Cubic critical, narrow range

Percentage Accurate: 55.7% → 90.7%

Time: 17.5s

Alternatives: 13

Speedup: 23.2×

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: 55.7% 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: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot 3\right)}\\ t_1 := \mathsf{fma}\left(\sqrt{a \cdot c}, \sqrt{3}, b\right)\\ \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\frac{{b}^{2} + t_1 \cdot \left(t_0 - b\right)}{\left(-b\right) - \sqrt{t_1 \cdot \left(b - t_0\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c (* a 3.0)))) (t_1 (fma (sqrt (* a c)) (sqrt 3.0) b)))
   (if (<= b 16.0)
     (/
      (/ (+ (pow b 2.0) (* t_1 (- t_0 b))) (- (- b) (sqrt (* t_1 (- b t_0)))))
      (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (/ (* -1.0546875 (pow (* a c) 4.0)) (* a (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * (a * 3.0)));
	double t_1 = fma(sqrt((a * c)), sqrt(3.0), b);
	double tmp;
	if (b <= 16.0) {
		tmp = ((pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + ((-1.0546875 * pow((a * c), 4.0)) / (a * pow(b, 7.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * Float64(a * 3.0)))
	t_1 = fma(sqrt(Float64(a * c)), sqrt(3.0), b)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) + Float64(t_1 * Float64(t_0 - b))) / Float64(Float64(-b) - sqrt(Float64(t_1 * Float64(b - t_0))))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(-1.0546875 * (Float64(a * c) ^ 4.0)) / Float64(a * (b ^ 7.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(a * c), $MachinePrecision]], $MachinePrecision] * N[Sqrt[3.0], $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[b, 16.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(t$95$1 * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 83.9%

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

      2. difference-of-squares 84.1%

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

      3. associate-*l* 84.1%

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

      4. associate-*l* 84.1%

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

   3. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

   5. Simplified 84.1%

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

      1. flip-+ 84.0%

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

      2. pow2 84.0%

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

      3. add-sqr-sqrt 85.1%

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

      4. +-commutative 85.1%

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

      5. sqrt-prod 85.1%

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

      6. fma-def 85.0%

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

      7. associate-*l* 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. unpow2 85.1%

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

      2. sqr-neg 85.1%

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

      3. unpow2 85.1%

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

      4. associate-*r* 85.1%

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

      5. *-commutative 85.1%

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

      6. associate-*l* 85.1%

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

      7. associate-*r* 85.1%

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

      8. *-commutative 85.1%

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

      9. associate-*l* 85.1%

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

   9. Simplified 85.1%

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

   if 16 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 94.0%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. *-commutative 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

      5. times-frac 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{1.265625 + 5.0625}{a}\right)}\right)\right) \]

   5. Simplified 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)}\right)\right) \]

      2. expm1-udef 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)} - 1\right)}\right)\right) \]

      3. frac-times 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)} - 1\right)\right)\right) \]

   7. Applied egg-rr 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)} - 1\right)}\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)\right)}\right)\right) \]

      2. expm1-log1p 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)\right) \]

      3. associate-*r/ 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{{b}^{7} \cdot a}}\right)\right) \]

      4. *-commutative 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(6.328125 \cdot {\left(a \cdot c\right)}^{4}\right)}}{{b}^{7} \cdot a}\right)\right) \]

      5. associate-*r* 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{\left(-0.16666666666666666 \cdot 6.328125\right) \cdot {\left(a \cdot c\right)}^{4}}}{{b}^{7} \cdot a}\right)\right) \]

      6. metadata-eval 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{-1.0546875} \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot a}\right)\right) \]

      7. *-commutative 94.0%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{a \cdot {b}^{7}}}\right)\right) \]

