Cubic critical, narrow range

Percentage Accurate: 55.8% → 91.5%

Time: 33.1s

Alternatives: 13

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.23:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \left(a \cdot -1.0546875\right) \cdot \frac{{c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (+ (/ (pow b 2.0) a) (* c -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.23)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (*
         a
         (+
          (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
          (* (* a -1.0546875) (/ (pow c 4.0) (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.23) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + ((a * -1.0546875) * (pow(c, 4.0) / pow(b, 7.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (((b ** 2.0d0) / a) + (c * (-3.0d0)))
    if (((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)) <= (-0.23d0)) then
        tmp = ((t_0 - (-b ** 2.0d0)) / (b + sqrt(t_0))) / (3.0d0 * a)
    else
        tmp = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((a * (-1.0546875d0)) * ((c ** 4.0d0) / (b ** 7.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * ((Math.pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.23) {
		tmp = ((t_0 - Math.pow(-b, 2.0)) / (b + Math.sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + ((a * -1.0546875) * (Math.pow(c, 4.0) / Math.pow(b, 7.0)))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * ((math.pow(b, 2.0) / a) + (c * -3.0))
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.23:
		tmp = ((t_0 - math.pow(-b, 2.0)) / (b + math.sqrt(t_0))) / (3.0 * a)
	else:
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + ((a * -1.0546875) * (math.pow(c, 4.0) / math.pow(b, 7.0)))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64((b ^ 2.0) / a) + Float64(c * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.23)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(Float64(a * -1.0546875) * Float64((c ^ 4.0) / (b ^ 7.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (((b ^ 2.0) / a) + (c * -3.0));
	tmp = 0.0;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.23)
		tmp = ((t_0 - (-b ^ 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	else
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-0.5625 * ((c ^ 3.0) / (b ^ 5.0))) + ((a * -1.0546875) * ((c ^ 4.0) / (b ^ 7.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision] + N[(c * -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], -0.23], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -1.0546875), $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.23000000000000001

   1. Initial program 84.2%

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

      1. sqr-neg 84.2%

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

      2. sqr-neg 84.2%

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

      3. associate-*l* 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in a around inf 84.2%

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

      1. flip-+ 83.9%

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

      2. pow2 83.9%

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

      3. add-sqr-sqrt 85.0%

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

      4. cancel-sign-sub-inv 85.0%

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

      5. metadata-eval 85.0%

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

      6. cancel-sign-sub-inv 85.0%

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

      7. metadata-eval 85.0%

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

   7. Applied egg-rr 85.0%

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

   if -0.23000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.3%

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

      1. sqr-neg 46.3%

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

      2. sqr-neg 46.3%

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

      3. associate-*l* 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in a around 0 95.0%

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

   6. Taylor expanded in c around 0 95.0%

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

      1. associate-/l* 95.0%

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

      2. associate-*r* 95.0%

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

   8. Simplified 95.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.23:\\ \;\;\;\;\frac{\frac{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right) - {\left(-b\right)}^{2}}{b + \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + \left(a \cdot -1.0546875\right) \cdot \frac{{c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, {a}^{3} \cdot \left(-1.0546875 \cdot \frac{{c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{a \cdot {b}^{5}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (+ (/ (pow b 2.0) a) (* c -3.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.4)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (*
      c
      (-
       (*
        c
        (fma
         -0.375
         (/ a (pow b 3.0))
         (*
          (pow a 3.0)
          (+
           (* -1.0546875 (/ (pow c 2.0) (pow b 7.0)))
           (* -0.5625 (/ c (* a (pow b 5.0))))))))
       (/ 0.5 b))))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.4) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = c * ((c * fma(-0.375, (a / pow(b, 3.0)), (pow(a, 3.0) * ((-1.0546875 * (pow(c, 2.0) / pow(b, 7.0))) + (-0.5625 * (c / (a * pow(b, 5.0)))))))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64((b ^ 2.0) / a) + Float64(c * -3.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.4)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-0.375, Float64(a / (b ^ 3.0)), Float64((a ^ 3.0) * Float64(Float64(-1.0546875 * Float64((c ^ 2.0) / (b ^ 7.0))) + Float64(-0.5625 * Float64(c / Float64(a * (b ^ 5.0)))))))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision] + N[(c * -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], -0.4], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-1.0546875 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / 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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.40000000000000002

