Cubic critical, narrow range

Percentage Accurate: 55.0% → 91.7%

Time: 15.3s

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: 55.0% 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: 91.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9 \cdot \left({b}^{4} \cdot c\right)\\ t_1 := \mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{t\_0}^{2}}{{b}^{6}} \cdot -0.25\right)\\ t_2 := \mathsf{fma}\left(-27, {c}^{3}, \left(\frac{t\_1}{{b}^{6}} \cdot t\_0\right) \cdot -0.5\right)\\ \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(0.5, \frac{t\_2}{{b}^{6}} \cdot t\_0, \frac{{t\_1}^{2}}{{b}^{6}} \cdot 0.25\right)}{{b}^{3}} \cdot a, \frac{t\_2}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{t\_1}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{0.16666666666666666 \cdot t\_0}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(\mathsf{fma}\left(-1.5, \frac{a}{b}, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot c}{{b}^{5}}, \frac{\left(a \cdot a\right) \cdot -1.125}{{b}^{3}}\right) \cdot c\right) \cdot c + b\right) + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -9.0 (* (pow b 4.0) c)))
        (t_1
         (fma
          (* (* c c) (* b b))
          27.0
          (* (/ (pow t_0 2.0) (pow b 6.0)) -0.25)))
        (t_2 (fma -27.0 (pow c 3.0) (* (* (/ t_1 (pow b 6.0)) t_0) -0.5))))
   (/
    (fma
     a
     (fma
      a
      (fma
       -0.16666666666666666
       (*
        (/
         (fma
          0.5
          (* (/ t_2 (pow b 6.0)) t_0)
          (* (/ (pow t_1 2.0) (pow b 6.0)) 0.25))
         (pow b 3.0))
        a)
       (* (/ t_2 (pow b 3.0)) 0.16666666666666666))
      (* (/ t_1 (pow b 3.0)) 0.16666666666666666))
     (/ (* 0.16666666666666666 t_0) (pow b 3.0)))
    (fma
     b
     (+
      (+
       (*
        (fma
         -1.5
         (/ a b)
         (*
          (fma
           -1.6875
           (/ (* (pow a 3.0) c) (pow b 5.0))
           (/ (* (* a a) -1.125) (pow b 3.0)))
          c))
        c)
       b)
      b)
     (fma (* c -3.0) a (* b b))))))

C

double code(double a, double b, double c) {
	double t_0 = -9.0 * (pow(b, 4.0) * c);
	double t_1 = fma(((c * c) * (b * b)), 27.0, ((pow(t_0, 2.0) / pow(b, 6.0)) * -0.25));
	double t_2 = fma(-27.0, pow(c, 3.0), (((t_1 / pow(b, 6.0)) * t_0) * -0.5));
	return fma(a, fma(a, fma(-0.16666666666666666, ((fma(0.5, ((t_2 / pow(b, 6.0)) * t_0), ((pow(t_1, 2.0) / pow(b, 6.0)) * 0.25)) / pow(b, 3.0)) * a), ((t_2 / pow(b, 3.0)) * 0.16666666666666666)), ((t_1 / pow(b, 3.0)) * 0.16666666666666666)), ((0.16666666666666666 * t_0) / pow(b, 3.0))) / fma(b, (((fma(-1.5, (a / b), (fma(-1.6875, ((pow(a, 3.0) * c) / pow(b, 5.0)), (((a * a) * -1.125) / pow(b, 3.0))) * c)) * c) + b) + b), fma((c * -3.0), a, (b * b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(-9.0 * Float64((b ^ 4.0) * c))
	t_1 = fma(Float64(Float64(c * c) * Float64(b * b)), 27.0, Float64(Float64((t_0 ^ 2.0) / (b ^ 6.0)) * -0.25))
	t_2 = fma(-27.0, (c ^ 3.0), Float64(Float64(Float64(t_1 / (b ^ 6.0)) * t_0) * -0.5))
	return Float64(fma(a, fma(a, fma(-0.16666666666666666, Float64(Float64(fma(0.5, Float64(Float64(t_2 / (b ^ 6.0)) * t_0), Float64(Float64((t_1 ^ 2.0) / (b ^ 6.0)) * 0.25)) / (b ^ 3.0)) * a), Float64(Float64(t_2 / (b ^ 3.0)) * 0.16666666666666666)), Float64(Float64(t_1 / (b ^ 3.0)) * 0.16666666666666666)), Float64(Float64(0.16666666666666666 * t_0) / (b ^ 3.0))) / fma(b, Float64(Float64(Float64(fma(-1.5, Float64(a / b), Float64(fma(-1.6875, Float64(Float64((a ^ 3.0) * c) / (b ^ 5.0)), Float64(Float64(Float64(a * a) * -1.125) / (b ^ 3.0))) * c)) * c) + b) + b), fma(Float64(c * -3.0), a, Float64(b * b))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-9.0 * N[(N[Power[b, 4.0], $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * c), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-27.0 * N[Power[c, 3.0], $MachinePrecision] + N[(N[(N[(t$95$1 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(a * N[(-0.16666666666666666 * N[(N[(N[(0.5 * N[(N[(t$95$2 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + N[(N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * t$95$0), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(N[(N[(-1.5 * N[(a / b), $MachinePrecision] + N[(N[(-1.6875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -1.125), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] + b), $MachinePrecision] + b), $MachinePrecision] + N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 56.9%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \frac{-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)}{{b}^{3}} + a \cdot \left(\frac{1}{6} \cdot \frac{\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}}{{b}^{3}} + a \cdot \left(\frac{-1}{6} \cdot \frac{a \cdot \left(\frac{1}{4} \cdot \frac{{\left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}^{2}}{{b}^{6}} + \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(-27 \cdot {c}^{3} - \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)}{{b}^{6}}\right)}{{b}^{3}} + \frac{1}{6} \cdot \frac{-27 \cdot {c}^{3} - \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}}{{b}^{3}}\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

