Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.9%

Time: 16.1s

Alternatives: 14

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: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(3 \cdot a\right)}\right)\\ \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t_0}^{1.5}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot \sqrt{t_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (* (fma (sqrt (* 3.0 a)) (sqrt c) b) (- b (sqrt (* c (* 3.0 a)))))))
   (if (<= b 0.028)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_0 1.5))
       (+ (pow (- b) 2.0) (+ t_0 (* b (sqrt t_0)))))
      (* 3.0 a))
     (+
      (* -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)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(sqrt((3.0 * a)), sqrt(c), b) * (b - sqrt((c * (3.0 * a))));
	double tmp;
	if (b <= 0.028) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 1.5)) / (pow(-b, 2.0) + (t_0 + (b * sqrt(t_0))))) / (3.0 * a);
	} 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))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(sqrt(Float64(3.0 * a)), sqrt(c), b) * Float64(b - sqrt(Float64(c * Float64(3.0 * a)))))
	tmp = 0.0
	if (b <= 0.028)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 1.5)) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	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(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.028], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0280000000000000006

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. difference-of-squares 89.5%

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

      3. associate-*l* 89.5%

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

      4. associate-*l* 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. flip3-+ 89.2%

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

   7. Applied egg-rr 90.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(a \cdot 3\right)}\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(a \cdot 3\right)}\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(\sqrt{a \cdot 3}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(a \cdot 3\right)}\right)}\right)}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. cancel-sign-sub 90.5%

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

   9. Simplified 90.5%

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

   if 0.0280000000000000006 < b

   1. Initial program 53.2%

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

   2. Taylor expanded in b around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. unpow-prod-down 92.9%

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

      3. pow-prod-down 92.9%

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

      4. pow-pow 92.9%

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

      5. metadata-eval 92.9%

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

      6. metadata-eval 92.9%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in c around 0 92.9%

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

      1. distribute-rgt-out 92.9%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

      4. times-frac 92.9%

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

   7. Simplified 92.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(3 \cdot a\right)}\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(3 \cdot a\right)}\right) + b \cdot \sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{c \cdot \left(3 \cdot a\right)}\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(3 \cdot c\right)}\\ t_1 := \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)\\ \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\frac{{b}^{2} + t_1 \cdot \left(t_0 - b\right)}{\left(-b\right) - \sqrt{t_1 \cdot \left(b - t_0\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* 3.0 c)))) (t_1 (fma (sqrt (* 3.0 a)) (sqrt c) b)))
   (if (<= b 0.028)
     (/
      (/ (+ (pow b 2.0) (* t_1 (- t_0 b))) (- (- b) (sqrt (* t_1 (- b t_0)))))
      (* 3.0 a))
     (+
      (* -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)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (3.0 * c)));
	double t_1 = fma(sqrt((3.0 * a)), sqrt(c), b);
	double tmp;
	if (b <= 0.028) {
		tmp = ((pow(b, 2.0) + (t_1 * (t_0 - b))) / (-b - sqrt((t_1 * (b - t_0))))) / (3.0 * a);
	} 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))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(3.0 * c)))
	t_1 = fma(sqrt(Float64(3.0 * a)), sqrt(c), b)
	tmp = 0.0
	if (b <= 0.028)
		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(3.0 * a));
	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(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[b, 0.028], 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[(3.0 * a), $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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0280000000000000006

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. difference-of-squares 89.5%

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

      3. associate-*l* 89.5%

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

      4. associate-*l* 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. flip-+ 89.4%

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

   7. Applied egg-rr 90.0%

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

      1. unpow2 90.0%

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

      2. sqr-neg 90.0%

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

      3. unpow2 90.0%

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

      4. *-commutative 90.0%

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

      5. associate-*r* 90.0%

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

      6. *-commutative 90.0%

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

      7. *-commutative 90.0%

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

      8. associate-*r* 90.0%

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

      9. *-commutative 90.0%

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

   9. Simplified 90.0%

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

   if 0.0280000000000000006 < b

   1. Initial program 53.2%

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

   2. Taylor expanded in b around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. unpow-prod-down 92.9%

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

      3. pow-prod-down 92.9%

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

      4. pow-pow 92.9%

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

      5. metadata-eval 92.9%

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

      6. metadata-eval 92.9%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in c around 0 92.9%