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. sqr-neg 85.2%

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

      3. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in a around inf 85.2%

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

      1. flip-+ 85.0%

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

      2. pow2 85.0%

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

      3. add-sqr-sqrt 86.0%

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

      4. cancel-sign-sub-inv 86.0%

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

      5. metadata-eval 86.0%

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

      6. cancel-sign-sub-inv 86.0%

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

      7. metadata-eval 86.0%

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

   7. Applied egg-rr 86.0%

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

   if -0.40000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.2%

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

      1. sqr-neg 47.2%

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

      2. sqr-neg 47.2%

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

      3. associate-*l* 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in c around 0 94.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

   7. Simplified 94.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

   8. Taylor expanded in c around 0 94.3%

      \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]
   9. Step-by-step derivation

      1. associate-*r/ 94.3%

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

   10. Simplified 94.3%

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

   11. Taylor expanded in a around inf 94.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\frac{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right) - {\left(-b\right)}^{2}}{b + \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, {a}^{3} \cdot \left(-1.0546875 \cdot \frac{{c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{a \cdot {b}^{5}}\right)\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)\\ \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{\frac{t\_0 - {\left(-b\right)}^{2}}{b + \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(-3 \cdot \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (+ (/ (pow b 2.0) a) (* c -3.0)))))
   (if (<= b 0.175)
     (/ (/ (- t_0 (pow (- b) 2.0)) (+ b (sqrt t_0))) (* 3.0 a))
     (/
      1.0
      (*
       b
       (+
        (* -3.0 (/ (* -0.375 (* c (pow a 2.0))) (pow b 4.0)))
        (fma 1.5 (/ a (pow b 2.0)) (/ -2.0 c))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((pow(b, 2.0) / a) + (c * -3.0));
	double tmp;
	if (b <= 0.175) {
		tmp = ((t_0 - pow(-b, 2.0)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * ((-3.0 * ((-0.375 * (c * pow(a, 2.0))) / pow(b, 4.0))) + fma(1.5, (a / pow(b, 2.0)), (-2.0 / c))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64((b ^ 2.0) / a) + Float64(c * -3.0)))
	tmp = 0.0
	if (b <= 0.175)
		tmp = Float64(Float64(Float64(t_0 - (Float64(-b) ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(-3.0 * Float64(Float64(-0.375 * Float64(c * (a ^ 2.0))) / (b ^ 4.0))) + fma(1.5, Float64(a / (b ^ 2.0)), Float64(-2.0 / c)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision] + N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.175], N[(N[(N[(t$95$0 - N[Power[(-b), 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(-0.375 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17499999999999999

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in a around inf 86.2%

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

      1. flip-+ 85.9%

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

      2. pow2 85.9%

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

      3. add-sqr-sqrt 87.4%

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

      4. cancel-sign-sub-inv 87.4%

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

      5. metadata-eval 87.4%

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

      6. cancel-sign-sub-inv 87.4%

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

      7. metadata-eval 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 0.17499999999999999 < b

   1. Initial program 48.4%

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

      1. sqr-neg 48.4%

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

      2. sqr-neg 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in a around inf 48.3%

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

      1. clear-num 48.3%

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

      2. inv-pow 48.3%

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

      3. *-commutative 48.3%

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

      4. neg-mul-1 48.3%

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

      5. fma-define 48.3%

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

      6. cancel-sign-sub-inv 48.3%

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

      7. metadata-eval 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. unpow-1 48.3%

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

      2. associate-/l* 48.3%

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

      3. +-commutative 48.3%

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

      4. fma-define 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in b around inf 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   11. Step-by-step derivation

