Cubic critical, narrow range

Percentage Accurate: 54.9% → 92.2%

Time: 18.0s

Alternatives: 17

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: 54.9% 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: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -4.4)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+
      (* c -0.5)
      (+
       (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))
       (/ (* (pow (* a c) 4.0) -1.0546875) (* a (pow b 6.0))))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -4.4) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))) + ((pow((a * c), 4.0) * -1.0546875) / (a * pow(b, 6.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -4.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) + Float64(Float64((Float64(a * c) ^ 4.0) * -1.0546875) / Float64(a * (b ^ 6.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -4.4], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -1.0546875), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -4.4000000000000004

   1. Initial program 87.8%

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

   3. Simplified 88.0%

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

   if -4.4000000000000004 < (/.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 48.5%

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

      1. sqr-neg 48.5%

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

      2. sqr-neg 48.5%

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

      3. associate-*l* 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in b around inf 93.5%

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

      1. associate-*r/ 93.5%

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

      2. distribute-rgt-out 93.5%

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

      3. pow-prod-down 93.5%

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

      4. metadata-eval 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. *-commutative 93.5%

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

      2. associate-*l* 93.5%

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

      3. metadata-eval 93.5%

         \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot \color{blue}{-1.0546875}}{a \cdot {b}^{6}}\right)\right)}{b} \]

   9. Simplified 93.5%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{{\left(a \cdot c\right)}^{4} \cdot -1.0546875}{a \cdot {b}^{6}}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -4.4)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 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)))
        (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -4.4) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (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))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -4.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / 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(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -4.4], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(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[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $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)) < -4.4000000000000004

   1. Initial program 87.8%

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

   3. Simplified 88.0%

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

   if -4.4000000000000004 < (/.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 48.5%

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

      1. sqr-neg 48.5%

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

      2. sqr-neg 48.5%

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

      3. associate-*l* 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in a around 0 93.4%

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

   6. Taylor expanded in c around 0 93.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)}{3 \cdot a}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (cbrt
    (pow
     (/ (fma b (sqrt (fma -3.0 (* (* a c) (pow b -2.0)) 1.0)) (- b)) (* 3.0 a))
     3.0))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = cbrt(pow((fma(b, sqrt(fma(-3.0, ((a * c) * pow(b, -2.0)), 1.0)), -b) / (3.0 * a)), 3.0));
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = cbrt((Float64(fma(b, sqrt(fma(-3.0, Float64(Float64(a * c) * (b ^ -2.0)), 1.0)), Float64(-b)) / Float64(3.0 * a)) ^ 3.0));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[Power[N[Power[N[(N[(b * N[Sqrt[N[(-3.0 * N[(N[(a * c), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. add-cbrt-cube 84.9%

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

       2. pow3 85.0%

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

       3. sqrt-pow1 85.0%

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

       4. metadata-eval 85.0%

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

       5. pow1 85.0%

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

       6. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in b around inf 92.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)}{3 \cdot a}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (/
    (log1p
     (expm1 (fma b (sqrt (fma -3.0 (* (* a c) (pow b -2.0)) 1.0)) (- b))))
    (* 3.0 a))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = log1p(expm1(fma(b, sqrt(fma(-3.0, ((a * c) * pow(b, -2.0)), 1.0)), -b))) / (3.0 * a);
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(log1p(expm1(fma(b, sqrt(fma(-3.0, Float64(Float64(a * c) * (b ^ -2.0)), 1.0)), Float64(-b)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(N[Log[1 + N[(Exp[N[(b * N[Sqrt[N[(-3.0 * N[(N[(a * c), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. log1p-expm1-u 85.0%

          \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sqrt{{b}^{2}}, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}}{3 \cdot a} \]

       2. sqrt-pow1 85.0%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{{b}^{\left(\frac{2}{2}\right)}}, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}{3 \cdot a} \]

       3. metadata-eval 85.0%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left({b}^{\color{blue}{1}}, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}{3 \cdot a} \]

       4. pow1 85.0%

          \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\color{blue}{b}, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}{3 \cdot a} \]

   11. Applied egg-rr 85.0%

       \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}}{3 \cdot a} \]

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in b around inf 92.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)\right)\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (*
    (fma b (sqrt (fma -3.0 (* (* a c) (pow b -2.0)) 1.0)) (- b))
    (/ 1.0 (* 3.0 a)))
   (/
    (+
     (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
     (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = fma(b, sqrt(fma(-3.0, ((a * c) * pow(b, -2.0)), 1.0)), -b) * (1.0 / (3.0 * a));
	} else {
		tmp = ((-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0))) + ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(fma(b, sqrt(fma(-3.0, Float64(Float64(a * c) * (b ^ -2.0)), 1.0)), Float64(-b)) * Float64(1.0 / Float64(3.0 * a)));
	else
		tmp = Float64(Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))) + Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(N[(b * N[Sqrt[N[(-3.0 * N[(N[(a * c), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. div-inv 85.0%

