Cubic critical, narrow range

Percentage Accurate: 55.6% → 91.6%

Time: 11.2s

Alternatives: 10

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: 55.6% 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: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ t_1 := {\left(a \cdot c\right)}^{4}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{1.265625 \cdot t_1 + t_1 \cdot 5.0625}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -3.0)))) (t_1 (pow (* a c) 4.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5625
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (fma
       -0.16666666666666666
       (/ (+ (* 1.265625 t_1) (* t_1 5.0625)) (* a (pow b 7.0)))
       (fma -0.5 (/ c b) (* -0.375 (/ (* c c) (/ (pow b 3.0) a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -3.0)));
	double t_1 = pow((a * c), 4.0);
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, (((1.265625 * t_1) + (t_1 * 5.0625)) / (a * pow(b, 7.0))), fma(-0.5, (c / b), (-0.375 * ((c * c) / (pow(b, 3.0) / a))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -3.0)))
	t_1 = Float64(a * c) ^ 4.0
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64(Float64(1.265625 * t_1) + Float64(t_1 * 5.0625)) / Float64(a * (b ^ 7.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(c * c) / Float64((b ^ 3.0) / a))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(1.265625 * t$95$1), $MachinePrecision] + N[(t$95$1 * 5.0625), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $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)) < -1

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. flip-+ 83.7%

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

      2. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. sqr-neg 85.1%

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

   7. Simplified 85.1%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 94.5%

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

      1. fma-def 94.5%

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

      2. associate-/l* 94.5%

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

      3. unpow2 94.5%

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

      4. fma-def 94.5%

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

   4. Simplified 94.5%

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

      1. pow1 94.5%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{{\left({c}^{4} \cdot {a}^{4}\right)}^{1}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

      2. pow-prod-down 94.5%

         \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot {\color{blue}{\left({\left(c \cdot a\right)}^{4}\right)}}^{1}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{{\left({\left(c \cdot a\right)}^{4}\right)}^{1}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. unpow1 94.5%

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

   8. Simplified 94.5%

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

   9. Taylor expanded in c around 0 94.5%

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

       1. metadata-eval 94.5%

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

       2. pow-sqr 94.5%

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

       3. metadata-eval 94.5%

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

       4. pow-sqr 94.5%

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

       5. unswap-sqr 94.5%

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

       6. unpow2 94.5%

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

       7. unpow2 94.5%

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

       8. swap-sqr 94.5%

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

       9. unpow2 94.5%

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

       10. unpow2 94.5%

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

       11. swap-sqr 94.5%

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

       12. unpow2 94.5%

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

       13. unpow2 94.5%

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

       14. pow-sqr 94.5%

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

       15. metadata-eval 94.5%

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

   11. Simplified 94.5%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{1.265625 \cdot {\left(c \cdot a\right)}^{4}} + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{1.265625 \cdot {\left(a \cdot c\right)}^{4} + {\left(a \cdot c\right)}^{4} \cdot 5.0625}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 89.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. flip-+ 83.7%

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

      2. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. sqr-neg 85.1%

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

   7. Simplified 85.1%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 92.3%

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

      1. fma-def 92.3%

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

      2. associate-/l* 92.3%

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

      3. unpow2 92.3%

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

      4. fma-def 92.3%

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

      5. associate-/l* 92.3%

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

      6. unpow2 92.3%

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

   4. Simplified 92.3%

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

4. Final simplification 91.3%

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

Alternative 3: 89.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. flip-+ 83.7%

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

      2. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. sqr-neg 85.1%

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

   7. Simplified 85.1%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 91.9%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 91.9%

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

      2. associate-/l* 91.9%

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

      3. associate-/r/ 91.9%

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

      4. unpow2 91.9%

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

      5. associate-/l* 91.9%

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

      6. unpow2 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}, -1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}{3 \cdot a} \]

      7. +-commutative 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \color{blue}{-1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + -1.5 \cdot \frac{c \cdot a}{b}}\right)}{3 \cdot a} \]

