Cubic critical, narrow range

Percentage Accurate: 55.5% → 92.2%

Time: 11.5s

Alternatives: 15

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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.045:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, {b}^{-5} \cdot \left(-0.5625 \cdot \left(c \cdot c\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.045)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (*
      0.3333333333333333
      (pow
       (fma
        (fma
         (fma
          (- a)
          (* (pow b -5.0) (* -0.5625 (* c c)))
          (* 0.375 (/ c (pow b 3.0))))
         a
         (/ 0.5 b))
        a
        (* (/ b c) -0.6666666666666666))
       -1.0)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.045) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = 0.3333333333333333 * pow(fma(fma(fma(-a, (pow(b, -5.0) * (-0.5625 * (c * c))), (0.375 * (c / pow(b, 3.0)))), a, (0.5 / b)), a, ((b / c) * -0.6666666666666666)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.045)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(0.3333333333333333 * (fma(fma(fma(Float64(-a), Float64((b ^ -5.0) * Float64(-0.5625 * Float64(c * c))), Float64(0.375 * Float64(c / (b ^ 3.0)))), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.6666666666666666)) ^ -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.045], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[(N[(N[((-a) * N[(N[Power[b, -5.0], $MachinePrecision] * N[(-0.5625 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.044999999999999998

   1. Initial program 89.1%

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

      1. 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 89.1%

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

   6. Applied rewrites 90.1%

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

   if 0.044999999999999998 < b

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 51.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 94.6%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{5}}, 0.5625, \frac{-0.75}{b} \cdot \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}}^{-1} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{1}{3} \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{\frac{-45}{32} \cdot {c}^{2} + \left(\frac{9}{32} \cdot {c}^{2} + \frac{9}{16} \cdot {c}^{2}\right)}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)\right)}^{-1} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.6%

       \[\leadsto 0.3333333333333333 \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{\left(c \cdot c\right) \cdot -0.5625}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \]
   11. Step-by-step derivation

   12. Applied rewrites 94.6%

       \[\leadsto 0.3333333333333333 \cdot {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, {b}^{-5} \cdot \left(-0.5625 \cdot \left(c \cdot c\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}^{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{t\_0}^{1.5} - {b}^{3}}{a \cdot \mathsf{fma}\left(b, \sqrt{t\_0} + b, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{-0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.1458)
     (*
      0.3333333333333333
      (/ (- (pow t_0 1.5) (pow b 3.0)) (* a (fma b (+ (sqrt t_0) b) t_0))))
     (pow
      (/
       (fma
        (fma (* c -3.0) (/ (* -0.375 (* a a)) (pow b 3.0)) (* 1.5 (/ a b)))
        c
        (* -2.0 b))
       c)
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.1458) {
		tmp = 0.3333333333333333 * ((pow(t_0, 1.5) - pow(b, 3.0)) / (a * fma(b, (sqrt(t_0) + b), t_0)));
	} else {
		tmp = pow((fma(fma((c * -3.0), ((-0.375 * (a * a)) / pow(b, 3.0)), (1.5 * (a / b))), c, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.1458)
		tmp = Float64(0.3333333333333333 * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(a * fma(b, Float64(sqrt(t_0) + b), t_0))));
	else
		tmp = Float64(fma(fma(Float64(c * -3.0), Float64(Float64(-0.375 * Float64(a * a)) / (b ^ 3.0)), Float64(1.5 * Float64(a / b))), c, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.1458], N[(0.3333333333333333 * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(b * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(N[(c * -3.0), $MachinePrecision] * N[(N[(-0.375 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.145800000000000013

   1. Initial program 87.9%

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

      1. 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 87.9%

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

   6. Applied rewrites 89.0%

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

   if 0.145800000000000013 < b

   1. Initial program 51.3%

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

      1. 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 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 92.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{\left(\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{-0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{-0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.1458)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (pow
      (/
       (fma
        (fma (* c -3.0) (/ (* -0.375 (* a a)) (pow b 3.0)) (* 1.5 (/ a b)))
        c
        (* -2.0 b))
       c)
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.1458) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = pow((fma(fma((c * -3.0), ((-0.375 * (a * a)) / pow(b, 3.0)), (1.5 * (a / b))), c, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.1458)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(fma(fma(Float64(c * -3.0), Float64(Float64(-0.375 * Float64(a * a)) / (b ^ 3.0)), Float64(1.5 * Float64(a / b))), c, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.1458], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(N[(c * -3.0), $MachinePrecision] * N[(N[(-0.375 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.145800000000000013

   1. Initial program 87.9%

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

      1. 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 87.9%

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

   6. Applied rewrites 88.9%

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

   if 0.145800000000000013 < b

   1. Initial program 51.3%

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

      1. 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 51.3%

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 92.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{-0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 0.375\right) \cdot c, {b}^{-3}, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.1458)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (/
      0.3333333333333333
      (/
       (fma
        (fma (* (* (* a a) 0.375) c) (pow b -3.0) (* (/ a b) 0.5))
        c
        (* -0.6666666666666666 b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.1458) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = 0.3333333333333333 / (fma(fma((((a * a) * 0.375) * c), pow(b, -3.0), ((a / b) * 0.5)), c, (-0.6666666666666666 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.1458)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(fma(Float64(Float64(Float64(a * a) * 0.375) * c), (b ^ -3.0), Float64(Float64(a / b) * 0.5)), c, Float64(-0.6666666666666666 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.1458], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.375), $MachinePrecision] * c), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * c + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.145800000000000013

   1. Initial program 87.9%

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

      1. 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 87.9%

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

   6. Applied rewrites 88.9%

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

   if 0.145800000000000013 < b

   1. Initial program 51.3%

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

      1. 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 51.3%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 94.7%