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

       2. sqrt-pow1 85.0%

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

       3. metadata-eval 85.0%

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

       4. pow1 85.0%

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

       5. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in b around inf 92.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (*
    (fma b (sqrt (fma -3.0 (* (* a c) (pow b -2.0)) 1.0)) (- b))
    (/ 1.0 (* 3.0 a)))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = fma(b, sqrt(fma(-3.0, ((a * c) * pow(b, -2.0)), 1.0)), -b) * (1.0 / (3.0 * a));
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(fma(b, sqrt(fma(-3.0, Float64(Float64(a * c) * (b ^ -2.0)), 1.0)), Float64(-b)) * Float64(1.0 / 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(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(N[(b * N[Sqrt[N[(-3.0 * N[(N[(a * c), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(3.0 * a), $MachinePrecision]), $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[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. div-inv 85.0%

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

       2. sqrt-pow1 85.0%

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

       3. metadata-eval 85.0%

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

       4. pow1 85.0%

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

       5. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in a around 0 92.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (*
    (fma b (sqrt (fma -3.0 (* (* a c) (pow b -2.0)) 1.0)) (- b))
    (/ 1.0 (* 3.0 a)))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
       (* -0.375 (/ a (pow b 3.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = fma(b, sqrt(fma(-3.0, ((a * c) * pow(b, -2.0)), 1.0)), -b) * (1.0 / (3.0 * a));
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(fma(b, sqrt(fma(-3.0, Float64(Float64(a * c) * (b ^ -2.0)), 1.0)), Float64(-b)) * Float64(1.0 / Float64(3.0 * a)));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(N[(b * N[Sqrt[N[(-3.0 * N[(N[(a * c), $MachinePrecision] * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $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 < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. div-inv 85.0%

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

       2. sqrt-pow1 85.0%

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

       3. metadata-eval 85.0%

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

       4. pow1 85.0%

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

       5. *-commutative 85.0%

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

   11. Applied egg-rr 85.0%

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

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in c around 0 92.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6)
   (/
    1.0
    (*
     a
     (/ 3.0 (* b (+ (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) -1.0)))))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
       (* -0.375 (/ a (pow b 3.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6) {
		tmp = 1.0 / (a * (3.0 / (b * (sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)) + -1.0))));
	} else {
		tmp = c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / Float64(b * Float64(sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)) + -1.0)))));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6], N[(1.0 / N[(a * N[(3.0 / N[(b * N[(N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $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 < 2.60000000000000009

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in b around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. fma-define 84.2%

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

   7. Simplified 84.2%

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

      1. +-commutative 84.2%

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

      2. sqrt-prod 84.1%

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

      3. fma-define 85.0%

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

      4. div-inv 84.9%

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

      5. pow-flip 85.0%

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

      6. metadata-eval 85.0%

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

   9. Applied egg-rr 85.0%

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

       1. clear-num 85.0%

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

       2. inv-pow 85.0%

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

       3. *-commutative 85.0%

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

       4. sqrt-pow1 85.0%

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

       5. metadata-eval 85.0%

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

       6. pow1 85.0%

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

   11. Applied egg-rr 85.0%

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

       1. unpow-1 85.0%

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

       2. associate-/l* 84.9%

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

       3. fma-define 84.0%

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

       4. neg-mul-1 84.0%

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

       5. *-commutative 84.0%

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

       6. distribute-lft-out 84.9%

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

       7. associate-*l* 84.9%

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

   13. Simplified 84.9%

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

   if 2.60000000000000009 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in c around 0 92.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.5)
   (/
    1.0
    (*
     a
     (/ 3.0 (* b (+ (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) -1.0)))))
   (/ (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5) {
		tmp = 1.0 / (a * (3.0 / (b * (sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)) + -1.0))));
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(1.0 / Float64(a * Float64(3.0 / Float64(b * Float64(sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)) + -1.0)))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.5], N[(1.0 / N[(a * N[(3.0 / N[(b * N[(N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-*l* 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in b around inf 81.4%

