Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.5%

Time: 14.4s

Alternatives: 16

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{3} \cdot -0.5625}{{b}^{5}}, c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375\right) \cdot -3, c, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 1.9)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (/
      1.0
      (/
       (fma
        (fma
         (*
          (fma
           (/ (* (pow a 3.0) -0.5625) (pow b 5.0))
           c
           (* (/ (* a a) (pow b 3.0)) -0.375))
          -3.0)
         c
         (* 1.5 (/ a b)))
        c
        (* -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 <= 1.9) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = 1.0 / (fma(fma((fma(((pow(a, 3.0) * -0.5625) / pow(b, 5.0)), c, (((a * a) / pow(b, 3.0)) * -0.375)) * -3.0), c, (1.5 * (a / b))), c, (-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 <= 1.9)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(1.0 / Float64(fma(fma(Float64(fma(Float64(Float64((a ^ 3.0) * -0.5625) / (b ^ 5.0)), c, Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375)) * -3.0), c, Float64(1.5 * Float64(a / b))), c, Float64(-2.0 * 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, 1.9], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(N[(N[(N[Power[a, 3.0], $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c + N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision] * c + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8999999999999999

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 1.8999999999999999 < b

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 48.3

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

      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 48.3

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

      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 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 95.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{{a}^{3} \cdot -0.5625}{{b}^{5}}, c, \frac{a \cdot a}{{b}^{3}} \cdot -0.375\right) \cdot -3, c, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 1.9)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (fma
      (/
       (fma
        (* -1.0546875 (* a a))
        (pow c 4.0)
        (*
         (fma (* -0.375 (* b b)) (* c c) (* (* (pow c 3.0) a) -0.5625))
         (* b b)))
       (pow b 7.0))
      a
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 1.9) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = fma((fma((-1.0546875 * (a * a)), pow(c, 4.0), (fma((-0.375 * (b * b)), (c * c), ((pow(c, 3.0) * a) * -0.5625)) * (b * b))) / pow(b, 7.0)), a, (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 1.9)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = fma(Float64(fma(Float64(-1.0546875 * Float64(a * a)), (c ^ 4.0), Float64(fma(Float64(-0.375 * Float64(b * b)), Float64(c * c), Float64(Float64((c ^ 3.0) * a) * -0.5625)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.9], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0546875 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8999999999999999

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 1.8999999999999999 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-0.375 \cdot \left(b \cdot b\right), c \cdot c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 2.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (/
      1.0
      (/
       (fma
        (fma (* c -3.0) (* (/ (* a a) (pow b 3.0)) -0.375) (* 1.5 (/ a b)))
        c
        (* -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 <= 2.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = 1.0 / (fma(fma((c * -3.0), (((a * a) / pow(b, 3.0)) * -0.375), (1.5 * (a / b))), c, (-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 <= 2.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(1.0 / Float64(fma(fma(Float64(c * -3.0), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.375), Float64(1.5 * Float64(a / b))), c, Float64(-2.0 * 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, 2.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * -3.0), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 2.7000000000000002 < b

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 48.3

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

      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 48.3

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

      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 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 93.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -3, \frac{a \cdot a}{{b}^{3}} \cdot -0.375, 1.5 \cdot \frac{a}{b}\right), c, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, -3, \frac{1.5}{b}\right), a, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 2.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (/
      1.0
      (fma
       (fma (* (* (/ c (pow b 3.0)) -0.375) a) -3.0 (/ 1.5 b))
       a
       (* (/ b c) -2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 2.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = 1.0 / fma(fma((((c / pow(b, 3.0)) * -0.375) * a), -3.0, (1.5 / b)), a, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 2.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(1.0 / fma(fma(Float64(Float64(Float64(c / (b ^ 3.0)) * -0.375) * a), -3.0, Float64(1.5 / b)), a, Float64(Float64(b / c) * -2.0)));
	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, 2.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 2.7000000000000002 < b

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 48.3

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

      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 48.3

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

      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 48.3

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

   4. Applied rewrites 48.3%

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

   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 93.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot a, -3, \frac{1.5}{b}\right), a, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 2.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (fma
      (/ -0.5 b)
      c
      (*
       (*
        (* (fma (* (* b b) a) -0.375 (* (* (* a a) c) -0.5625)) (pow b -5.0))
        c)
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 2.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = fma((-0.5 / b), c, (((fma(((b * b) * a), -0.375, (((a * a) * c) * -0.5625)) * pow(b, -5.0)) * c) * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 2.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = fma(Float64(-0.5 / b), c, Float64(Float64(Float64(fma(Float64(Float64(b * b) * a), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) * (b ^ -5.0)) * c) * c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 2.7000000000000002 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 93.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, \left(\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}\right) \cdot c\right) \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 2.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (*
      (fma
       (* (fma (* (* b b) a) -0.375 (* (* (* a a) c) -0.5625)) (pow b -5.0))
       c
       (/ -0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 2.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = fma((fma(((b * b) * a), -0.375, (((a * a) * c) * -0.5625)) * pow(b, -5.0)), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 2.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64(b * b) * a), -0.375, Float64(Float64(Float64(a * a) * c) * -0.5625)) * (b ^ -5.0)), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * -0.375 + N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 85.1