      \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{5}}, 0.5625, \frac{-0.75}{b} \cdot \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)\right)}}^{-1} \]

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 92.1%

       \[\leadsto 0.3333333333333333 \cdot {\color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}\right)}}^{-1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow-1 N/A

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

       4. un-div-inv N/A

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

       5. lower-/.f64 92.2

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.375 \cdot \left(a \cdot a\right)}{{b}^{3}}, c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}}} \]

   12. Applied rewrites 92.2%

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

Alternative 5: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\left({b}^{-5} \cdot \mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right)\right) \cdot c\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.1458)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (fma
      (/ -0.5 b)
      c
      (*
       (*
        (* (pow b -5.0) (fma (* (* b b) a) -0.375 (* (* (* a a) c) -0.5625)))
        c)
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.1458) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = fma((-0.5 / b), c, (((pow(b, -5.0) * fma(((b * b) * a), -0.375, (((a * a) * c) * -0.5625))) * c) * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.1458)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = fma(Float64(-0.5 / b), c, Float64(Float64(Float64((b ^ -5.0) * fma(Float64(Float64(b * b) * a), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625))) * c) * c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.1458], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(N[(N[Power[b, -5.0], $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.145800000000000013

   1. Initial program 87.9%

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

      1. 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 87.9%

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

   6. Applied rewrites 88.9%

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

   if 0.145800000000000013 < b

   1. Initial program 51.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 91.9%

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

   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 91.9%

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

   10. Applied rewrites 92.0%

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

Alternative 6: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.1458:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 0.1458)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (*
      (fma
       (* (fma (* (* b b) a) -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((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 0.1458) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = fma((fma(((b * b) * a), -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(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.1458)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(b * b) * a), -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[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.1458], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * a), $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 < 0.145800000000000013

   1. Initial program 87.9%

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

      1. 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 87.9%

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

   6. Applied rewrites 88.9%

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

   if 0.145800000000000013 < b

   1. Initial program 51.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 91.9%

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

   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 91.9%

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

   10. Applied rewrites 91.9%

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

Alternative 7: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 31.0)
     (/ (* (- (* b b) t_0) (/ 0.3333333333333333 a)) (- (- b) (sqrt t_0)))
     (pow (/ (fma (* a (/ c b)) 1.5 (* -2.0 b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 31.0) {
		tmp = (((b * b) - t_0) * (0.3333333333333333 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = pow((fma((a * (c / b)), 1.5, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 31.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 31.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 31

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 81.9%

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

   6. Applied rewrites 82.7%

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

   if 31 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 31.0)
     (/ (- (* b b) t_0) (* (* 3.0 a) (- (- b) (sqrt t_0))))
     (pow (/ (fma (* a (/ c b)) 1.5 (* -2.0 b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 31.0) {
		tmp = ((b * b) - t_0) / ((3.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = pow((fma((a * (c / b)), 1.5, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 31.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(3.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 31.0], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 31

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 81.9%

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

   6. Applied rewrites 82.7%

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

   if 31 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 31:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_0 - b \cdot b}{a \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= b 31.0)
     (* 0.3333333333333333 (/ (- t_0 (* b b)) (* a (+ (sqrt t_0) b))))
     (pow (/ (fma (* a (/ c b)) 1.5 (* -2.0 b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (b <= 31.0) {
		tmp = 0.3333333333333333 * ((t_0 - (b * b)) / (a * (sqrt(t_0) + b)));
	} else {
		tmp = pow((fma((a * (c / b)), 1.5, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 31.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(t_0 - Float64(b * b)) / Float64(a * Float64(sqrt(t_0) + b))));
	else
		tmp = Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 31.0], N[(0.3333333333333333 * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 31

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 81.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{a}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}\right)}^{-1}} \]
   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 82.7%

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

   if 31 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 31:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{a \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 30.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (pow (/ (fma (* a (/ c b)) 1.5 (* -2.0 b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 30.0) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = pow((fma((a * (c / b)), 1.5, (-2.0 * b)) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 30.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(a * Float64(c / b)), 1.5, Float64(-2.0 * b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 30.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 30

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 82.1

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

   4. Applied rewrites 82.1%

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

   if 30 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 30.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (pow (fma 1.5 (/ a b) (* -2.0 (/ b c))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 30.0) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = pow(fma(1.5, (a / b), (-2.0 * (b / c))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 30.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = fma(1.5, Float64(a / b), Float64(-2.0 * Float64(b / c))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 30.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 30

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 82.1

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

   4. Applied rewrites 82.1%

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

   if 30 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 30.0)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (pow (fma 1.5 (/ a b) (* -2.0 (/ b c))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 30.0) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = pow(fma(1.5, (a / b), (-2.0 * (b / c))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 30.0)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = fma(1.5, Float64(a / b), Float64(-2.0 * Float64(b / c))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 30.0], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[Power[N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 30

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 81.9%

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

   if 30 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 30.0)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
   (pow (fma 1.5 (/ a b) (* -2.0 (/ b c))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 30.0) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = pow(fma(1.5, (a / b), (-2.0 * (b / c))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 30.0)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = fma(1.5, Float64(a / b), Float64(-2.0 * Float64(b / c))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 30.0], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 30

   1. Initial program 81.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 81.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.8

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

   4. Applied rewrites 81.9%

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

   if 30 < b

   1. Initial program 46.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 46.0%

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

   6. Applied rewrites 46.0%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 30:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(1.5, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

   1. 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 54.7%

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

6. Applied rewrites 54.7%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 83.4

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

9. Applied rewrites 83.4%

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

10. Final simplification 83.4%

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

Alternative 15: 64.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 65.2

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

5. Applied rewrites 65.2%

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

Reproduce

herbie shell --seed 2024299 
(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)))