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

      1. distribute-rgt-out 92.9%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

      4. times-frac 92.9%

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

   7. Simplified 92.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\frac{{b}^{2} + \mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(\sqrt{a \cdot \left(3 \cdot c\right)} - b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right) \cdot \left(b - \sqrt{a \cdot \left(3 \cdot c\right)}\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)}, \sqrt{b - \sqrt{c \cdot \left(3 \cdot a\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.028)
   (/
    (fma
     (sqrt (fma (sqrt (* 3.0 a)) (sqrt c) b))
     (sqrt (- b (sqrt (* c (* 3.0 a)))))
     (- b))
    (* 3.0 a))
   (+
    (* -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)))
      (*
       -0.16666666666666666
       (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.028) {
		tmp = fma(sqrt(fma(sqrt((3.0 * a)), sqrt(c), b)), sqrt((b - sqrt((c * (3.0 * a))))), -b) / (3.0 * a);
	} 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))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.028)
		tmp = Float64(fma(sqrt(fma(sqrt(Float64(3.0 * a)), sqrt(c), b)), sqrt(Float64(b - sqrt(Float64(c * Float64(3.0 * a))))), Float64(-b)) / Float64(3.0 * a));
	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(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.028], N[(N[(N[Sqrt[N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(b - N[Sqrt[N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0280000000000000006

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. difference-of-squares 89.5%

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

      3. associate-*l* 89.5%

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

      4. associate-*l* 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. +-commutative 89.5%

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

      2. sqrt-prod 89.3%

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

      3. fma-def 90.0%

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

      4. +-commutative 90.0%

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

      5. sqrt-prod 89.8%

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

      6. fma-def 89.8%

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

      7. *-commutative 89.8%

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

      8. *-commutative 89.8%

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

      9. *-commutative 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 0.0280000000000000006 < b

   1. Initial program 53.2%

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

   2. Taylor expanded in b around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. unpow-prod-down 92.9%

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

      3. pow-prod-down 92.9%

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

      4. pow-pow 92.9%

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

      5. metadata-eval 92.9%

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

      6. metadata-eval 92.9%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in c around 0 92.9%

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

      1. distribute-rgt-out 92.9%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

      4. times-frac 92.9%

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

   7. Simplified 92.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)}, \sqrt{b - \sqrt{c \cdot \left(3 \cdot a\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

Alternative 4: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)}, \sqrt{b - \sqrt{c \cdot \left(3 \cdot a\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.029)
   (/
    (fma
     (sqrt (fma (sqrt (* 3.0 a)) (sqrt c) b))
     (sqrt (- b (sqrt (* c (* 3.0 a)))))
     (- b))
    (* 3.0 a))
   (+
    (* -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 <= 0.029) {
		tmp = fma(sqrt(fma(sqrt((3.0 * a)), sqrt(c), b)), sqrt((b - sqrt((c * (3.0 * a))))), -b) / (3.0 * a);
	} 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 <= 0.029)
		tmp = Float64(fma(sqrt(fma(sqrt(Float64(3.0 * a)), sqrt(c), b)), sqrt(Float64(b - sqrt(Float64(c * Float64(3.0 * a))))), Float64(-b)) / Float64(3.0 * a));
	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, 0.029], N[(N[(N[Sqrt[N[(N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(b - N[Sqrt[N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $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 < 0.0290000000000000015

   1. Initial program 89.3%

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

      1. add-sqr-sqrt 89.3%

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

      2. difference-of-squares 89.5%

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

      3. associate-*l* 89.5%

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

      4. associate-*l* 89.5%

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

   3. Applied egg-rr 89.5%

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

      1. associate-*r* 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*r* 89.5%

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

      4. *-commutative 89.5%

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

   5. Simplified 89.5%

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

      1. +-commutative 89.5%

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

      2. sqrt-prod 89.3%

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

      3. fma-def 90.0%

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

      4. +-commutative 90.0%

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

      5. sqrt-prod 89.8%

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

      6. fma-def 89.8%

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

      7. *-commutative 89.8%

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

      8. *-commutative 89.8%

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

      9. *-commutative 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 0.0290000000000000015 < b

   1. Initial program 53.2%

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

   2. Taylor expanded in b around inf 90.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\sqrt{3 \cdot a}, \sqrt{c}, b\right)}, \sqrt{b - \sqrt{c \cdot \left(3 \cdot a\right)}}, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.028)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (+
    (* -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 <= 0.028) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} 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 <= 0.028)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	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, 0.028], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $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 < 0.0280000000000000006

   1. Initial program 89.3%

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

   3. Simplified 89.7%

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

   if 0.0280000000000000006 < b

   1. Initial program 53.2%

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

   2. Taylor expanded in b around inf 90.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.028:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.075)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ 1.0 (fma 1.5 (/ a b) (/ -2.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.075) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma(1.5, (a / b), (-2.0 / (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.075)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(-2.0 / Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.075], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0749999999999999972

