow
       (fma
        (fma (* (/ c (pow b 3.0)) 0.375) a (/ 0.5 b))
        a
        (* (/ b c) -0.6666666666666666))
       -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(fma(fma(((c / pow(b, 3.0)) * 0.375), a, (0.5 / b)), a, ((b / c) * -0.6666666666666666)), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((fma(fma(Float64(Float64(c / (b ^ 3.0)) * 0.375), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.6666666666666666)) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\frac{-2}{3} \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}}^{-1} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right) + \frac{-2}{3} \cdot \frac{b}{c}\right)}}^{-1} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{3} \cdot {\left(\color{blue}{\left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot a} + \frac{-2}{3} \cdot \frac{b}{c}\right)}^{-1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\mathsf{fma}\left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}, a, \frac{-2}{3} \cdot \frac{b}{c}\right)\right)}}^{-1} \]

   7. Applied rewrites 94.1%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.375, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{{b}^{3}} \cdot 0.375, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (fma
      (/ -0.5 b)
      c
      (*
       (*
        (* (fma (* a (* b b)) -0.375 (* (* (* a a) c) -0.5625)) (pow b -5.0))
        c)
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma((-0.5 / b), c, (((fma((a * (b * b)), -0.375, (((a * a) * c) * -0.5625)) * pow(b, -5.0)) * c) * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = fma(Float64(-0.5 / b), c, Float64(Float64(Float64(fma(Float64(a * Float64(b * b)), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) * (b ^ -5.0)) * c) * c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(N[(N[(N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}}, c, \frac{\frac{-1}{2}}{b}\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c \]
   9. Step-by-step derivation

   10. Applied rewrites 93.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c, c, \frac{-0.5}{b} \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (fma
      (*
       (* (fma (* a (* b b)) -0.375 (* (* (* a a) c) -0.5625)) (pow b -5.0))
       c)
      c
      (* (/ -0.5 b) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(((fma((a * (b * b)), -0.375, (((a * a) * c) * -0.5625)) * pow(b, -5.0)) * c), c, ((-0.5 / b) * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = fma(Float64(Float64(fma(Float64(a * Float64(b * b)), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) * (b ^ -5.0)) * c), c, Float64(Float64(-0.5 / b) * c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}}, c, \frac{\frac{-1}{2}}{b}\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c \]
   9. Step-by-step derivation

   10. Applied rewrites 93.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c, c, \frac{-0.5}{b} \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.4)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (fma
       (* (fma (* a (* b b)) -0.375 (* (* (* a a) c) -0.5625)) (pow b -5.0))
       c
       (/ -0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.4) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma((fma((a * (b * b)), -0.375, (((a * a) * c) * -0.5625)) * pow(b, -5.0)), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(Float64(fma(Float64(a * Float64(b * b)), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) * (b ^ -5.0)), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.4], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.4000000000000004 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}}, c, \frac{\frac{-1}{2}}{b}\right) \cdot c \]
   7. Step-by-step derivation

   8. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)}{{b}^{5}}, c, \frac{-0.5}{b}\right) \cdot c \]
   9. Step-by-step derivation

   10. Applied rewrites 93.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \left(b \cdot b\right), -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(\frac{c}{b} \cdot a\right) \cdot 0.5\right)}{c}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (pow (/ (fma -0.6666666666666666 b (* (* (/ c b) a) 0.5)) c) -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow((fma(-0.6666666666666666, b, (((c / b) * a) * 0.5)) / c), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((Float64(fma(-0.6666666666666666, b, Float64(Float64(Float64(c / b) * a) * 0.5)) / c) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(-0.6666666666666666 * b + N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(b \cdot \left(-1 \cdot \frac{\frac{-3}{4} \cdot \left(a \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \left({a}^{2} \cdot c\right) + \frac{3}{8} \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(\frac{-2}{9} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + \frac{9}{16} \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{6}} - \left(\frac{-3}{4} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \left(\frac{-1}{2} \cdot \frac{a}{{b}^{2}} + \left(\frac{3}{8} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{2}{3} \cdot \frac{1}{c}\right)\right)\right)\right)\right)}}^{-1} \]