   9. Simplified 94.0%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\frac{{b}^{2} + \mathsf{fma}\left(\sqrt{a \cdot c}, \sqrt{3}, b\right) \cdot \left(\sqrt{c \cdot \left(a \cdot 3\right)} - b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, \sqrt{3}, b\right) \cdot \left(b - \sqrt{c \cdot \left(a \cdot 3\right)}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)\\ t_1 := {b}^{4} + {\left(a \cdot c\right)}^{2} \cdot -9\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{{b}^{2} - \frac{t_1}{t_0}}{\left(-b\right) - {\left(\frac{t_0}{t_1}\right)}^{-0.5}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a 3.0))))
        (t_1 (+ (pow b 4.0) (* (pow (* a c) 2.0) -9.0))))
   (if (<= b 4.8)
     (/
      (/ (- (pow b 2.0) (/ t_1 t_0)) (- (- b) (pow (/ t_0 t_1) -0.5)))
      (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (/ (* -1.0546875 (pow (* a c) 4.0)) (* a (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * 3.0)));
	double t_1 = pow(b, 4.0) + (pow((a * c), 2.0) * -9.0);
	double tmp;
	if (b <= 4.8) {
		tmp = ((pow(b, 2.0) - (t_1 / t_0)) / (-b - pow((t_0 / t_1), -0.5))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + ((-1.0546875 * pow((a * c), 4.0)) / (a * pow(b, 7.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * 3.0)))
	t_1 = Float64((b ^ 4.0) + Float64((Float64(a * c) ^ 2.0) * -9.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64((b ^ 2.0) - Float64(t_1 / t_0)) / Float64(Float64(-b) - (Float64(t_0 / t_1) ^ -0.5))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(-1.0546875 * (Float64(a * c) ^ 4.0)) / Float64(a * (b ^ 7.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * -9.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Power[N[(t$95$0 / t$95$1), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

      1. flip-- 84.8%

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

      2. clear-num 84.4%

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

      3. fma-def 84.1%

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

      4. associate-*l* 84.1%

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

      5. pow2 84.1%

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

      6. pow2 84.1%

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

      7. pow-prod-up 84.8%

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

      8. metadata-eval 84.8%

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

      9. pow2 84.8%

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

      10. associate-*l* 84.8%

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

   3. Applied egg-rr 84.8%

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

      1. flip-+ 84.7%

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

   5. Applied egg-rr 85.2%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)} \cdot \left({b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}\right)}^{-0.5}}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. unpow2 85.2%

         \[\leadsto \frac{\frac{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)} \cdot \left({b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}\right)}^{-0.5}}}{3 \cdot a} \]

      2. sqr-neg 85.2%

         \[\leadsto \frac{\frac{\color{blue}{b \cdot b} - \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)} \cdot \left({b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}\right)}^{-0.5}}}{3 \cdot a} \]

      3. unpow2 85.2%

         \[\leadsto \frac{\frac{\color{blue}{{b}^{2}} - \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)} \cdot \left({b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}\right)}^{-0.5}}}{3 \cdot a} \]

      4. associate-*l/ 85.7%

         \[\leadsto \frac{\frac{{b}^{2} - \color{blue}{\frac{1 \cdot \left({b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot 3\right)\right)}{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 9}\right)}^{-0.5}}}{3 \cdot a} \]

      5. *-lft-identity 85.7%

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

      6. sub-neg 85.7%

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

      7. distribute-rgt-neg-in 85.7%

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

      8. metadata-eval 85.7%

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

      9. associate-*r* 85.7%

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

      10. *-commutative 85.7%

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

      11. associate-*l* 85.7%

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

   7. Simplified 85.7%

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

   if 4.79999999999999982 < b

   1. Initial program 49.2%

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

   2. Taylor expanded in b around inf 93.2%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

      5. times-frac 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{1.265625 + 5.0625}{a}\right)}\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)}\right)\right) \]

      2. expm1-udef 92.5%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)} - 1\right)}\right)\right) \]

      3. frac-times 92.5%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)} - 1\right)\right)\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)} - 1\right)}\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)\right)}\right)\right) \]

      2. expm1-log1p 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)\right) \]

      3. associate-*r/ 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{{b}^{7} \cdot a}}\right)\right) \]

      4. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(6.328125 \cdot {\left(a \cdot c\right)}^{4}\right)}}{{b}^{7} \cdot a}\right)\right) \]