       1. associate--l+ 91.5%

          \[\leadsto \frac{1}{b \cdot \color{blue}{\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)}} \]

       2. distribute-rgt-out 91.5%

          \[\leadsto \frac{1}{b \cdot \left(-3 \cdot \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

       3. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(-3 \cdot \frac{\left({a}^{2} \cdot c\right) \cdot \color{blue}{-0.375}}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

       4. fma-neg 91.5%

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

       5. associate-*r/ 91.5%

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

       6. metadata-eval 91.5%

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

       7. distribute-neg-frac 91.5%

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

       8. metadata-eval 91.5%

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

   12. Simplified 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(-3 \cdot \frac{\left({a}^{2} \cdot c\right) \cdot -0.375}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{\frac{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right) - {\left(-b\right)}^{2}}{b + \sqrt{a \cdot \left(\frac{{b}^{2}}{a} + c \cdot -3\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(-3 \cdot \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(-3 \cdot \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.175)
   (/ 1.0 (* a (/ 3.0 (- (sqrt (* a (fma -3.0 c (/ (pow b 2.0) a)))) b))))
   (/
    1.0
    (*
     b
     (+
      (* -3.0 (/ (* -0.375 (* c (pow a 2.0))) (pow b 4.0)))
      (fma 1.5 (/ a (pow b 2.0)) (/ -2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.175) {
		tmp = 1.0 / (a * (3.0 / (sqrt((a * fma(-3.0, c, (pow(b, 2.0) / a)))) - b)));
	} else {
		tmp = 1.0 / (b * ((-3.0 * ((-0.375 * (c * pow(a, 2.0))) / pow(b, 4.0))) + fma(1.5, (a / pow(b, 2.0)), (-2.0 / c))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.175)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / Float64(sqrt(Float64(a * fma(-3.0, c, Float64((b ^ 2.0) / a)))) - b))));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(-3.0 * Float64(Float64(-0.375 * Float64(c * (a ^ 2.0))) / (b ^ 4.0))) + fma(1.5, Float64(a / (b ^ 2.0)), Float64(-2.0 / c)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.175], N[(1.0 / N[(a * N[(3.0 / N[(N[Sqrt[N[(a * N[(-3.0 * c + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(-3.0 * N[(N[(-0.375 * N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17499999999999999

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in a around inf 86.2%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. *-commutative 86.3%

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

      4. neg-mul-1 86.3%

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

      5. fma-define 86.3%

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

      6. cancel-sign-sub-inv 86.3%

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

      7. metadata-eval 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. associate-/l* 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.4%

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

   9. Simplified 86.4%

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

       1. fma-undefine 86.4%

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

   11. Applied egg-rr 86.4%

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

       1. +-commutative 86.4%

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

       2. neg-mul-1 86.4%

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

       3. unsub-neg 86.4%

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

   13. Simplified 86.4%

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

   if 0.17499999999999999 < b

   1. Initial program 48.4%

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

      1. sqr-neg 48.4%

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

      2. sqr-neg 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in a around inf 48.3%

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

      1. clear-num 48.3%

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

      2. inv-pow 48.3%

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

      3. *-commutative 48.3%

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

      4. neg-mul-1 48.3%

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

      5. fma-define 48.3%

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

      6. cancel-sign-sub-inv 48.3%

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

      7. metadata-eval 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. unpow-1 48.3%

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

      2. associate-/l* 48.3%

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

      3. +-commutative 48.3%

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

      4. fma-define 48.3%

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

   9. Simplified 48.3%

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

   10. Taylor expanded in b around inf 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + 1.5 \cdot \frac{a}{{b}^{2}}\right) - 2 \cdot \frac{1}{c}\right)}} \]
   11. Step-by-step derivation

       1. associate--l+ 91.5%

          \[\leadsto \frac{1}{b \cdot \color{blue}{\left(-3 \cdot \frac{-0.75 \cdot \left({a}^{2} \cdot c\right) + 0.375 \cdot \left({a}^{2} \cdot c\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)}} \]