   7. Applied rewrites 96.3%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -0.75, \mathsf{fma}\left(\frac{-0.5}{b}, \frac{a}{b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, 0.375, \frac{0.6666666666666666}{c}\right)\right)\right)\right) \cdot b\right)}}^{-1} \]

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.0

         \[\leadsto 0.3333333333333333 \cdot {\left(\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(a \cdot \color{blue}{\frac{c}{b}}\right) \cdot 0.5\right)}{c}\right)}^{-1} \]

   10. Applied rewrites 89.0%

       \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(a \cdot \frac{c}{b}\right) \cdot 0.5\right)}{c}\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(\frac{c}{b} \cdot a\right) \cdot 0.5\right)}{c}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{b}{c}, -0.6666666666666666, \frac{a}{b} \cdot 0.5\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (pow (fma (/ b c) -0.6666666666666666 (* (/ a b) 0.5)) -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(fma((b / c), -0.6666666666666666, ((a / b) * 0.5)), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((fma(Float64(b / c), -0.6666666666666666, Float64(Float64(a / b) * 0.5)) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(b / c), $MachinePrecision] * -0.6666666666666666 + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in b around inf

      \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(b \cdot \left(-1 \cdot \frac{\frac{-3}{4} \cdot \left(a \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \left({a}^{2} \cdot c\right) + \frac{3}{8} \cdot \left({a}^{2} \cdot c\right)\right)\right)\right) + \left(\frac{-2}{9} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {c}^{2}} + \frac{9}{16} \cdot \left({a}^{3} \cdot {c}^{2}\right)\right)}{{b}^{6}} - \left(\frac{-3}{4} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \left(\frac{-1}{2} \cdot \frac{a}{{b}^{2}} + \left(\frac{3}{8} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{2}{3} \cdot \frac{1}{c}\right)\right)\right)\right)\right)}}^{-1} \]

   7. Applied rewrites 96.3%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\left(\frac{\mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{c \cdot c} \cdot \frac{6.328125}{a}, -0.2222222222222222, \mathsf{fma}\left(0.5625 \cdot {a}^{3}, c \cdot c, \left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot -0.375\right) \cdot c\right) \cdot a\right) \cdot -0.75\right)\right)}{-{b}^{6}} - \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -0.75, \mathsf{fma}\left(\frac{-0.5}{b}, \frac{a}{b}, \mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, 0.375, \frac{0.6666666666666666}{c}\right)\right)\right)\right) \cdot b\right)}}^{-1} \]

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.0

         \[\leadsto 0.3333333333333333 \cdot {\left(\mathsf{fma}\left(\frac{b}{c}, -0.6666666666666666, \color{blue}{\frac{a}{b}} \cdot 0.5\right)\right)}^{-1} \]

   10. Applied rewrites 89.0%

       \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{b}{c}, -0.6666666666666666, \frac{a}{b} \cdot 0.5\right)\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{b}{c}, -0.6666666666666666, \frac{a}{b} \cdot 0.5\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (*
      (pow (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666)) -1.0)
      0.3333333333333333))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(fma((a / b), 0.5, ((b / c) * -0.6666666666666666)), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.9

         \[\leadsto 0.3333333333333333 \cdot {\left(\mathsf{fma}\left(\frac{a}{b}, 0.5, \color{blue}{\frac{b}{c}} \cdot -0.6666666666666666\right)\right)}^{-1} \]

   7. Applied rewrites 88.9%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.1e-7)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.1e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.1e-7], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.1000000000000001e-7

   1. Initial program 70.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 70.6

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

   4. Applied rewrites 70.6%

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

   if -1.1000000000000001e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.1e-7)
   (* (- (sqrt (fma (* a -3.0) c (* b b))) b) (/ 0.3333333333333333 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.1e-7)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.1e-7], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.1000000000000001e-7

   1. Initial program 70.6%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 70.6

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

   6. Applied rewrites 70.6%

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

   if -1.1000000000000001e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.1e-7)
   (* (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) a) 0.3333333333333333)
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7) {
		tmp = ((sqrt(fma((a * -3.0), c, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.1e-7)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.1e-7], N[(N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.1000000000000001e-7