      5. associate-*r* 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{\left(-0.16666666666666666 \cdot 6.328125\right) \cdot {\left(a \cdot c\right)}^{4}}}{{b}^{7} \cdot a}\right)\right) \]

      6. metadata-eval 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{-1.0546875} \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot a}\right)\right) \]

      7. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{a \cdot {b}^{7}}}\right)\right) \]

   9. Simplified 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{{b}^{2} - \frac{{b}^{4} + {\left(a \cdot c\right)}^{2} \cdot -9}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}{{b}^{4} + {\left(a \cdot c\right)}^{2} \cdot -9}\right)}^{-0.5}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 3: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/
    (-
     (sqrt (+ (fma b b (* a (* c -3.0))) (fma (* a -3.0) c (* (* a c) 3.0))))
     b)
    (* a 3.0))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+
     (* -0.5 (/ c b))
     (+
      (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
      (/ (* -1.0546875 (pow (* a c) 4.0)) (* a (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (sqrt((fma(b, b, (a * (c * -3.0))) + fma((a * -3.0), c, ((a * c) * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + ((-1.0546875 * pow((a * c), 4.0)) / (a * pow(b, 7.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(a * Float64(c * -3.0))) + fma(Float64(a * -3.0), c, Float64(Float64(a * c) * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(Float64(-1.0546875 * (Float64(a * c) ^ 4.0)) / Float64(a * (b ^ 7.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(N[(N[Sqrt[N[(N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -3.0), $MachinePrecision] * c + N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. prod-diff 85.2%

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

      3. associate-*l* 85.2%

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

      4. distribute-lft-neg-in 85.2%

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

      5. metadata-eval 85.2%

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

      6. *-commutative 85.2%

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

      7. associate-*r* 85.2%

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

      8. *-commutative 85.2%

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

      9. distribute-rgt-neg-in 85.2%

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

      10. metadata-eval 85.2%

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

      11. associate-*l* 85.2%

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

   3. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.2%

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

   2. Taylor expanded in b around inf 93.2%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

   3. Taylor expanded in c around 0 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-out 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

      5. times-frac 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{1.265625 + 5.0625}{a}\right)}\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)}\right)\right) \]

      2. expm1-udef 92.5%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)} - 1\right)}\right)\right) \]

      3. frac-times 92.5%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)} - 1\right)\right)\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)} - 1\right)}\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}\right)\right)}\right)\right) \]

      2. expm1-log1p 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{{b}^{7} \cdot a}}\right)\right) \]

      3. associate-*r/ 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{{b}^{7} \cdot a}}\right)\right) \]

      4. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot \color{blue}{\left(6.328125 \cdot {\left(a \cdot c\right)}^{4}\right)}}{{b}^{7} \cdot a}\right)\right) \]

      5. associate-*r* 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{\left(-0.16666666666666666 \cdot 6.328125\right) \cdot {\left(a \cdot c\right)}^{4}}}{{b}^{7} \cdot a}\right)\right) \]

      6. metadata-eval 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{\color{blue}{-1.0546875} \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot a}\right)\right) \]

      7. *-commutative 93.2%

         \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{a \cdot {b}^{7}}}\right)\right) \]

   9. Simplified 93.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1.0546875 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 4: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot 3\right)}\\ \mathbf{if}\;b \leq 16:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\left(b + t_0\right) \cdot \left(b - t_0\right)}\right)}{a \cdot 3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c 3.0)))))
   (if (<= b 16.0)
     (pow (cbrt (/ (fma -1.0 b (sqrt (* (+ b t_0) (- b t_0)))) (* a 3.0))) 3.0)
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * 3.0)));
	double tmp;
	if (b <= 16.0) {
		tmp = pow(cbrt((fma(-1.0, b, sqrt(((b + t_0) * (b - t_0)))) / (a * 3.0))), 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * 3.0)))
	tmp = 0.0
	if (b <= 16.0)
		tmp = cbrt(Float64(fma(-1.0, b, sqrt(Float64(Float64(b + t_0) * Float64(b - t_0)))) / Float64(a * 3.0))) ^ 3.0;
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 16.0], N[Power[N[Power[N[(N[(-1.0 * b + N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 83.9%