       2. distribute-rgt-out 91.5%

          \[\leadsto \frac{1}{b \cdot \left(-3 \cdot \frac{\color{blue}{\left({a}^{2} \cdot c\right) \cdot \left(-0.75 + 0.375\right)}}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

       3. metadata-eval 91.5%

          \[\leadsto \frac{1}{b \cdot \left(-3 \cdot \frac{\left({a}^{2} \cdot c\right) \cdot \color{blue}{-0.375}}{{b}^{4}} + \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)\right)} \]

       4. fma-neg 91.5%

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

       5. associate-*r/ 91.5%

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

       6. metadata-eval 91.5%

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

       7. distribute-neg-frac 91.5%

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

       8. metadata-eval 91.5%

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

   12. Simplified 91.5%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(-3 \cdot \frac{\left({a}^{2} \cdot c\right) \cdot -0.375}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(-3 \cdot \frac{-0.375 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{4}} + \mathsf{fma}\left(1.5, \frac{a}{{b}^{2}}, \frac{-2}{c}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.175)
   (/ 1.0 (* a (/ 3.0 (- (sqrt (* a (fma -3.0 c (/ (pow b 2.0) a)))) b))))
   (*
    c
    (+
     (*
      c
      (* a (+ (/ -0.375 (pow b 3.0)) (* a (* c (* -0.5625 (pow b -5.0)))))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.175) {
		tmp = 1.0 / (a * (3.0 / (sqrt((a * fma(-3.0, c, (pow(b, 2.0) / a)))) - b)));
	} else {
		tmp = c * ((c * (a * ((-0.375 / pow(b, 3.0)) + (a * (c * (-0.5625 * pow(b, -5.0))))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.175)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / Float64(sqrt(Float64(a * fma(-3.0, c, Float64((b ^ 2.0) / a)))) - b))));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.375 / (b ^ 3.0)) + Float64(a * Float64(c * Float64(-0.5625 * (b ^ -5.0))))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.175], N[(1.0 / N[(a * N[(3.0 / N[(N[Sqrt[N[(a * N[(-3.0 * c + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(-0.5625 * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17499999999999999

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in a around inf 86.2%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. *-commutative 86.3%

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

      4. neg-mul-1 86.3%

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

      5. fma-define 86.3%

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

      6. cancel-sign-sub-inv 86.3%

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

      7. metadata-eval 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. associate-/l* 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.4%

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

   9. Simplified 86.4%

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

       1. fma-undefine 86.4%

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

   11. Applied egg-rr 86.4%

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

       1. +-commutative 86.4%

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

       2. neg-mul-1 86.4%

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

       3. unsub-neg 86.4%

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

   13. Simplified 86.4%

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

   if 0.17499999999999999 < b

   1. Initial program 48.4%

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

      1. sqr-neg 48.4%

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

      2. sqr-neg 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 91.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 90.6%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      2. log1p-undefine 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{-3}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   7. Applied egg-rr 82.7%

      \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 82.7%

         \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

      2. *-commutative 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \color{blue}{\left(\frac{{a}^{2} \cdot c}{{b}^{5}} \cdot -0.5625\right)} + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left(\left({a}^{2} \cdot c\right) \cdot \frac{1}{{b}^{5}}\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot \color{blue}{{b}^{\left(-5\right)}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot {b}^{\color{blue}{-5}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      6. associate-*r* 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      7. log1p-define 90.6%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      8. log1p-expm1-u 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      9. metadata-eval 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{\left(-3\right)}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      10. pow-flip 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot \color{blue}{\frac{1}{{b}^{3}}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      11. div-inv 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \color{blue}{\frac{a}{{b}^{3}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied egg-rr 91.3%

      \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
   10. Step-by-step derivation

       1. +-commutative 91.3%

          \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right) + c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       2. distribute-lft-out 91.3%

