Cubic critical, narrow range

Percentage Accurate: 55.8% → 91.9%

Time: 11.7s

Alternatives: 14

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: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-0.5625 \cdot {a}^{3}}{{b}^{5}}, \frac{\frac{a \cdot a}{b \cdot b}}{-b} \cdot -0.375\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -3.0 (* b b))))
   (if (<= b 0.34)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (/
       (fma
        (fma
         (fma
          (- c)
          (/ (* -0.5625 (pow a 3.0)) (pow b 5.0))
          (* (/ (/ (* a a) (* b b)) (- b)) -0.375))
         c
         (* (/ a b) 0.5))
        c
        (* -0.6666666666666666 b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -3.0, (b * b));
	double tmp;
	if (b <= 0.34) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / (fma(fma(fma(-c, ((-0.5625 * pow(a, 3.0)) / pow(b, 5.0)), ((((a * a) / (b * b)) / -b) * -0.375)), c, ((a / b) * 0.5)), c, (-0.6666666666666666 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -3.0, Float64(b * b))
	tmp = 0.0
	if (b <= 0.34)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(fma(fma(Float64(-c), Float64(Float64(-0.5625 * (a ^ 3.0)) / (b ^ 5.0)), Float64(Float64(Float64(Float64(a * a) / Float64(b * b)) / Float64(-b)) * -0.375)), c, 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[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.34], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(N[((-c) * N[(N[(-0.5625 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c + 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 < 0.340000000000000024

   1. Initial program 87.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 88.0

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 89.8%

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

   if 0.340000000000000024 < b

   1. Initial program 49.9%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.9%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.3%

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

   12. Applied rewrites 93.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right) - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-0.5625 \cdot {a}^{3}}{{b}^{5}}, \frac{\frac{a \cdot a}{b \cdot b}}{-b} \cdot -0.375\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{5}}, 0.5625, \frac{c}{{b}^{3}} \cdot 0.375\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -3.0 (* b b))))
   (if (<= b 0.34)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma
       (fma
        (fma (/ (* (* c c) a) (pow b 5.0)) 0.5625 (* (/ c (pow b 3.0)) 0.375))
        a
        (/ 0.5 b))
       a
       (* (/ b c) -0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -3.0, (b * b));
	double tmp;
	if (b <= 0.34) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma(fma(fma((((c * c) * a) / pow(b, 5.0)), 0.5625, ((c / pow(b, 3.0)) * 0.375)), a, (0.5 / b)), a, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -3.0, Float64(b * b))
	tmp = 0.0
	if (b <= 0.34)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(fma(fma(Float64(Float64(Float64(c * c) * a) / (b ^ 5.0)), 0.5625, Float64(Float64(c / (b ^ 3.0)) * 0.375)), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.34], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * 0.5625 + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.340000000000000024

   1. Initial program 87.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 88.0

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 89.8%

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

   if 0.340000000000000024 < b

   1. Initial program 49.9%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.9%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.3%

       \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-0.5625 \cdot {a}^{3}}{{b}^{5}}, \left(-\frac{a \cdot a}{{b}^{3}}\right) \cdot -0.375\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 93.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right) - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{5}}, 0.5625, \frac{c}{{b}^{3}} \cdot 0.375\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, a \cdot \mathsf{fma}\left(-1, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.375\right), \frac{0.5}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -3.0 (* b b))))
   (if (<= b 0.35)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma
       -0.6666666666666666
       (/ b c)
       (* a (fma -1.0 (* a (* (/ c (pow b 3.0)) -0.375)) (/ 0.5 b))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -3.0, (b * b));
	double tmp;
	if (b <= 0.35) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), (a * fma(-1.0, (a * ((c / pow(b, 3.0)) * -0.375)), (0.5 / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -3.0, Float64(b * b))
	tmp = 0.0
	if (b <= 0.35)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(a * fma(-1.0, Float64(a * Float64(Float64(c / (b ^ 3.0)) * -0.375)), Float64(0.5 / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.35], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(a * N[(-1.0 * N[(a * N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.34999999999999998

   1. Initial program 87.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 88.0

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 89.8%

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

   if 0.34999999999999998 < b

   1. Initial program 49.9%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 49.9

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 49.9

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

   6. Applied rewrites 49.9%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 91.0

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

   9. Applied rewrites 91.0%

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

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{3}}, 0.375, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -3.0 (* b b))))
   (if (<= b 0.35)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma
       (fma (* a (/ c (pow b 3.0))) 0.375 (/ 0.5 b))
       a
       (* (/ b c) -0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -3.0, (b * b));
	double tmp;
	if (b <= 0.35) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma(fma((a * (c / pow(b, 3.0))), 0.375, (0.5 / b)), a, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -3.0, Float64(b * b))
	tmp = 0.0
	if (b <= 0.35)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(fma(Float64(a * Float64(c / (b ^ 3.0))), 0.375, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.35], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375 + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.34999999999999998

   1. Initial program 87.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 88.0

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 88.0

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

   6. Applied rewrites 88.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 89.8%

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

   if 0.34999999999999998 < b

   1. Initial program 49.9%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.9%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 93.3%