9. Applied rewrites 92.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, a \cdot \frac{\mathsf{fma}\left(0.5, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, 0.25 \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{0.16666666666666666 \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \color{blue}{\left(b + c \cdot \left(\frac{-3}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \color{blue}{\left(b + c \cdot \left(\frac{-3}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + \color{blue}{c \cdot \left(\frac{-3}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \color{blue}{\mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \color{blue}{\frac{a}{b}}, c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, \color{blue}{c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)}\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \color{blue}{\mathsf{fma}\left(\frac{-27}{16}, \frac{{a}^{3} \cdot c}{{b}^{5}}, \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)}\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    7. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \color{blue}{\frac{{a}^{3} \cdot c}{{b}^{5}}}, \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \frac{\color{blue}{{a}^{3} \cdot c}}{{b}^{5}}, \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    9. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \frac{\color{blue}{{a}^{3}} \cdot c}{{b}^{5}}, \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    10. lower-pow.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \frac{{a}^{3} \cdot c}{\color{blue}{{b}^{5}}}, \frac{-9}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    11. associate-*r/ N/A

        \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \frac{{a}^{3} \cdot c}{{b}^{5}}, \color{blue}{\frac{\frac{-9}{8} \cdot {a}^{2}}{{b}^{3}}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

    12. lower-/.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, \frac{1}{4} \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(b + c \cdot \mathsf{fma}\left(\frac{-3}{2}, \frac{a}{b}, c \cdot \mathsf{fma}\left(\frac{-27}{16}, \frac{{a}^{3} \cdot c}{{b}^{5}}, \color{blue}{\frac{\frac{-9}{8} \cdot {a}^{2}}{{b}^{3}}}\right)\right)\right) + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

12. Applied rewrites 92.1%

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, a \cdot \frac{\mathsf{fma}\left(0.5, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, 0.25 \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{0.16666666666666666 \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \color{blue}{\left(b + c \cdot \mathsf{fma}\left(-1.5, \frac{a}{b}, c \cdot \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot c}{{b}^{5}}, \frac{-1.125 \cdot \left(a \cdot a\right)}{{b}^{3}}\right)\right)\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

13. Final simplification 92.1%

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {c}^{3}, \left(\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)}{{b}^{6}} \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{6}} \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right), \frac{{\left(\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)\right)}^{2}}{{b}^{6}} \cdot 0.25\right)}{{b}^{3}} \cdot a, \frac{\mathsf{fma}\left(-27, {c}^{3}, \left(\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)}{{b}^{6}} \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{0.16666666666666666 \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \left(\mathsf{fma}\left(-1.5, \frac{a}{b}, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot c}{{b}^{5}}, \frac{\left(a \cdot a\right) \cdot -1.125}{{b}^{3}}\right) \cdot c\right) \cdot c + b\right) + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)} \]
14. Add Preprocessing