          \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       3. *-commutative 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a}{{b}^{3}} \cdot -0.375} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       4. associate-*l/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a \cdot -0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       5. associate-/l* 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{a \cdot \frac{-0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       6. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \frac{\color{blue}{-0.375}}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       7. distribute-neg-frac 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \color{blue}{\left(-\frac{0.375}{{b}^{3}}\right)} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       8. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\frac{\color{blue}{0.375 \cdot 1}}{{b}^{3}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       9. associate-*r/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\color{blue}{0.375 \cdot \frac{1}{{b}^{3}}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       10. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{{a}^{2} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       11. unpow2 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

       12. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{a \cdot \left(a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       13. distribute-lft-out 91.3%

           \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(\left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   11. Simplified 91.3%

       \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left({b}^{-5} \cdot -0.5625\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.175:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.19:\\ \;\;\;\;\left(\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.19)
   (* (- (sqrt (* a (fma -3.0 c (/ (pow b 2.0) a)))) b) (/ 1.0 (* 3.0 a)))
   (*
    c
    (+
     (*
      c
      (* a (+ (/ -0.375 (pow b 3.0)) (* a (* c (* -0.5625 (pow b -5.0)))))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.19) {
		tmp = (sqrt((a * fma(-3.0, c, (pow(b, 2.0) / a)))) - b) * (1.0 / (3.0 * a));
	} else {
		tmp = c * ((c * (a * ((-0.375 / pow(b, 3.0)) + (a * (c * (-0.5625 * pow(b, -5.0))))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.19)
		tmp = Float64(Float64(sqrt(Float64(a * fma(-3.0, c, Float64((b ^ 2.0) / a)))) - b) * Float64(1.0 / Float64(3.0 * a)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.375 / (b ^ 3.0)) + Float64(a * Float64(c * Float64(-0.5625 * (b ^ -5.0))))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.19], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(-0.5625 * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.19

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in a around inf 86.2%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. *-commutative 86.3%

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

      4. neg-mul-1 86.3%

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

      5. fma-define 86.3%

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

      6. cancel-sign-sub-inv 86.3%

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

      7. metadata-eval 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. associate-/l* 86.4%

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

      3. +-commutative 86.4%

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

      4. fma-define 86.4%

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

   9. Simplified 86.4%

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

       1. *-un-lft-identity 86.4%

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

       2. associate-*r/ 86.3%

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

   11. Applied egg-rr 86.3%

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

       1. *-lft-identity 86.3%

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

       2. associate-/r/ 86.3%

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

       3. fma-define 86.3%

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

       4. +-commutative 86.3%

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

       5. neg-mul-1 86.3%

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

       6. unsub-neg 86.3%

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

   13. Simplified 86.3%

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

   if 0.19 < b

   1. Initial program 48.4%

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

      1. sqr-neg 48.4%

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

      2. sqr-neg 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 91.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 90.6%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      2. log1p-undefine 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{-3}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   7. Applied egg-rr 82.7%

      \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 82.7%

         \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

      2. *-commutative 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \color{blue}{\left(\frac{{a}^{2} \cdot c}{{b}^{5}} \cdot -0.5625\right)} + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left(\left({a}^{2} \cdot c\right) \cdot \frac{1}{{b}^{5}}\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot \color{blue}{{b}^{\left(-5\right)}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot {b}^{\color{blue}{-5}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      6. associate-*r* 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      7. log1p-define 90.6%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      8. log1p-expm1-u 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      9. metadata-eval 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{\left(-3\right)}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      10. pow-flip 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot \color{blue}{\frac{1}{{b}^{3}}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      11. div-inv 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \color{blue}{\frac{a}{{b}^{3}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied egg-rr 91.3%

      \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
   10. Step-by-step derivation

       1. +-commutative 91.3%

          \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right) + c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       2. distribute-lft-out 91.3%

          \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       3. *-commutative 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a}{{b}^{3}} \cdot -0.375} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       4. associate-*l/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a \cdot -0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       5. associate-/l* 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{a \cdot \frac{-0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       6. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \frac{\color{blue}{-0.375}}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       7. distribute-neg-frac 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \color{blue}{\left(-\frac{0.375}{{b}^{3}}\right)} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       8. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\frac{\color{blue}{0.375 \cdot 1}}{{b}^{3}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       9. associate-*r/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\color{blue}{0.375 \cdot \frac{1}{{b}^{3}}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       10. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{{a}^{2} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       11. unpow2 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