       \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-c, \frac{-0.5625 \cdot {a}^{3}}{{b}^{5}}, \left(-\frac{a \cdot a}{{b}^{3}}\right) \cdot -0.375\right), c, \frac{a}{b} \cdot 0.5\right), c, -0.6666666666666666 \cdot b\right)}{c}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 90.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.35:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right) - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{3}}, 0.375, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}^{-1} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.52)
   (/
    0.3333333333333333
    (* (pow (- (sqrt (fma b b (* (* c a) -3.0))) b) -1.0) a))
   (/
    0.3333333333333333
    (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.52) {
		tmp = 0.3333333333333333 / (pow((sqrt(fma(b, b, ((c * a) * -3.0))) - b), -1.0) * a);
	} else {
		tmp = 0.3333333333333333 / (fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.52)
		tmp = Float64(0.3333333333333333 / Float64((Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b) ^ -1.0) * a));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.52], N[(0.3333333333333333 / N[(N[Power[N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.52000000000000002

   1. Initial program 86.7%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 86.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 86.9

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 86.9

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

   6. Applied rewrites 86.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

   if 0.52000000000000002 < b

   1. Initial program 49.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b\right)}^{-1} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -3.0 (* b b))))
   (if (<= b 130.0)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -3.0, (b * b));
	double tmp;
	if (b <= 130.0) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -3.0, Float64(b * b))
	tmp = 0.0
	if (b <= 130.0)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 130.0], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 130

   1. Initial program 78.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 78.8

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 78.8

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

   6. Applied rewrites 78.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.0

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

   8. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   10. Applied rewrites 80.6%

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

   if 130 < b

   1. Initial program 45.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right) - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{t\_0 - b \cdot b} \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 130.0)
     (/ 0.3333333333333333 (* (/ a (- t_0 (* b b))) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 130.0) {
		tmp = 0.3333333333333333 / ((a / (t_0 - (b * b))) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 130.0)
		tmp = Float64(0.3333333333333333 / Float64(Float64(a / Float64(t_0 - Float64(b * b))) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 130.0], N[(0.3333333333333333 / N[(N[(a / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 130

   1. Initial program 78.8%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 78.8%

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

   6. Applied rewrites 80.6%

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

   if 130 < b

   1. Initial program 45.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{a}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 130.0)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ (sqrt t_0) b)))
     (/
      0.3333333333333333
      (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 130.0) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (sqrt(t_0) + b));
	} else {
		tmp = 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 130.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 130.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 130

   1. Initial program 78.8%

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

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

   4. Applied rewrites 78.8%

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

   6. Applied rewrites 80.5%

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

   if 130 < b

   1. Initial program 45.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 130:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.52)
   (/ (/ (- (sqrt (fma b b (* (* c a) -3.0))) b) a) 3.0)
   (/
    0.3333333333333333
    (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.52) {
		tmp = ((sqrt(fma(b, b, ((c * a) * -3.0))) - b) / a) / 3.0;
	} else {
		tmp = 0.3333333333333333 / (fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.52)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -3.0))) - b) / a) / 3.0);
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.52], N[(N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(0.3333333333333333 / N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.52000000000000002

   1. Initial program 86.7%

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

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

   4. Applied rewrites 86.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.1

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

   6. Applied rewrites 87.1%

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

   if 0.52000000000000002 < b

   1. Initial program 49.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.52)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* c a) -3.0 (* b b))) b))
   (/
    0.3333333333333333
    (/ (fma 0.5 (* a (/ c b)) (* -0.6666666666666666 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.52) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((c * a), -3.0, (b * b))) - b);
	} else {
		tmp = 0.3333333333333333 / (fma(0.5, (a * (c / b)), (-0.6666666666666666 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.52)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) - b));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(0.5, Float64(a * Float64(c / b)), Float64(-0.6666666666666666 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.52], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(0.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.52000000000000002

   1. Initial program 86.7%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 86.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 86.9

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 86.9

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

   6. Applied rewrites 86.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. associate-/r/ N/A

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

   10. Applied rewrites 86.9%

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

   if 0.52000000000000002 < b

   1. Initial program 49.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(0.5, a \cdot \frac{c}{b}, -0.6666666666666666 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.52)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* c a) -3.0 (* b b))) b))
   (/ 0.3333333333333333 (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.52) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((c * a), -3.0, (b * b))) - b);
	} else {
		tmp = 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.52)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) - b));
	else
		tmp = Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.52], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.52000000000000002

   1. Initial program 86.7%

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

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 86.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 86.9

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

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

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 86.9

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

   6. Applied rewrites 86.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.1

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

   8. Applied rewrites 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lift--.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. +-commutative N/A

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

      14. associate-/r/ N/A

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

   10. Applied rewrites 86.9%

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

   if 0.52000000000000002 < b

   1. Initial program 49.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.52)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
   (/ 0.3333333333333333 (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.52) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.52)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.52], 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[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.52000000000000002

   1. Initial program 86.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. 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 86.9

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

      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 86.9

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

   4. Applied rewrites 86.9%

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

   if 0.52000000000000002 < b

   1. Initial program 49.5%

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

      1. 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. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma (/ a b) 0.5 (* (/ b c) -0.6666666666666666))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma((a / b), 0.5, ((b / c) * -0.6666666666666666));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.6666666666666666)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

4. Applied rewrites 54.3%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 81.2

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

7. Applied rewrites 81.2%

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

8. Final simplification 81.2%

   \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.6666666666666666\right)} \]
9. Add Preprocessing

Alternative 14: 64.0% 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 54.3%

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

3. Taylor expanded in 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 64.8

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

5. Applied rewrites 64.8%

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

Reproduce

herbie shell --seed 2024322 
(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)))