Cubic critical, narrow range

Percentage Accurate: 55.5% → 91.9%

Time: 16.5s

Alternatives: 11

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: 55.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, \mathsf{fma}\left(-c, a \cdot 3, 3 \cdot \left(c \cdot a\right)\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)\\ \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (fma
          2.0
          (fma (- c) (* a 3.0) (* 3.0 (* c a)))
          (fma b b (* a (* c -3.0))))))
   (if (<= b 0.3)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* c a) 4.0) (pow b 7.0)) (/ 6.328125 a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(2.0, fma(-c, (a * 3.0), (3.0 * (c * a))), fma(b, b, (a * (c * -3.0))));
	double tmp;
	if (b <= 0.3) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) / pow(b, 7.0)) * (6.328125 / a)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(2.0, fma(Float64(-c), Float64(a * 3.0), Float64(3.0 * Float64(c * a))), fma(b, b, Float64(a * Float64(c * -3.0))))
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[((-c) * N[(a * 3.0), $MachinePrecision] + N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.3], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 87.3%

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

      1. prod-diff 87.2%

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

      2. *-commutative 87.2%

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

      3. fma-neg 87.3%

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

      4. prod-diff 87.3%

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

      5. *-commutative 87.3%

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

      6. fma-neg 87.4%

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

      7. associate-+l+ 87.4%

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

      8. fma-neg 87.3%

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

      9. associate-*l* 87.3%

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

      10. distribute-lft-neg-in 87.3%

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

      11. metadata-eval 87.3%

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

      12. *-commutative 87.3%

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

      13. associate-*r* 87.3%

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

   4. Applied egg-rr 87.2%

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

      1. flip-+ 87.2%

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

   6. Applied egg-rr 88.8%

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

   if 0.299999999999999989 < b

   1. Initial program 53.6%

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

   3. Taylor expanded in b around inf 92.7%

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

   4. Taylor expanded in c around 0 92.7%

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

      1. distribute-rgt-out 92.7%

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

      2. associate-*r* 92.7%

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

      3. *-commutative 92.7%

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

      4. *-commutative 92.7%

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

      5. times-frac 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \mathsf{fma}\left(2, \mathsf{fma}\left(-c, a \cdot 3, 3 \cdot \left(c \cdot a\right)\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(-c, a \cdot 3, 3 \cdot \left(c \cdot a\right)\right), \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)\\ \mathbf{if}\;b \leq 0.24:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma 2.0 (* (* a 3.0) (- c c)) (fma b b (* c (* a -3.0))))))
   (if (<= b 0.24)
     (/ (/ (- (pow b 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* c a) 4.0) (pow b 7.0)) (/ 6.328125 a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(2.0, ((a * 3.0) * (c - c)), fma(b, b, (c * (a * -3.0))));
	double tmp;
	if (b <= 0.24) {
		tmp = ((pow(b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) / pow(b, 7.0)) * (6.328125 / a)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(2.0, Float64(Float64(a * 3.0) * Float64(c - c)), fma(b, b, Float64(c * Float64(a * -3.0))))
	tmp = 0.0
	if (b <= 0.24)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(N[(a * 3.0), $MachinePrecision] * N[(c - c), $MachinePrecision]), $MachinePrecision] + N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.24], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.23999999999999999