Alternative 2: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6.75 \cdot \left(c \cdot c\right)\\ t_1 := -9 \cdot \left({b}^{4} \cdot c\right)\\ t_2 := \mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{t\_1}^{2}}{{b}^{6}} \cdot -0.25\right)\\ t_3 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{\mathsf{fma}\left(-4.5, \mathsf{fma}\left(-27, {c}^{3}, \left(t\_0 \cdot c\right) \cdot 4.5\right) \cdot c, {t\_0}^{2} \cdot 0.25\right)}{b \cdot b}}{{b}^{3}} \cdot a, \frac{\mathsf{fma}\left(-27, {c}^{3}, \left(\frac{t\_2}{{b}^{6}} \cdot t\_1\right) \cdot -0.5\right)}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{t\_2}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{0.16666666666666666 \cdot t\_1}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \sqrt{t\_3} + b, t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 6.75 (* c c)))
        (t_1 (* -9.0 (* (pow b 4.0) c)))
        (t_2
         (fma
          (* (* c c) (* b b))
          27.0
          (* (/ (pow t_1 2.0) (pow b 6.0)) -0.25)))
        (t_3 (fma (* c -3.0) a (* b b))))
   (/
    (fma
     a
     (fma
      a
      (fma
       -0.16666666666666666
       (*
        (/
         (/
          (fma
           -4.5
           (* (fma -27.0 (pow c 3.0) (* (* t_0 c) 4.5)) c)
           (* (pow t_0 2.0) 0.25))
          (* b b))
         (pow b 3.0))
        a)
       (*
        (/
         (fma -27.0 (pow c 3.0) (* (* (/ t_2 (pow b 6.0)) t_1) -0.5))
         (pow b 3.0))
        0.16666666666666666))
      (* (/ t_2 (pow b 3.0)) 0.16666666666666666))
     (/ (* 0.16666666666666666 t_1) (pow b 3.0)))
    (fma b (+ (sqrt t_3) b) t_3))))

C

double code(double a, double b, double c) {
	double t_0 = 6.75 * (c * c);
	double t_1 = -9.0 * (pow(b, 4.0) * c);
	double t_2 = fma(((c * c) * (b * b)), 27.0, ((pow(t_1, 2.0) / pow(b, 6.0)) * -0.25));
	double t_3 = fma((c * -3.0), a, (b * b));
	return fma(a, fma(a, fma(-0.16666666666666666, (((fma(-4.5, (fma(-27.0, pow(c, 3.0), ((t_0 * c) * 4.5)) * c), (pow(t_0, 2.0) * 0.25)) / (b * b)) / pow(b, 3.0)) * a), ((fma(-27.0, pow(c, 3.0), (((t_2 / pow(b, 6.0)) * t_1) * -0.5)) / pow(b, 3.0)) * 0.16666666666666666)), ((t_2 / pow(b, 3.0)) * 0.16666666666666666)), ((0.16666666666666666 * t_1) / pow(b, 3.0))) / fma(b, (sqrt(t_3) + b), t_3);
}

Julia

function code(a, b, c)
	t_0 = Float64(6.75 * Float64(c * c))
	t_1 = Float64(-9.0 * Float64((b ^ 4.0) * c))
	t_2 = fma(Float64(Float64(c * c) * Float64(b * b)), 27.0, Float64(Float64((t_1 ^ 2.0) / (b ^ 6.0)) * -0.25))
	t_3 = fma(Float64(c * -3.0), a, Float64(b * b))
	return Float64(fma(a, fma(a, fma(-0.16666666666666666, Float64(Float64(Float64(fma(-4.5, Float64(fma(-27.0, (c ^ 3.0), Float64(Float64(t_0 * c) * 4.5)) * c), Float64((t_0 ^ 2.0) * 0.25)) / Float64(b * b)) / (b ^ 3.0)) * a), Float64(Float64(fma(-27.0, (c ^ 3.0), Float64(Float64(Float64(t_2 / (b ^ 6.0)) * t_1) * -0.5)) / (b ^ 3.0)) * 0.16666666666666666)), Float64(Float64(t_2 / (b ^ 3.0)) * 0.16666666666666666)), Float64(Float64(0.16666666666666666 * t_1) / (b ^ 3.0))) / fma(b, Float64(sqrt(t_3) + b), t_3))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(6.75 * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-9.0 * N[(N[Power[b, 4.0], $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * c), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(a * N[(-0.16666666666666666 * N[(N[(N[(N[(-4.5 * N[(N[(-27.0 * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$0 * c), $MachinePrecision] * 4.5), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(-27.0 * N[Power[c, 3.0], $MachinePrecision] + N[(N[(N[(t$95$2 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(0.16666666666666666 * t$95$1), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[Sqrt[t$95$3], $MachinePrecision] + b), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 56.9%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \frac{-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)}{{b}^{3}} + a \cdot \left(\frac{1}{6} \cdot \frac{\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}}{{b}^{3}} + a \cdot \left(\frac{-1}{6} \cdot \frac{a \cdot \left(\frac{1}{4} \cdot \frac{{\left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}^{2}}{{b}^{6}} + \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(-27 \cdot {c}^{3} - \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)}{{b}^{6}}\right)}{{b}^{3}} + \frac{1}{6} \cdot \frac{-27 \cdot {c}^{3} - \frac{1}{2} \cdot \frac{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({b}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot \frac{{\left(-6 \cdot \left({b}^{4} \cdot c\right) + -3 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}}{{b}^{3}}\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