      8. fma-def 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{fma}\left(-1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, -1.5 \cdot \frac{c \cdot a}{b}\right)}\right)}{3 \cdot a} \]

      9. cube-prod 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{\color{blue}{{\left(c \cdot a\right)}^{3}}}{{b}^{5}}, -1.5 \cdot \frac{c \cdot a}{b}\right)\right)}{3 \cdot a} \]

      10. associate-/l* 91.9%

          \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right)\right)}{3 \cdot a} \]

      11. associate-/r/ 91.9%

          \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right)\right)}{3 \cdot a} \]

   4. Simplified 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in c around 0 92.3%

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

      1. +-commutative 92.3%

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

      2. associate-+l+ 92.3%

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

      3. +-commutative 92.3%

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

      4. fma-def 92.3%

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

      5. +-commutative 92.3%

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

      6. fma-def 92.3%

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

   7. Simplified 92.3%

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

4. Final simplification 91.4%

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

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{b \cdot b - t_0}{\left(-b\right) - \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \frac{c}{\frac{{b}^{3}}{c}}, \mathsf{fma}\left(-1.6875, \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)\right)}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       -1.125
       (* (* a a) (/ c (/ (pow b 3.0) c)))
       (fma
        -1.6875
        (/ (* (* a c) (* (* a c) (* a c))) (pow b 5.0))
        (* -1.5 (* a (/ c b)))))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -3.0)));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-1.125, ((a * a) * (c / (pow(b, 3.0) / c))), fma(-1.6875, (((a * c) * ((a * c) * (a * c))) / pow(b, 5.0)), (-1.5 * (a * (c / b))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-1.125, Float64(Float64(a * a) * Float64(c / Float64((b ^ 3.0) / c))), fma(-1.6875, Float64(Float64(Float64(a * c) * Float64(Float64(a * c) * Float64(a * c))) / (b ^ 5.0)), Float64(-1.5 * Float64(a * Float64(c / b))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.125 * N[(N[(a * a), $MachinePrecision] * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.6875 * N[(N[(N[(a * c), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -1

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. flip-+ 83.7%

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

      2. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. sqr-neg 85.1%

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

   7. Simplified 85.1%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 91.9%

      \[\leadsto \frac{\color{blue}{-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. fma-def 91.9%

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

      2. associate-/l* 91.9%

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

      3. associate-/r/ 91.9%

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

      4. unpow2 91.9%

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

      5. associate-/l* 91.9%

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

      6. unpow2 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}, -1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)}{3 \cdot a} \]

      7. +-commutative 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \color{blue}{-1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + -1.5 \cdot \frac{c \cdot a}{b}}\right)}{3 \cdot a} \]

      8. fma-def 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \color{blue}{\mathsf{fma}\left(-1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, -1.5 \cdot \frac{c \cdot a}{b}\right)}\right)}{3 \cdot a} \]

      9. cube-prod 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{\color{blue}{{\left(c \cdot a\right)}^{3}}}{{b}^{5}}, -1.5 \cdot \frac{c \cdot a}{b}\right)\right)}{3 \cdot a} \]

      10. associate-/l* 91.9%

          \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right)\right)}{3 \cdot a} \]

      11. associate-/r/ 91.9%

          \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right)\right)}{3 \cdot a} \]

   4. Simplified 91.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)\right)}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow3 91.9%

         \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{{b}^{5}}, -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)\right)}{3 \cdot a} \]

   6. Applied egg-rr 91.9%

      \[\leadsto \frac{\mathsf{fma}\left(-1.125, \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}}{{b}^{5}}, -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)\right)}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \frac{c}{\frac{{b}^{3}}{c}}, \mathsf{fma}\left(-1.6875, \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 5: 84.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

      1. flip-+ 83.7%

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

      2. add-sqr-sqrt 85.1%

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

   5. Applied egg-rr 85.1%

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

      1. sqr-neg 85.1%

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

   7. Simplified 85.1%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 87.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   3. Step-by-step derivation

      1. +-commutative 87.5%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

      2. fma-def 87.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{{c}^{2} \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