   1. Initial program 82.0%

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

      1. neg-sub0 82.0%

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

      2. sqr-neg 82.0%

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

      3. associate-+l- 82.0%

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

      4. sub0-neg 82.0%

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

   3. Simplified 82.2%

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

   if -0.0749999999999999972 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 49.3%

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

   2. Taylor expanded in b around inf 86.3%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. fma-def 86.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      4. associate-/r/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      5. *-commutative 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{a \cdot c}{b} \cdot -1.5}\right)}{3 \cdot a} \]

      6. associate-*l/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}\right)}{3 \cdot a} \]

   4. Simplified 86.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. clear-num 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}} \]

      2. inv-pow 86.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1}} \]

      3. *-commutative 86.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      4. associate-*l/ 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      5. pow-prod-down 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      6. associate-/l* 86.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \color{blue}{\frac{a \cdot c}{\frac{b}{-1.5}}}\right)}\right)}^{-1} \]

      7. div-inv 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{\color{blue}{b \cdot \frac{1}{-1.5}}}\right)}\right)}^{-1} \]

      8. metadata-eval 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot \color{blue}{-0.6666666666666666}}\right)}\right)}^{-1} \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   9. Taylor expanded in a around 0 87.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. +-commutative 87.0%

          \[\leadsto \frac{1}{\color{blue}{1.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c}}} \]

       2. fma-def 87.0%

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

       3. associate-*r/ 87.0%

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

       4. associate-/l* 87.1%

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

   11. Simplified 87.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 7: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.075)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (/ 1.0 (fma 1.5 (/ a b) (/ -2.0 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.075) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / fma(1.5, (a / b), (-2.0 / (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.075)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(-2.0 / Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.075], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0749999999999999972

   1. Initial program 82.0%

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

   3. Simplified 82.2%

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

   if -0.0749999999999999972 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 49.3%

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

   2. Taylor expanded in b around inf 86.3%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. fma-def 86.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      4. associate-/r/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      5. *-commutative 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{a \cdot c}{b} \cdot -1.5}\right)}{3 \cdot a} \]

      6. associate-*l/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}\right)}{3 \cdot a} \]

   4. Simplified 86.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. clear-num 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}} \]

      2. inv-pow 86.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1}} \]

      3. *-commutative 86.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      4. associate-*l/ 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      5. pow-prod-down 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      6. associate-/l* 86.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \color{blue}{\frac{a \cdot c}{\frac{b}{-1.5}}}\right)}\right)}^{-1} \]

      7. div-inv 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{\color{blue}{b \cdot \frac{1}{-1.5}}}\right)}\right)}^{-1} \]

      8. metadata-eval 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot \color{blue}{-0.6666666666666666}}\right)}\right)}^{-1} \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   9. Taylor expanded in a around 0 87.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. +-commutative 87.0%

          \[\leadsto \frac{1}{\color{blue}{1.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c}}} \]

       2. fma-def 87.0%

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

       3. associate-*r/ 87.0%

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

       4. associate-/l* 87.1%

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

   11. Simplified 87.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 8: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.075:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -0.075) t_0 (/ 1.0 (fma 1.5 (/ a b) (/ -2.0 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.075) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(1.5, (a / b), (-2.0 / (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.075)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(1.5, Float64(a / b), Float64(-2.0 / Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.075], t$95$0, N[(1.0 / N[(1.5 * N[(a / b), $MachinePrecision] + N[(-2.0 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.0749999999999999972

   1. Initial program 82.0%

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

   if -0.0749999999999999972 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 49.3%

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

   2. Taylor expanded in b around inf 86.3%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. +-commutative 86.3%

         \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

      2. fma-def 86.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. associate-/l* 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      4. associate-/r/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

      5. *-commutative 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{a \cdot c}{b} \cdot -1.5}\right)}{3 \cdot a} \]

      6. associate-*l/ 86.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}\right)}{3 \cdot a} \]

   4. Simplified 86.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. clear-num 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}} \]

      2. inv-pow 86.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1}} \]

      3. *-commutative 86.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      4. associate-*l/ 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      5. pow-prod-down 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

      6. associate-/l* 86.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \color{blue}{\frac{a \cdot c}{\frac{b}{-1.5}}}\right)}\right)}^{-1} \]

      7. div-inv 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{\color{blue}{b \cdot \frac{1}{-1.5}}}\right)}\right)}^{-1} \]

      8. metadata-eval 86.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot \color{blue}{-0.6666666666666666}}\right)}\right)}^{-1} \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 86.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   8. Simplified 86.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

   9. Taylor expanded in a around 0 87.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
   10. Step-by-step derivation

       1. +-commutative 87.0%

          \[\leadsto \frac{1}{\color{blue}{1.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c}}} \]