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

      2. difference-of-squares 84.1%

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

      3. associate-*l* 84.1%

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

      4. associate-*l* 84.1%

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

   3. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

   5. Simplified 84.1%

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

      1. add-log-exp 78.5%

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

      2. neg-mul-1 78.5%

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

      3. fma-def 78.5%

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

      4. +-commutative 78.5%

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

      5. sqrt-prod 78.5%

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

      6. fma-def 78.4%

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

      7. associate-*l* 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. add-cube-cbrt 78.4%

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

      2. pow3 78.4%

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

   9. Applied egg-rr 84.1%

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

   if 16 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 91.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}{a \cdot 3}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/
    (-
     (sqrt (+ (fma b b (* a (* c -3.0))) (fma (* a -3.0) c (* (* a c) 3.0))))
     b)
    (* a 3.0))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (sqrt((fma(b, b, (a * (c * -3.0))) + fma((a * -3.0), c, ((a * c) * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(a * Float64(c * -3.0))) + fma(Float64(a * -3.0), c, Float64(Float64(a * c) * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 16.0], N[(N[(N[Sqrt[N[(N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -3.0), $MachinePrecision] * c + N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. prod-diff 84.1%

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

      3. associate-*l* 84.1%

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

      4. distribute-lft-neg-in 84.1%

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

      5. metadata-eval 84.1%

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

      6. *-commutative 84.1%

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

      7. associate-*r* 84.1%

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

      8. *-commutative 84.1%

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

      9. distribute-rgt-neg-in 84.1%

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

      10. metadata-eval 84.1%

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

      11. associate-*l* 84.1%

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

   3. Applied egg-rr 84.1%

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

   if 16 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 91.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -8e-6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -8e-6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -8e-6], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -7.99999999999999964e-6

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. sqr-neg 74.0%

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

      3. unsub-neg 74.0%

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

      4. div-sub 73.4%

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

      5. --rgt-identity 73.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 74.0%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 74.1%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 36.6%

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

   2. Taylor expanded in b around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l/ 80.0%

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

   4. Simplified 80.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/
    (-
     (sqrt (+ (fma b b (* a (* c -3.0))) (fma (* a -3.0) c (* (* a c) 3.0))))
     b)
    (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (sqrt((fma(b, b, (a * (c * -3.0))) + fma((a * -3.0), c, ((a * c) * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(sqrt(Float64(fma(b, b, Float64(a * Float64(c * -3.0))) + fma(Float64(a * -3.0), c, Float64(Float64(a * c) * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 16.0], N[(N[(N[Sqrt[N[(N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -3.0), $MachinePrecision] * c + N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. prod-diff 84.1%

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

      3. associate-*l* 84.1%

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

      4. distribute-lft-neg-in 84.1%

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

      5. metadata-eval 84.1%

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

      6. *-commutative 84.1%

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

      7. associate-*r* 84.1%

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

      8. *-commutative 84.1%

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

      9. distribute-rgt-neg-in 84.1%

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

      10. metadata-eval 84.1%

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

      11. associate-*l* 84.1%

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

   3. Applied egg-rr 84.1%

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

   if 16 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 87.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + \mathsf{fma}\left(a \cdot -3, c, \left(a \cdot c\right) \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 8: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot 3\right)}\\ \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(b + t_0\right) \cdot \left(b - t_0\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c 3.0)))))
   (if (<= b 16.0)
     (* (/ 0.3333333333333333 a) (- (sqrt (* (+ b t_0) (- b t_0))) b))
     (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * 3.0)));
	double tmp;
	if (b <= 16.0) {
		tmp = (0.3333333333333333 / a) * (sqrt(((b + t_0) * (b - t_0))) - b);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a * (c * 3.0d0)))
    if (b <= 16.0d0) then
        tmp = (0.3333333333333333d0 / a) * (sqrt(((b + t_0) * (b - t_0))) - b)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (c * 3.0)));
	double tmp;
	if (b <= 16.0) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt(((b + t_0) * (b - t_0))) - b);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (c * 3.0)))
	tmp = 0
	if b <= 16.0:
		tmp = (0.3333333333333333 / a) * (math.sqrt(((b + t_0) * (b - t_0))) - b)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * 3.0)))
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(Float64(b + t_0) * Float64(b - t_0))) - b));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * (c * 3.0)));
	tmp = 0.0;
	if (b <= 16.0)
		tmp = (0.3333333333333333 / a) * (sqrt(((b + t_0) * (b - t_0))) - b);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 16.0], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b + t$95$0), $MachinePrecision] * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 83.9%