      3. associate-/l* 87.5%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, -0.5 \cdot \frac{c}{b}\right) \]

      4. associate-/r/ 87.5%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot a}, -0.5 \cdot \frac{c}{b}\right) \]

      5. unpow2 87.5%

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

      6. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 87.2%

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

Alternative 6: 84.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma -0.375 (* a (/ c (/ (pow b 3.0) c))) (* -0.5 (/ c b)))))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], 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[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $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)) < -1

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 87.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
   3. Step-by-step derivation

      1. +-commutative 87.5%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

      2. fma-def 87.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{{c}^{2} \cdot a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

      3. associate-/l* 87.5%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}}, -0.5 \cdot \frac{c}{b}\right) \]

      4. associate-/r/ 87.5%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot a}, -0.5 \cdot \frac{c}{b}\right) \]

      5. unpow2 87.5%

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

      6. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 87.0%

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

Alternative 7: 84.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (*
    (/ 0.3333333333333333 a)
    (* a (+ (* (/ c b) -1.5) (* -1.125 (* (* a c) (/ c (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (0.3333333333333333 / a) * (a * (((c / b) * -1.5) + (-1.125 * ((a * c) * (c / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(a * Float64(Float64(Float64(c / b) * -1.5) + Float64(-1.125 * Float64(Float64(a * c) * Float64(c / (b ^ 3.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], 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[(0.3333333333333333 / a), $MachinePrecision] * N[(a * N[(N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(-1.125 * N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -1

   1. Initial program 83.6%

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

      1. fma-neg 83.9%

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

      2. *-commutative 83.9%

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

      3. distribute-rgt-neg-in 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-neg-in 83.9%

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

      6. metadata-eval 83.9%

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

   3. Applied egg-rr 83.9%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 87.2%

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

      1. fma-def 87.2%

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

      2. associate-/l* 87.2%

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

      3. associate-/r/ 87.2%

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

      4. unpow2 87.2%

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

      5. associate-/l* 87.2%

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

      6. unpow2 87.2%

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

      7. associate-/l* 87.2%

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

      8. associate-/r/ 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in c around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*l/ 87.1%

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

      3. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

      1. div-inv 87.2%

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

      2. associate-/r/ 87.2%

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

   9. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right), \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right) \cdot \frac{1}{3 \cdot a}} \]
   10. Step-by-step derivation

       1. *-commutative 87.2%

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

       2. associate-/r* 87.1%

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

       3. metadata-eval 87.1%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right), \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right) \]

       4. fma-udef 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right)} \]

       5. associate-*r* 87.0%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \frac{\color{blue}{\left(c \cdot a\right) \cdot -1.5}}{b}\right) \]

       6. associate-*l/ 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \color{blue}{\frac{c \cdot a}{b} \cdot -1.5}\right) \]

       7. *-commutative 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \color{blue}{-1.5 \cdot \frac{c \cdot a}{b}}\right) \]

       8. +-commutative 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)} \]

       9. associate-*l/ 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \]

       10. associate-*l* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right) \cdot a} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \]

       11. associate-*r* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + -1.125 \cdot \color{blue}{\left(\left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot a\right) \cdot a\right)}\right) \]

       12. associate-*l/ 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + -1.125 \cdot \left(\left(\color{blue}{\frac{c \cdot c}{{b}^{3}}} \cdot a\right) \cdot a\right)\right) \]

       13. associate-*r* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + \color{blue}{\left(-1.125 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)\right) \cdot a}\right) \]

       14. distribute-rgt-out 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)\right)\right)} \]