       12. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{a \cdot \left(a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       13. distribute-lft-out 91.3%

           \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(\left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   11. Simplified 91.3%

       \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left({b}^{-5} \cdot -0.5625\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.19:\\ \;\;\;\;\left(\sqrt{a \cdot \mathsf{fma}\left(-3, c, \frac{{b}^{2}}{a}\right)} - b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.4)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.4) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / c)));
	}
	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)) <= (-0.4d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    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)) <= -0.4) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.4:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	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)) <= -0.4)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	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)) <= -0.4)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	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], -0.4], 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[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $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 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.40000000000000002

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. sqr-neg 85.2%

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

      3. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

   if -0.40000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.2%

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

      1. sqr-neg 47.2%

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

      2. sqr-neg 47.2%

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

      3. associate-*l* 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in a around inf 47.1%

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

      1. clear-num 47.1%

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

      2. inv-pow 47.1%

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

      3. *-commutative 47.1%

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

      4. neg-mul-1 47.1%

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

      5. fma-define 47.1%

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

      6. cancel-sign-sub-inv 47.1%

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

      7. metadata-eval 47.1%

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

   7. Applied egg-rr 47.1%

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

      1. unpow-1 47.1%

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

      2. associate-/l* 47.1%

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

      3. +-commutative 47.1%

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

      4. fma-define 47.1%

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

   9. Simplified 47.1%

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

   10. Taylor expanded in b around inf 88.3%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 88.3%

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

       2. *-commutative 88.3%

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

       3. associate-*r/ 88.3%

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

       4. metadata-eval 88.3%

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

   12. Simplified 88.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.18:\\ \;\;\;\;\frac{\sqrt{a \cdot \left({b}^{2} \cdot \frac{1}{a} - 3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.18)
   (/ (- (sqrt (* a (- (* (pow b 2.0) (/ 1.0 a)) (* 3.0 c)))) b) (* 3.0 a))
   (*
    c
    (+
     (*
      c
      (* a (+ (/ -0.375 (pow b 3.0)) (* a (* c (* -0.5625 (pow b -5.0)))))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.18) {
		tmp = (sqrt((a * ((pow(b, 2.0) * (1.0 / a)) - (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((c * (a * ((-0.375 / pow(b, 3.0)) + (a * (c * (-0.5625 * pow(b, -5.0))))))) + (0.5 * (-1.0 / 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 (b <= 0.18d0) then
        tmp = (sqrt((a * (((b ** 2.0d0) * (1.0d0 / a)) - (3.0d0 * c)))) - b) / (3.0d0 * a)
    else
        tmp = c * ((c * (a * (((-0.375d0) / (b ** 3.0d0)) + (a * (c * ((-0.5625d0) * (b ** (-5.0d0)))))))) + (0.5d0 * ((-1.0d0) / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.18) {
		tmp = (Math.sqrt((a * ((Math.pow(b, 2.0) * (1.0 / a)) - (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((c * (a * ((-0.375 / Math.pow(b, 3.0)) + (a * (c * (-0.5625 * Math.pow(b, -5.0))))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.18:
		tmp = (math.sqrt((a * ((math.pow(b, 2.0) * (1.0 / a)) - (3.0 * c)))) - b) / (3.0 * a)
	else:
		tmp = c * ((c * (a * ((-0.375 / math.pow(b, 3.0)) + (a * (c * (-0.5625 * math.pow(b, -5.0))))))) + (0.5 * (-1.0 / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.18)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64((b ^ 2.0) * Float64(1.0 / a)) - Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.375 / (b ^ 3.0)) + Float64(a * Float64(c * Float64(-0.5625 * (b ^ -5.0))))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.18)
		tmp = (sqrt((a * (((b ^ 2.0) * (1.0 / a)) - (3.0 * c)))) - b) / (3.0 * a);
	else
		tmp = c * ((c * (a * ((-0.375 / (b ^ 3.0)) + (a * (c * (-0.5625 * (b ^ -5.0))))))) + (0.5 * (-1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.18], N[(N[(N[Sqrt[N[(a * N[(N[(N[Power[b, 2.0], $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(a * N[(N[(-0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(-0.5625 * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17999999999999999