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

      2. difference-of-squares 84.1%

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

      3. associate-*l* 84.1%

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

      4. associate-*l* 84.1%

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

   3. Applied egg-rr 84.1%

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

      1. *-commutative 84.1%

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

      2. *-commutative 84.1%

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

   5. Simplified 84.1%

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

      1. add-log-exp 78.5%

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

      2. neg-mul-1 78.5%

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

      3. fma-def 78.5%

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

      4. +-commutative 78.5%

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

      5. sqrt-prod 78.5%

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

      6. fma-def 78.4%

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

      7. associate-*l* 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. expm1-log1p-u 57.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, \sqrt{3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 55.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(e^{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, \sqrt{3}, b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}\right)}{3 \cdot a}\right)} - 1} \]

   9. Applied egg-rr 58.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\left(\sqrt{a \cdot \left(c \cdot 3\right)} + b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}{a \cdot 3}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 62.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\left(\sqrt{a \cdot \left(c \cdot 3\right)} + b\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}{a \cdot 3}\right)\right)} \]

       2. expm1-log1p 84.1%

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

       3. *-lft-identity 84.1%

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

       4. associate-*l/ 84.1%

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

       5. fma-def 84.1%

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

       6. +-commutative 84.1%

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

       7. *-commutative 84.1%

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

       8. associate-/r* 84.1%

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

       9. metadata-eval 84.1%

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

       10. mul-1-neg 84.1%

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

       11. unsub-neg 84.1%

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

   11. Simplified 84.1%

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

   if 16 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 87.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\left(b + \sqrt{a \cdot \left(c \cdot 3\right)}\right) \cdot \left(b - \sqrt{a \cdot \left(c \cdot 3\right)}\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 9: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -8e-6) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -8e-6) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-8d-6)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -8e-6) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -8e-6:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -8e-6)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -8e-6)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -8e-6], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -7.99999999999999964e-6

   1. Initial program 74.0%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 36.6%

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

   2. Taylor expanded in b around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l/ 80.0%

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

   4. Simplified 80.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -8e-6)
   (/ (- (sqrt (- (* b b) (* (* a c) 3.0))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-6) {
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-8d-6)) then
        tmp = (sqrt(((b * b) - ((a * c) * 3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-6) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-6:
		tmp = (math.sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -8e-6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -8e-6)
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -8e-6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -7.99999999999999964e-6

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0 74.0%

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

      1. *-commutative 74.0%

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

   4. Simplified 74.0%

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

   if -7.99999999999999964e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 36.6%

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

   2. Taylor expanded in b around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l/ 80.0%

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

   4. Simplified 80.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 19.5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 19.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. sqr-neg 83.9%

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

      3. unsub-neg 83.9%

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

      4. div-sub 83.5%

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

      5. --rgt-identity 83.5%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 83.9%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 84.1%

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

   if 19.5 < b

   1. Initial program 47.5%

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

   2. Taylor expanded in b around inf 87.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 12: 64.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in b around inf 63.7%

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

   1. *-commutative 63.7%

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

   2. associate-*l/ 63.7%

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

4. Simplified 63.7%

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

5. Final simplification 63.7%

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

Alternative 13: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

   1. add-sqr-sqrt 56.0%

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

   2. difference-of-squares 56.1%

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

   3. associate-*l* 56.1%

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

   4. associate-*l* 56.1%

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

3. Applied egg-rr 56.1%

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

   1. *-commutative 56.1%

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

   2. *-commutative 56.1%

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

5. Simplified 56.1%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023338 
(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)))