   11. Simplified 87.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(c \cdot a\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Alternative 8: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (*
    (/ 0.3333333333333333 a)
    (* a (+ (* (/ c b) -1.5) (* -1.125 (* (* a c) (/ c (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (0.3333333333333333 / a) * (a * (((c / b) * -1.5) + (-1.125 * ((a * c) * (c / 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 (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-1.0d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (0.3333333333333333d0 / a) * (a * (((c / b) * (-1.5d0)) + ((-1.125d0) * ((a * c) * (c / (b ** 3.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (0.3333333333333333 / a) * (a * (((c / b) * -1.5) + (-1.125 * ((a * c) * (c / Math.pow(b, 3.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (0.3333333333333333 / a) * (a * (((c / b) * -1.5) + (-1.125 * ((a * c) * (c / math.pow(b, 3.0))))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(a * Float64(Float64(Float64(c / b) * -1.5) + Float64(-1.125 * Float64(Float64(a * c) * Float64(c / (b ^ 3.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.0)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (0.3333333333333333 / a) * (a * (((c / b) * -1.5) + (-1.125 * ((a * c) * (c / (b ^ 3.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(a * N[(N[(N[(c / b), $MachinePrecision] * -1.5), $MachinePrecision] + N[(-1.125 * N[(N[(a * c), $MachinePrecision] * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -1

   1. Initial program 83.6%

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

   2. Taylor expanded in a around 0 83.6%

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

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

   1. Initial program 46.9%

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

   2. Taylor expanded in b around inf 87.2%

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

      1. fma-def 87.2%

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

      2. associate-/l* 87.2%

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

      3. associate-/r/ 87.2%

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

      4. unpow2 87.2%

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

      5. associate-/l* 87.2%

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

      6. unpow2 87.2%

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

      7. associate-/l* 87.2%

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

      8. associate-/r/ 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in c around 0 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-*l/ 87.1%

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

      3. associate-*l* 87.3%

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

   7. Simplified 87.3%

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

      1. div-inv 87.2%

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

      2. associate-/r/ 87.2%

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

   9. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right), \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right) \cdot \frac{1}{3 \cdot a}} \]
   10. Step-by-step derivation

       1. *-commutative 87.2%

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

       2. associate-/r* 87.1%

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

       3. metadata-eval 87.1%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1.125, \left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right), \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right) \]

       4. fma-udef 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \frac{c \cdot \left(a \cdot -1.5\right)}{b}\right)} \]

       5. associate-*r* 87.0%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \frac{\color{blue}{\left(c \cdot a\right) \cdot -1.5}}{b}\right) \]

       6. associate-*l/ 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \color{blue}{\frac{c \cdot a}{b} \cdot -1.5}\right) \]

       7. *-commutative 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right) + \color{blue}{-1.5 \cdot \frac{c \cdot a}{b}}\right) \]

       8. +-commutative 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1.5 \cdot \frac{c \cdot a}{b} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)} \]

       9. associate-*l/ 87.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(-1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \]

       10. associate-*l* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right) \cdot a} + -1.125 \cdot \left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \]

       11. associate-*r* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + -1.125 \cdot \color{blue}{\left(\left(\left(\frac{c}{{b}^{3}} \cdot c\right) \cdot a\right) \cdot a\right)}\right) \]

       12. associate-*l/ 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + -1.125 \cdot \left(\left(\color{blue}{\frac{c \cdot c}{{b}^{3}}} \cdot a\right) \cdot a\right)\right) \]

       13. associate-*r* 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\left(-1.5 \cdot \frac{c}{b}\right) \cdot a + \color{blue}{\left(-1.125 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)\right) \cdot a}\right) \]

       14. distribute-rgt-out 87.1%

           \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a\right)\right)\right)} \]

   11. Simplified 87.1%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(c \cdot a\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(a \cdot \left(\frac{c}{b} \cdot -1.5 + -1.125 \cdot \left(\left(a \cdot c\right) \cdot \frac{c}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Alternative 9: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -1.1e-7)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-1.1d-7)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -1.1e-7)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -1.1e-7)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.1e-7], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -1.1000000000000001e-7

   1. Initial program 71.1%

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

   2. Taylor expanded in a around 0 71.2%

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

   if -1.1000000000000001e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 27.3%

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

   2. Taylor expanded in b around inf 86.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 64.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

2. Taylor expanded in b around inf 67.0%

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

3. Final simplification 67.0%

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

Reproduce

herbie shell --seed 2023187 
(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)))