   1. Initial program 86.2%

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

      1. sqr-neg 86.2%

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

      2. sqr-neg 86.2%

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

      3. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in a around inf 86.2%

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

      1. div-inv 86.3%

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

   7. Applied egg-rr 86.3%

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

   if 0.17999999999999999 < b

   1. Initial program 48.4%

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

      1. sqr-neg 48.4%

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

      2. sqr-neg 48.4%

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

      3. associate-*l* 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 91.3%

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 90.6%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      2. log1p-undefine 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot \color{blue}{{b}^{\left(-3\right)}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{-3}}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   7. Applied egg-rr 82.7%

      \[\leadsto c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in 82.7%

         \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

      2. *-commutative 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \color{blue}{\left(\frac{{a}^{2} \cdot c}{{b}^{5}} \cdot -0.5625\right)} + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      3. div-inv 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left(\left({a}^{2} \cdot c\right) \cdot \frac{1}{{b}^{5}}\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      4. pow-flip 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot \color{blue}{{b}^{\left(-5\right)}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      5. metadata-eval 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left(\left({a}^{2} \cdot c\right) \cdot {b}^{\color{blue}{-5}}\right) \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      6. associate-*r* 82.7%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\color{blue}{\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right)} \cdot -0.5625\right) + c \cdot \log \left(1 + \mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      7. log1p-define 90.6%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      8. log1p-expm1-u 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \color{blue}{\left(-0.375 \cdot \left(a \cdot {b}^{-3}\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

      9. metadata-eval 91.3%

         \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot {b}^{\color{blue}{\left(-3\right)}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      10. pow-flip 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \left(a \cdot \color{blue}{\frac{1}{{b}^{3}}}\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

      11. div-inv 91.3%

          \[\leadsto c \cdot \left(\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \color{blue}{\frac{a}{{b}^{3}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   9. Applied egg-rr 91.3%

      \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) + c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
   10. Step-by-step derivation

       1. +-commutative 91.3%

          \[\leadsto c \cdot \left(\color{blue}{\left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right) + c \cdot \left(\left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       2. distribute-lft-out 91.3%

          \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right)} - 0.5 \cdot \frac{1}{b}\right) \]

       3. *-commutative 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a}{{b}^{3}} \cdot -0.375} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       4. associate-*l/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{a \cdot -0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       5. associate-/l* 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{a \cdot \frac{-0.375}{{b}^{3}}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       6. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \frac{\color{blue}{-0.375}}{{b}^{3}} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       7. distribute-neg-frac 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \color{blue}{\left(-\frac{0.375}{{b}^{3}}\right)} + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       8. metadata-eval 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\frac{\color{blue}{0.375 \cdot 1}}{{b}^{3}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       9. associate-*r/ 91.3%

          \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-\color{blue}{0.375 \cdot \frac{1}{{b}^{3}}}\right) + \left({a}^{2} \cdot \left(c \cdot {b}^{-5}\right)\right) \cdot -0.5625\right) - 0.5 \cdot \frac{1}{b}\right) \]

       10. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{{a}^{2} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       11. unpow2 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

       12. associate-*l* 91.3%

           \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + \color{blue}{a \cdot \left(a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)}\right) - 0.5 \cdot \frac{1}{b}\right) \]

       13. distribute-lft-out 91.3%

           \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(\left(-0.375 \cdot \frac{1}{{b}^{3}}\right) + a \cdot \left(\left(c \cdot {b}^{-5}\right) \cdot -0.5625\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]