   1. Initial program 70.6%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. remove-double-div 70.6

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

      5. lower-/.f64 N/A

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

      6. *-lft-identity N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 70.5

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

   6. Applied rewrites 70.5%

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

   if -1.1000000000000001e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/
      (/ 1.0 (/ (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b))) c))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = (1.0 / (fma(-0.6666666666666666, (b / a), (0.5 * (c / b))) / c)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b))) / c)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.8

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{3 \cdot a} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 3.0 a)))
     (/
      (/ 1.0 (/ (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b))) c))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (3.0 * a));
	} else {
		tmp = (1.0 / (fma(-0.6666666666666666, (b / a), (0.5 * (c / b))) / c)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(3.0 * a)));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b))) / c)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

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

   6. Applied rewrites 85.7%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.8

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{3 \cdot a} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 7.8)
     (* (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) a)) 0.3333333333333333)
     (/
      (/ 1.0 (/ (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b))) c))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 7.8) {
		tmp = ((t_0 - (b * b)) / ((sqrt(t_0) + b) * a)) * 0.3333333333333333;
	} else {
		tmp = (1.0 / (fma(-0.6666666666666666, (b / a), (0.5 * (c / b))) / c)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * a)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b))) / c)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}\right)}^{-1}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-*.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 85.6%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.8

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{3 \cdot a} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/
    (/ 1.0 (/ (fma -0.6666666666666666 (/ b a) (* 0.5 (/ c b))) c))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = (1.0 / (fma(-0.6666666666666666, (b / a), (0.5 * (c / b))) / c)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(-0.6666666666666666, Float64(b / a), Float64(0.5 * Float64(c / b))) / c)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 84.1

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

   4. Applied rewrites 84.1%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{-2}{3} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.8

         \[\leadsto \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \color{blue}{\frac{c}{b}}\right)}{c}}}{3 \cdot a} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)}{c}}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{0.5}{b \cdot b} - \frac{0.6666666666666666}{a \cdot c}\right) \cdot b}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/
    (/ 1.0 (* (- (/ 0.5 (* b b)) (/ 0.6666666666666666 (* a c))) b))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = (1.0 / (((0.5 / (b * b)) - (0.6666666666666666 / (a * c))) * b)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(0.5 / Float64(b * b)) - Float64(0.6666666666666666 / Float64(a * c))) * b)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(0.5 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 / N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 84.1

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

   4. Applied rewrites 84.1%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 88.8

          \[\leadsto \frac{\frac{1}{b \cdot \left(\frac{0.5}{b \cdot b} - \frac{0.6666666666666666}{\color{blue}{a \cdot c}}\right)}}{3 \cdot a} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{b \cdot \left(\frac{0.5}{b \cdot b} - \frac{0.6666666666666666}{a \cdot c}\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(\frac{0.5}{b \cdot b} - \frac{0.6666666666666666}{a \cdot c}\right) \cdot b}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot \left(c \cdot c\right), \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ (fma (* -0.375 (* c c)) (/ a (* b b)) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = fma((-0.375 * (c * c)), (a / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(-0.375 * Float64(c * c)), Float64(a / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 84.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 84.1

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

   4. Applied rewrites 84.1%

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

   if 7.79999999999999982 < b

   1. Initial program 49.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}} + \frac{-1}{2} \cdot c}{b} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b \cdot b}} + \frac{-1}{2} \cdot c}{b} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{-3}{8} \cdot \color{blue}{\left({c}^{2} \cdot a\right)}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(\frac{-3}{8} \cdot {c}^{2}\right) \cdot a}}{b \cdot b} + \frac{-1}{2} \cdot c}{b} \]

      7. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{b} \cdot \frac{a}{b}} + \frac{-1}{2} \cdot c}{b} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-3}{8} \cdot {c}^{2}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}}{b} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{b}}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{b}, \frac{a}{b}, \frac{-1}{2} \cdot c\right)}{b} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b}, \color{blue}{\frac{a}{b}}, \frac{-1}{2} \cdot c\right)}{b} \]

      15. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 88.6%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot \left(c \cdot c\right), \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375 \cdot \left(c \cdot c\right), \frac{a}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.8

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

5. Applied rewrites 64.8%

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

Alternative 21: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 89.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{-1}{2}}{b} \cdot c \]
7. Step-by-step derivation

8. Applied rewrites 64.8%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(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)))