9. Applied rewrites 92.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, a \cdot \frac{\mathsf{fma}\left(0.5, \left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{6}}, 0.25 \cdot \frac{{\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}, 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{0.16666666666666666 \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

10. Taylor expanded in b around 0

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{6}, a \cdot \frac{\frac{\frac{-9}{2} \cdot \left(c \cdot \left(-27 \cdot {c}^{3} + \frac{9}{2} \cdot \left(c \cdot \left(\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}\right)\right)\right)\right) + \frac{1}{4} \cdot {\left(\frac{-81}{4} \cdot {c}^{2} + 27 \cdot {c}^{2}\right)}^{2}}{{b}^{2}}}{{\color{blue}{b}}^{3}}, \frac{1}{6} \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, \frac{-1}{2} \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), \frac{1}{6} \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, \frac{-1}{4} \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{\frac{1}{6} \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]
11. Step-by-step derivation

12. Applied rewrites 92.0%

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, a \cdot \frac{\frac{\mathsf{fma}\left(-4.5, c \cdot \mathsf{fma}\left(-27, {c}^{3}, 4.5 \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 6.75\right)\right)\right), 0.25 \cdot {\left(\left(c \cdot c\right) \cdot 6.75\right)}^{2}\right)}{b \cdot b}}{{\color{blue}{b}}^{3}}, 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(-27, {c}^{3}, -0.5 \cdot \left(\left(\left({b}^{4} \cdot c\right) \cdot -9\right) \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{6}}\right)\right)}{{b}^{3}}\right), 0.16666666666666666 \cdot \frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot \frac{{\left(\left({b}^{4} \cdot c\right) \cdot -9\right)}^{2}}{{b}^{6}}\right)}{{b}^{3}}\right), \frac{0.16666666666666666 \cdot \left(\left({b}^{4} \cdot c\right) \cdot -9\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

13. Final simplification 92.0%

    \[\leadsto \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{\mathsf{fma}\left(-4.5, \mathsf{fma}\left(-27, {c}^{3}, \left(\left(6.75 \cdot \left(c \cdot c\right)\right) \cdot c\right) \cdot 4.5\right) \cdot c, {\left(6.75 \cdot \left(c \cdot c\right)\right)}^{2} \cdot 0.25\right)}{b \cdot b}}{{b}^{3}} \cdot a, \frac{\mathsf{fma}\left(-27, {c}^{3}, \left(\frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)}{{b}^{6}} \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right)\right) \cdot -0.5\right)}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \left(b \cdot b\right), 27, \frac{{\left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}^{2}}{{b}^{6}} \cdot -0.25\right)}{{b}^{3}} \cdot 0.16666666666666666\right), \frac{0.16666666666666666 \cdot \left(-9 \cdot \left({b}^{4} \cdot c\right)\right)}{{b}^{3}}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)} \]
14. Add Preprocessing