   11. Simplified 91.3%

       \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left({b}^{-5} \cdot -0.5625\right)\right)\right)\right)} - 0.5 \cdot \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.18:\\ \;\;\;\;\frac{\sqrt{a \cdot \left({b}^{2} \cdot \frac{1}{a} - 3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(a \cdot \left(\frac{-0.375}{{b}^{3}} + a \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-5}\right)\right)\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. sqr-neg 51.6%

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

   2. sqr-neg 51.6%

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

   3. associate-*l* 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in a around inf 51.5%

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

   1. clear-num 51.6%

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

   2. inv-pow 51.6%

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

   3. *-commutative 51.6%

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

   4. neg-mul-1 51.6%

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

   5. fma-define 51.6%

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

   6. cancel-sign-sub-inv 51.6%

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

   7. metadata-eval 51.6%

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

7. Applied egg-rr 51.6%

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

   1. unpow-1 51.6%

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

   2. associate-/l* 51.6%

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

   3. +-commutative 51.6%

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

   4. fma-define 51.6%

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

9. Simplified 51.6%

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

10. Taylor expanded in b around inf 84.5%

    \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
11. Step-by-step derivation

    1. associate-*r/ 84.5%

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

    2. *-commutative 84.5%

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

    3. associate-*r/ 84.5%

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

    4. metadata-eval 84.5%

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

12. Simplified 84.5%

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

Alternative 10: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ (* -0.375 (* a c)) (pow b 3.0)) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * (((-0.375 * (a * c)) / pow(b, 3.0)) - (0.5 / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((((-0.375d0) * (a * c)) / (b ** 3.0d0)) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * (((-0.375 * (a * c)) / Math.pow(b, 3.0)) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * (((-0.375 * (a * c)) / math.pow(b, 3.0)) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(-0.375 * Float64(a * c)) / (b ^ 3.0)) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((-0.375 * (a * c)) / (b ^ 3.0)) - (0.5 / b));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. sqr-neg 51.6%

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

   2. sqr-neg 51.6%

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

   3. associate-*l* 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in a around 0 84.2%

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

6. Taylor expanded in c around 0 84.0%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 84.0%

      \[\leadsto c \cdot \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 84.0%

      \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 84.0%

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

8. Simplified 84.0%

   \[\leadsto \color{blue}{c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{0.5}{b}\right)} \]
9. Add Preprocessing

Alternative 11: 64.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. sqr-neg 51.6%

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

   2. sqr-neg 51.6%

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

   3. associate-*l* 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in b around inf 68.2%

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

   1. associate-*r/ 68.2%

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

   2. *-commutative 68.2%

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

7. Simplified 68.2%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
8. Add Preprocessing

Alternative 12: 64.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. sqr-neg 51.6%

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

   2. sqr-neg 51.6%

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

   3. associate-*l* 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in a around 0 84.2%

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

6. Taylor expanded in c around inf 83.9%

   \[\leadsto \color{blue}{{c}^{2} \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} - 0.5 \cdot \frac{1}{b \cdot c}\right)} \]

7. Taylor expanded in c around 0 68.2%

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

   1. associate-*r/ 68.2%

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

   2. *-commutative 68.2%

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

   3. associate-/l* 68.1%

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

9. Simplified 68.1%

   \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b}} \]
10. Add Preprocessing

Alternative 13: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

   1. sqr-neg 51.6%

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

   2. sqr-neg 51.6%

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

   3. associate-*l* 51.6%

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

3. Simplified 51.6%

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

5. Taylor expanded in a around inf 51.5%

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

   1. clear-num 51.6%

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

   2. inv-pow 51.6%

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

   3. *-commutative 51.6%

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

   4. neg-mul-1 51.6%

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

   5. fma-define 51.6%

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

   6. cancel-sign-sub-inv 51.6%

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

   7. metadata-eval 51.6%

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

7. Applied egg-rr 51.6%

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

   1. unpow-1 51.6%

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

   2. associate-/l* 51.6%

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

   3. +-commutative 51.6%

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

   4. fma-define 51.6%

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

9. Simplified 51.6%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
11. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

12. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))