   1. Initial program 87.3%

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

      1. prod-diff 87.2%

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

      2. *-commutative 87.2%

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

      3. fma-neg 87.3%

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

      4. prod-diff 87.3%

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

      5. *-commutative 87.3%

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

      6. fma-neg 87.4%

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

      7. associate-+l+ 87.4%

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

      8. fma-neg 87.3%

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

      9. associate-*l* 87.3%

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

      10. distribute-lft-neg-in 87.3%

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

      11. metadata-eval 87.3%

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

      12. *-commutative 87.3%

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

      13. associate-*r* 87.3%

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

   4. Applied egg-rr 87.2%

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

      1. flip-+ 87.2%

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

   6. Applied egg-rr 88.8%

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

      1. unpow2 88.8%

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

      2. sqr-neg 88.8%

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

      3. unpow2 88.8%

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

      4. fma-def 88.8%

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

      5. +-commutative 88.8%

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

      6. associate-*r* 88.8%

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

      7. *-commutative 88.8%

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

      8. *-commutative 88.8%

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

      9. distribute-lft-out 88.8%

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

      10. associate-*r* 88.9%

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

      11. *-commutative 88.9%

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

      12. associate-*r* 88.9%

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

   8. Simplified 88.7%

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

   if 0.23999999999999999 < b

   1. Initial program 53.6%

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

   3. Taylor expanded in b around inf 92.7%

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

   4. Taylor expanded in c around 0 92.7%

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

      1. distribute-rgt-out 92.7%

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

      2. associate-*r* 92.7%

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

      3. *-commutative 92.7%

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

      4. *-commutative 92.7%

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

      5. times-frac 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.24:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)\\ \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma 2.0 (* (* a 3.0) (- c c)) (fma b b (* c (* a -3.0))))))
   (if (<= b 10.5)
     (/ (/ (- (pow b 2.0) t_0) (- (- b) (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(2.0, ((a * 3.0) * (c - c)), fma(b, b, (c * (a * -3.0))));
	double tmp;
	if (b <= 10.5) {
		tmp = ((pow(b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(2.0, Float64(Float64(a * 3.0) * Float64(c - c)), fma(b, b, Float64(c * Float64(a * -3.0))))
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(N[(a * 3.0), $MachinePrecision] * N[(c - c), $MachinePrecision]), $MachinePrecision] + N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 10.5], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-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 < 10.5

   1. Initial program 85.1%

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

      1. prod-diff 85.1%

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

      2. *-commutative 85.1%

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

      3. fma-neg 85.1%

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

      4. prod-diff 85.1%

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

      5. *-commutative 85.1%

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

      6. fma-neg 85.1%

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

      7. associate-+l+ 85.1%

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

      8. fma-neg 85.1%

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

      9. associate-*l* 85.1%

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

      10. distribute-lft-neg-in 85.1%

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

      11. metadata-eval 85.1%

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

      12. *-commutative 85.1%

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

      13. associate-*r* 85.1%

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

   4. Applied egg-rr 85.0%

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

      1. flip-+ 84.6%

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

   6. Applied egg-rr 85.7%

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

      1. unpow2 85.7%

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

      2. sqr-neg 85.7%

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

      3. unpow2 85.7%

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

      4. fma-def 85.7%

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

      5. +-commutative 85.7%

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

      6. associate-*r* 85.8%

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

      7. *-commutative 85.8%

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

      8. *-commutative 85.8%

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

      9. distribute-lft-out 85.8%

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

      10. associate-*r* 85.8%

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

      11. *-commutative 85.8%

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

      12. associate-*r* 85.8%

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

   8. Simplified 85.7%

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

   if 10.5 < b

   1. Initial program 50.3%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\frac{{b}^{2} - \mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(2, \left(a \cdot 3\right) \cdot \left(c - c\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (/ (- (sqrt (fma -3.0 (* c a) (pow b 2.0))) b) (* a 3.0))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (sqrt(fma(-3.0, (c * a), pow(b, 2.0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 85.1%

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

      1. fma-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-lft-neg-in 85.0%

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

      4. metadata-eval 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*r* 85.0%

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

      7. add-cbrt-cube 84.3%

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

      8. pow3 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in b around 0 85.1%

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

      1. fma-def 85.1%

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

   7. Simplified 85.1%

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

   if 10.5 < b

   1. Initial program 50.3%

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

   3. Taylor expanded in b around inf 91.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -3\\ t_1 := t_0 + 2 \cdot \left(3 \cdot \left(c \cdot a\right) + t_0\right)\\ \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{{t_1}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{t_1}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t_1}{b}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) -3.0)) (t_1 (+ t_0 (* 2.0 (+ (* 3.0 (* c a)) t_0)))))
   (if (<= b 10.5)
     (/ (- (sqrt (fma -3.0 (* c a) (pow b 2.0))) b) (* a 3.0))
     (/
      (+
       (* -0.125 (/ (pow t_1 2.0) (pow b 3.0)))
       (+ (* 0.0625 (/ (pow t_1 3.0) (pow b 5.0))) (* 0.5 (/ t_1 b))))
      (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * -3.0;
	double t_1 = t_0 + (2.0 * ((3.0 * (c * a)) + t_0));
	double tmp;
	if (b <= 10.5) {
		tmp = (sqrt(fma(-3.0, (c * a), pow(b, 2.0))) - b) / (a * 3.0);
	} else {
		tmp = ((-0.125 * (pow(t_1, 2.0) / pow(b, 3.0))) + ((0.0625 * (pow(t_1, 3.0) / pow(b, 5.0))) + (0.5 * (t_1 / b)))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * -3.0)
	t_1 = Float64(t_0 + Float64(2.0 * Float64(Float64(3.0 * Float64(c * a)) + t_0)))
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-0.125 * Float64((t_1 ^ 2.0) / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_1 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_1 / b)))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(2.0 * N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 10.5], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.125 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$1, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 85.1%