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

      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 85.1

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

      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 85.1

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

   4. Applied rewrites 84.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 86.3

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

   6. Applied rewrites 86.3%

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

   if 2.7000000000000002 < b

   1. Initial program 48.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.9%

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

   10. Applied rewrites 92.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(b \cdot b\right) \cdot a, -0.375, \left(\left(a \cdot a\right) \cdot c\right) \cdot -0.5625\right) \cdot {b}^{-5}, c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0} + b\\ \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{t\_1}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(0.5 \cdot a, 6.75 \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right), -4.5 \cdot c\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, t\_1, t\_0\right) \cdot a}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))) (t_1 (+ (sqrt t_0) b)))
   (if (<= b 3.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ t_1 (- t_0 (* b b))))))
     (/
      (/
       (* (* (fma (* 0.5 a) (* 6.75 (* (/ c b) (/ c b))) (* -4.5 c)) a) b)
       (* (fma b t_1 t_0) a))
      3.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double t_1 = sqrt(t_0) + b;
	double tmp;
	if (b <= 3.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / (t_1 / (t_0 - (b * b)))));
	} else {
		tmp = (((fma((0.5 * a), (6.75 * ((c / b) * (c / b))), (-4.5 * c)) * a) * b) / (fma(b, t_1, t_0) * a)) / 3.0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	t_1 = Float64(sqrt(t_0) + b)
	tmp = 0.0
	if (b <= 3.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(t_1 / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(0.5 * a), Float64(6.75 * Float64(Float64(c / b) * Float64(c / b))), Float64(-4.5 * c)) * a) * b) / Float64(fma(b, t_1, t_0) * a)) / 3.0);
	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]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[b, 3.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(t$95$1 / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * a), $MachinePrecision] * N[(6.75 * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * t$95$1 + t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 84.1

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

      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 84.1

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

      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 84.1

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

   4. Applied rewrites 84.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

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

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

      5. associate-/l/ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-1 N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 48.7%

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

   7. Taylor expanded in b around inf

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

   9. Applied rewrites 95.2%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.2%

       \[\leadsto \frac{\frac{b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot a, \left(\frac{c}{b} \cdot \frac{c}{b}\right) \cdot 6.75, -4.5 \cdot c\right)}\right)}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b, \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\right) \cdot a}}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{fma}\left(0.5 \cdot a, 6.75 \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right), -4.5 \cdot c\right) \cdot a\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b, \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\right) \cdot a}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{t\_0} + b}{t\_0 - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 3.7)
     (/ 1.0 (/ (* 3.0 a) (/ 1.0 (/ (+ (sqrt t_0) b) (- t_0 (* b b))))))
     (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 3.7) {
		tmp = 1.0 / ((3.0 * a) / (1.0 / ((sqrt(t_0) + b) / (t_0 - (b * b)))));
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 3.7)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(t_0 - Float64(b * b))))));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(Float64(b / c) * -2.0)));
	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, 3.7], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 84.1

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

      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 84.1

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

      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 84.1

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

   4. Applied rewrites 84.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(t\_0 - b \cdot b\right)}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 3.7)
     (/ (* (/ 0.3333333333333333 a) (- t_0 (* b b))) (+ (sqrt t_0) b))
     (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 3.7) {
		tmp = ((0.3333333333333333 / a) * (t_0 - (b * b))) / (sqrt(t_0) + b);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 3.7)
		tmp = Float64(Float64(Float64(0.3333333333333333 / a) * Float64(t_0 - Float64(b * b))) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(Float64(b / c) * -2.0)));
	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, 3.7], N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 84.1

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

      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 84.1

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

      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 84.1

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

   4. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}} \]
   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. 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 85.3%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{a} \cdot \left(\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{t\_0} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= b 3.7)
     (/ (- t_0 (* b b)) (* (* 3.0 a) (+ (sqrt t_0) b)))
     (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (b <= 3.7) {
		tmp = (t_0 - (b * b)) / ((3.0 * a) * (sqrt(t_0) + b));
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 3.7)
		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(Float64(b / c) * -2.0)));
	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, 3.7], 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[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 84.1

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

      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 84.1

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

      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 84.1

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

   4. Applied rewrites 84.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}}} \]
   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. 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 85.3%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right) - b \cdot b}{\left(3 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* 3.0 a))
   (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7)
		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(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7], 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[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 84.1

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

   4. Applied rewrites 84.1%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7)
   (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) (* 3.0 a))
   (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

   4. Applied rewrites 84.0%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.7)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma (/ a b) 1.5 (* (/ b c) -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.7) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, ((b / c) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.7)
		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(Float64(b / c) * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.7], 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[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7000000000000002

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. 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 84.0

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

      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 84.0

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

   4. Applied rewrites 83.9%

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

   if 3.7000000000000002 < b

   1. Initial program 48.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 48.0

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

      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 48.0

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

      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 48.0

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

   4. Applied rewrites 48.0%

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

   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 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 52.4

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

   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 52.4

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

   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 52.4

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

4. Applied rewrites 52.3%

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

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 84.1

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

7. Applied rewrites 84.1%

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

Alternative 15: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 67.2

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

5. Applied rewrites 67.2%

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

6. Final simplification 67.2%

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

Alternative 16: 64.3% 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 52.4%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.3%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 67.1%

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

Reproduce

herbie shell --seed 2024267 
(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)))