Alternative 3: 91.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -9\\ t_1 := \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, {t\_0}^{2} \cdot -0.25\right)\\ t_2 := \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(t\_1 \cdot t\_0\right) \cdot -0.5\right)\\ t_3 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(t\_0 \cdot 0.5, t\_2, {t\_1}^{2} \cdot 0.25\right)}{{b}^{6} \cdot a}, \mathsf{fma}\left(0.16666666666666666, \frac{t\_1}{\left(b \cdot b\right) \cdot a} + \frac{t\_2}{{b}^{4} \cdot a}, \frac{t\_0}{a} \cdot 0.16666666666666666\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{t\_3} + b, t\_3\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) -9.0))
        (t_1 (fma (* (* a a) (* c c)) 27.0 (* (pow t_0 2.0) -0.25)))
        (t_2 (fma (* (pow a 3.0) -27.0) (pow c 3.0) (* (* t_1 t_0) -0.5)))
        (t_3 (fma (* c -3.0) a (* b b))))
   (/
    (*
     (fma
      -0.16666666666666666
      (/ (fma (* t_0 0.5) t_2 (* (pow t_1 2.0) 0.25)) (* (pow b 6.0) a))
      (fma
       0.16666666666666666
       (+ (/ t_1 (* (* b b) a)) (/ t_2 (* (pow b 4.0) a)))
       (* (/ t_0 a) 0.16666666666666666)))
     b)
    (fma b (+ (sqrt t_3) b) t_3))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * -9.0;
	double t_1 = fma(((a * a) * (c * c)), 27.0, (pow(t_0, 2.0) * -0.25));
	double t_2 = fma((pow(a, 3.0) * -27.0), pow(c, 3.0), ((t_1 * t_0) * -0.5));
	double t_3 = fma((c * -3.0), a, (b * b));
	return (fma(-0.16666666666666666, (fma((t_0 * 0.5), t_2, (pow(t_1, 2.0) * 0.25)) / (pow(b, 6.0) * a)), fma(0.16666666666666666, ((t_1 / ((b * b) * a)) + (t_2 / (pow(b, 4.0) * a))), ((t_0 / a) * 0.16666666666666666))) * b) / fma(b, (sqrt(t_3) + b), t_3);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * -9.0)
	t_1 = fma(Float64(Float64(a * a) * Float64(c * c)), 27.0, Float64((t_0 ^ 2.0) * -0.25))
	t_2 = fma(Float64((a ^ 3.0) * -27.0), (c ^ 3.0), Float64(Float64(t_1 * t_0) * -0.5))
	t_3 = fma(Float64(c * -3.0), a, Float64(b * b))
	return Float64(Float64(fma(-0.16666666666666666, Float64(fma(Float64(t_0 * 0.5), t_2, Float64((t_1 ^ 2.0) * 0.25)) / Float64((b ^ 6.0) * a)), fma(0.16666666666666666, Float64(Float64(t_1 / Float64(Float64(b * b) * a)) + Float64(t_2 / Float64((b ^ 4.0) * a))), Float64(Float64(t_0 / a) * 0.16666666666666666))) * b) / fma(b, Float64(sqrt(t_3) + b), t_3))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -9.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * 27.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * -27.0), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.16666666666666666 * N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] * t$95$2 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(t$95$1 / N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / N[(N[Power[b, 4.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 / a), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(b * N[(N[Sqrt[t$95$3], $MachinePrecision] + b), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 56.9%

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

7. Taylor expanded in b around inf

   \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{6} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{6}} + \left(\frac{1}{6} \cdot \frac{-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)}{a} + \left(\frac{1}{6} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{a \cdot {b}^{4}} + \frac{1}{6} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{a \cdot {b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

9. Applied rewrites 91.9%

   \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{a \cdot {b}^{6}}, \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(-27 \cdot {a}^{3}, {c}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{a \cdot {b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{a \cdot \left(b \cdot b\right)}, 0.16666666666666666 \cdot \frac{\left(a \cdot c\right) \cdot -9}{a}\right)\right)}}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right)} \]

10. Final simplification 91.9%

    \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(\left(\left(c \cdot a\right) \cdot -9\right) \cdot 0.5, \mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, {\left(\left(c \cdot a\right) \cdot -9\right)}^{2} \cdot -0.25\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right) \cdot -0.5\right), {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, {\left(\left(c \cdot a\right) \cdot -9\right)}^{2} \cdot -0.25\right)\right)}^{2} \cdot 0.25\right)}{{b}^{6} \cdot a}, \mathsf{fma}\left(0.16666666666666666, \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, {\left(\left(c \cdot a\right) \cdot -9\right)}^{2} \cdot -0.25\right)}{\left(b \cdot b\right) \cdot a} + \frac{\mathsf{fma}\left({a}^{3} \cdot -27, {c}^{3}, \left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, {\left(\left(c \cdot a\right) \cdot -9\right)}^{2} \cdot -0.25\right) \cdot \left(\left(c \cdot a\right) \cdot -9\right)\right) \cdot -0.5\right)}{{b}^{4} \cdot a}, \frac{\left(c \cdot a\right) \cdot -9}{a} \cdot 0.16666666666666666\right)\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right)} \]
11. Add Preprocessing