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

      1. fma-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-lft-neg-in 85.0%

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

      4. metadata-eval 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*r* 85.0%

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

      7. add-cbrt-cube 84.3%

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

      8. pow3 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in b around 0 85.1%

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

      1. fma-def 85.1%

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

   7. Simplified 85.1%

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

   if 10.5 < b

   1. Initial program 50.3%

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

      1. prod-diff 50.4%

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

      2. *-commutative 50.4%

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

      3. fma-neg 50.3%

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

      4. prod-diff 50.4%

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

      5. *-commutative 50.4%

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

      6. fma-neg 50.3%

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

      7. associate-+l+ 50.3%

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

      8. fma-neg 50.4%

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

      9. associate-*l* 50.3%

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

      10. distribute-lft-neg-in 50.3%

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

      11. metadata-eval 50.3%

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

      12. *-commutative 50.3%

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

      13. associate-*r* 50.3%

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

   4. Applied egg-rr 50.3%

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

   5. Taylor expanded in b around inf 91.1%

      \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(-3 \cdot \left(a \cdot c\right) + 2 \cdot \left(-3 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-3 \cdot \left(a \cdot c\right) + 2 \cdot \left(-3 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-3 \cdot \left(a \cdot c\right) + 2 \cdot \left(-3 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)\right)}{b}\right)}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.125 \cdot \frac{{\left(\left(c \cdot a\right) \cdot -3 + 2 \cdot \left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -3\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(\left(c \cdot a\right) \cdot -3 + 2 \cdot \left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -3\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(c \cdot a\right) \cdot -3 + 2 \cdot \left(3 \cdot \left(c \cdot a\right) + \left(c \cdot a\right) \cdot -3\right)}{b}\right)}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1e-8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -1e-8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -1e-8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1e-8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1e-8

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. sqr-neg 72.9%

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

      3. unsub-neg 72.9%

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

      4. div-sub 72.0%

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

      5. --rgt-identity 72.0%

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

      6. div-sub 72.9%

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

   3. Simplified 73.0%

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

   if -1e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 23.4%

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

   3. Taylor expanded in b around inf 89.3%

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

      1. associate-*r/ 89.3%

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

   5. Simplified 89.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (/ (- (sqrt (fma -3.0 (* c a) (pow b 2.0))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (sqrt(fma(-3.0, (c * a), pow(b, 2.0))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 85.1%

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

      1. fma-neg 85.0%

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

      2. associate-*l* 85.0%

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

      3. distribute-lft-neg-in 85.0%

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

      4. metadata-eval 85.0%

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

      5. *-commutative 85.0%

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

      6. associate-*r* 85.0%

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

      7. add-cbrt-cube 84.3%

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

      8. pow3 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in b around 0 85.1%

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

      1. fma-def 85.1%

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

   7. Simplified 85.1%

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

   if 10.5 < b

   1. Initial program 50.3%

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

   3. Taylor expanded in b around inf 86.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -1e-8) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1e-8) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-1d-8)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1e-8) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -1e-8:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -1e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -1e-8)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-8], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1e-8

   1. Initial program 72.9%

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

   if -1e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 23.4%

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

   3. Taylor expanded in b around inf 89.3%

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

      1. associate-*r/ 89.3%

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

   5. Simplified 89.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 10.5d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 10.5:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 10.5)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 85.1%

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

   if 10.5 < b

   1. Initial program 50.3%

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

   3. Taylor expanded in b around inf 86.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 58.4%

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

3. Taylor expanded in b around inf 61.8%

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

   1. associate-*r/ 61.8%

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

   2. associate-/l* 61.7%

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

5. Simplified 61.7%

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

   1. associate-/r/ 61.7%

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

7. Applied egg-rr 61.7%

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

8. Final simplification 61.7%

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

Alternative 11: 64.3% 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 58.4%

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

3. Taylor expanded in b around inf 61.8%

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

   1. associate-*r/ 61.8%

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

5. Simplified 61.8%

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

6. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2024020 
(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)))