       2. fma-def 87.0%

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

       3. associate-*r/ 87.0%

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

       4. associate-/l* 87.1%

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

   11. Simplified 87.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1.5, \frac{a}{b}, \frac{-2}{\frac{c}{b}}\right)}\\ \end{array} \]

Alternative 9: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf 80.8%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. +-commutative 80.8%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

   2. fma-def 80.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   3. associate-/l* 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

   4. associate-/r/ 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

   5. *-commutative 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{a \cdot c}{b} \cdot -1.5}\right)}{3 \cdot a} \]

   6. associate-*l/ 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}\right)}{3 \cdot a} \]

4. Simplified 80.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 80.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}} \]

   2. inv-pow 80.8%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1}} \]

   3. *-commutative 80.8%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   4. associate-*l/ 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   5. pow-prod-down 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   6. associate-/l* 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \color{blue}{\frac{a \cdot c}{\frac{b}{-1.5}}}\right)}\right)}^{-1} \]

   7. div-inv 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{\color{blue}{b \cdot \frac{1}{-1.5}}}\right)}\right)}^{-1} \]

   8. metadata-eval 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot \color{blue}{-0.6666666666666666}}\right)}\right)}^{-1} \]

6. Applied egg-rr 80.8%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 80.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

8. Simplified 80.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

9. Taylor expanded in a around 0 81.7%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
10. Step-by-step derivation

    1. +-commutative 81.7%

       \[\leadsto \frac{1}{\color{blue}{1.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c}}} \]

    2. fma-def 81.7%

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

    3. associate-*r/ 81.7%

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

    4. associate-/l* 81.7%

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

11. Simplified 81.7%

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

12. Final simplification 81.7%

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

Alternative 10: 82.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

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

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf 80.8%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. +-commutative 80.8%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]

   2. fma-def 80.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   3. associate-/l* 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

   4. associate-/r/ 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a} \]

   5. *-commutative 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{a \cdot c}{b} \cdot -1.5}\right)}{3 \cdot a} \]

   6. associate-*l/ 80.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \color{blue}{\frac{\left(a \cdot c\right) \cdot -1.5}{b}}\right)}{3 \cdot a} \]

4. Simplified 80.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}{3 \cdot a} \]
5. Step-by-step derivation

   1. clear-num 80.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}}} \]

   2. inv-pow 80.8%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1}} \]

   3. *-commutative 80.8%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\mathsf{fma}\left(-1.125, \frac{{a}^{2}}{{b}^{3}} \cdot {c}^{2}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   4. associate-*l/ 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   5. pow-prod-down 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}, \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right)}\right)}^{-1} \]

   6. associate-/l* 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \color{blue}{\frac{a \cdot c}{\frac{b}{-1.5}}}\right)}\right)}^{-1} \]

   7. div-inv 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{\color{blue}{b \cdot \frac{1}{-1.5}}}\right)}\right)}^{-1} \]

   8. metadata-eval 80.8%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot \color{blue}{-0.6666666666666666}}\right)}\right)}^{-1} \]

6. Applied egg-rr 80.8%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 80.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

8. Simplified 80.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}, \frac{a \cdot c}{b \cdot -0.6666666666666666}\right)}}} \]

9. Taylor expanded in a around 0 81.7%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]

10. Final simplification 81.7%

    \[\leadsto \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 11: 64.3% accurate, 23.2× speedup

Math

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

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(c * Float64(-0.5 / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

   2. associate-/l* 63.7%

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

4. Simplified 63.7%

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

   1. associate-/r/ 63.7%

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

6. Applied egg-rr 63.7%

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

7. Final simplification 63.7%

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

Alternative 12: 64.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

   2. associate-/l* 63.7%

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

4. Simplified 63.7%

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

5. Final simplification 63.7%

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

Alternative 13: 64.4% 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 55.8%

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

2. Taylor expanded in b around inf 63.8%

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

   1. associate-*r/ 63.8%

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

4. Simplified 63.8%

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

5. Final simplification 63.8%

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

Alternative 14: 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 55.8%

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

   1. add-sqr-sqrt 55.8%

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

   2. difference-of-squares 56.0%

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

   3. associate-*l* 56.0%

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

   4. associate-*l* 56.0%

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

3. Applied egg-rr 56.0%

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

   1. associate-*r* 56.0%

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

   2. *-commutative 56.0%

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

   3. associate-*r* 56.0%

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

   4. *-commutative 56.0%

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

5. Simplified 56.0%

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

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. *-commutative 3.2%

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

   2. associate-*l/ 3.2%

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

   3. distribute-lft1-in 3.2%

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

   4. metadata-eval 3.2%

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

   5. mul0-lft 3.2%

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

   6. 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 2023311 
(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)))