Alternative 4: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, a, \frac{c}{{b}^{3}} \cdot -0.375\right) \cdot -3, a, \frac{1.5}{b}\right), a, -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   (fma
    (*
     (fma (/ (* -0.5625 (* c c)) (pow b 5.0)) a (* (/ c (pow b 3.0)) -0.375))
     -3.0)
    a
    (/ 1.5 b))
   a
   (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(fma((fma(((-0.5625 * (c * c)) / pow(b, 5.0)), a, ((c / pow(b, 3.0)) * -0.375)) * -3.0), a, (1.5 / b)), a, (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(fma(Float64(fma(Float64(Float64(-0.5625 * Float64(c * c)) / (b ^ 5.0)), a, Float64(Float64(c / (b ^ 3.0)) * -0.375)) * -3.0), a, Float64(1.5 / b)), a, Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(N[(N[(N[(-0.5625 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] * a + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 91.9%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-3 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c \cdot c}, -0.2222222222222222, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{5}}, 0.5625, \frac{-0.75}{b} \cdot \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}\right)\right), a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{1.5}{b}\right), a, \frac{b}{c} \cdot -2\right)}} \]

8. Taylor expanded in c around 0

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

10. Applied rewrites 91.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-3 \cdot \mathsf{fma}\left(\frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, a, \frac{c}{{b}^{3}} \cdot -0.375\right), a, \frac{1.5}{b}\right), a, \frac{b}{c} \cdot -2\right)} \]

11. Final simplification 91.9%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, a, \frac{c}{{b}^{3}} \cdot -0.375\right) \cdot -3, a, \frac{1.5}{b}\right), a, -2 \cdot \frac{b}{c}\right)} \]
12. Add Preprocessing

Alternative 5: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, -3, \frac{1.5}{b}\right), a, -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   (fma (* (* (/ c (pow b 3.0)) -0.375) a) -3.0 (/ 1.5 b))
   a
   (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(fma((((c / pow(b, 3.0)) * -0.375) * a), -3.0, (1.5 / b)), a, (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(fma(Float64(Float64(Float64(c / (b ^ 3.0)) * -0.375) * a), -3.0, Float64(1.5 / b)), a, Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] * a), $MachinePrecision] * -3.0 + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 88.5%

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

8. Final simplification 88.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, -3, \frac{1.5}{b}\right), a, -2 \cdot \frac{b}{c}\right)} \]
9. Add Preprocessing

Alternative 6: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 33.5)
     (/ (* (- t_0 (* b b)) (/ 0.3333333333333333 a)) (+ (sqrt t_0) b))
     (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 33.5) {
		tmp = ((t_0 - (b * b)) * (0.3333333333333333 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.3333333333333333 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 33.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 81.1%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 33.5)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ (sqrt t_0) b)))
     (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 33.5) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (sqrt(t_0) + b));
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(3.0 * a) * Float64(sqrt(t_0) + b)));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 33.5], 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[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 79.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-*.f64 N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 81.0%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 33.5)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* 3.0 a))
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 33.5) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 33.5], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 79.5

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

   4. Applied rewrites 79.5%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 33.5)
   (/ (* (- (sqrt (fma (* c -3.0) a (* b b))) b) 0.3333333333333333) a)
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 33.5) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 33.5], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. 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-/r* N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.2%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 33.5)
   (* (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a) 0.3333333333333333)
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 33.5) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 33.5], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 79.2%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 33.5)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 33.5) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 33.5)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 33.5], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 33.5

   1. Initial program 79.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 79.2

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

      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 79.2

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

   4. Applied rewrites 79.2%

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

   if 33.5 < b

   1. Initial program 47.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 47.2

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 47.2

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

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 33.5:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

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

   4. lower-/.f64 56.0

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 56.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 56.0

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

4. Applied rewrites 56.0%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 82.3

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

7. Applied rewrites 82.3%

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

8. Final simplification 82.3%

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

Alternative 13: 64.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.3

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

5. Applied rewrites 64.3%

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

Alternative 14: 64.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.3

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

5. Applied rewrites 64.3%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 64.2%

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

8. Final simplification 64.2%

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

Reproduce

herbie shell --seed 2024241 
(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)))