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

      1. +-commutative 81.4%

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

      2. fma-define 81.6%

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

   7. Simplified 81.6%

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

      1. +-commutative 81.6%

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

      2. sqrt-prod 81.5%

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

      3. fma-define 82.4%

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

      4. div-inv 82.4%

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

      5. pow-flip 82.4%

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

      6. metadata-eval 82.4%

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

   9. Applied egg-rr 82.4%

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

       1. clear-num 82.4%

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

       2. inv-pow 82.4%

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

       3. *-commutative 82.4%

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

       4. sqrt-pow1 82.4%

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

       5. metadata-eval 82.4%

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

       6. pow1 82.4%

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

   11. Applied egg-rr 82.4%

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

       1. unpow-1 82.4%

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

       2. associate-/l* 82.4%

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

       3. fma-define 81.4%

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

       4. neg-mul-1 81.4%

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

       5. *-commutative 81.4%

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

       6. distribute-lft-out 82.4%

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

       7. associate-*l* 82.4%

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

   13. Simplified 82.4%

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

   if 7.5 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{1}{a \cdot \frac{3}{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (*
    0.3333333333333333
    (/ (* b (+ (sqrt (fma -3.0 (* a (* c (pow b -2.0))) 1.0)) -1.0)) a))
   (/ (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = 0.3333333333333333 * ((b * (sqrt(fma(-3.0, (a * (c * pow(b, -2.0))), 1.0)) + -1.0)) / a);
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b * Float64(sqrt(fma(-3.0, Float64(a * Float64(c * (b ^ -2.0))), 1.0)) + -1.0)) / a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(0.3333333333333333 * N[(N[(b * N[(N[Sqrt[N[(-3.0 * N[(a * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-*l* 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in b around inf 81.4%

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

      1. +-commutative 81.4%

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

      2. fma-define 81.6%

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

   7. Simplified 81.6%

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

      1. +-commutative 81.6%

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

      2. sqrt-prod 81.5%

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

      3. fma-define 82.4%

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

      4. div-inv 82.4%

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

      5. pow-flip 82.4%

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

      6. metadata-eval 82.4%

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

   9. Applied egg-rr 82.4%

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

       1. *-un-lft-identity 82.4%

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

       2. sqrt-pow1 82.4%

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

       3. metadata-eval 82.4%

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

       4. pow1 82.4%

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

       5. *-commutative 82.4%

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

   11. Applied egg-rr 82.4%

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

       1. associate-*r/ 82.4%

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

       2. *-commutative 82.4%

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

       3. times-frac 82.4%

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

       4. metadata-eval 82.4%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)}, -b\right)}{a} \]

       5. fma-define 81.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{b \cdot \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)} + \left(-b\right)}}{a} \]

       6. neg-mul-1 81.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{b \cdot \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)} + \color{blue}{-1 \cdot b}}{a} \]

       7. *-commutative 81.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{b \cdot \sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)} + \color{blue}{b \cdot -1}}{a} \]

       8. distribute-lft-out 82.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, \left(a \cdot c\right) \cdot {b}^{-2}, 1\right)} + -1\right)}}{a} \]

       9. associate-*l* 82.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, \color{blue}{a \cdot \left(c \cdot {b}^{-2}\right)}, 1\right)} + -1\right)}{a} \]

   13. Simplified 82.4%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}{a}} \]

   if 7.79999999999999982 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b \cdot \left(\sqrt{\mathsf{fma}\left(-3, a \cdot \left(c \cdot {b}^{-2}\right), 1\right)} + -1\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.9)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.9) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.9)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.9], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.9000000000000004

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if 7.9000000000000004 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in a around 0 88.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ (+ (* c -0.5) (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = ((c * -0.5) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if 7.79999999999999982 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 81.7%

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

   3. Simplified 81.8%

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

   if 7.79999999999999982 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

   6. Taylor expanded in c around 0 88.6%

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

      1. associate-/l* 88.6%

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

      2. associate-*r/ 88.6%

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

      3. metadata-eval 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.5)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.5d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.5:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.5)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.5], 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[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-*l* 81.7%

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

   3. Simplified 81.7%

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

   if 7.5 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

   6. Taylor expanded in c around 0 88.6%

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

      1. associate-/l* 88.6%

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

      2. associate-*r/ 88.6%

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

      3. metadata-eval 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.8d0) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.8:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.8)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], 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], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 81.7%

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

   if 7.79999999999999982 < b

   1. Initial program 44.0%

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

      1. sqr-neg 44.0%

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

      2. sqr-neg 44.0%

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

      3. associate-*l* 44.0%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in b around inf 88.8%

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

   6. Taylor expanded in c around 0 88.6%

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

      1. associate-/l* 88.6%

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

      2. associate-*r/ 88.6%

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

      3. metadata-eval 88.6%

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

   8. Simplified 88.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.6% accurate, 1.0× speedup

Math

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

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(-0.375 * Float64(a * Float64(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[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

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

   1. sqr-neg 51.7%

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

   2. sqr-neg 51.7%

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

   3. associate-*l* 51.7%

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

3. Simplified 51.7%

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

5. Taylor expanded in b around inf 82.9%

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

6. Taylor expanded in c around 0 82.7%

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

   1. associate-/l* 82.7%

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

   2. associate-*r/ 82.7%

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

   3. metadata-eval 82.7%

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

8. Simplified 82.7%

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

9. Final simplification 82.7%

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

Alternative 17: 64.8% 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 51.7%

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

   1. sqr-neg 51.7%

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

   2. sqr-neg 51.7%

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

   3. associate-*l* 51.7%

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

3. Simplified 51.7%

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

5. Taylor expanded in b around inf 67.1%

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

   1. associate-*r/ 67.1%

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

   2. *-commutative 67.1%

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

7. Simplified 67.1%

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

8. Final simplification 67.1%

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

Reproduce

herbie shell --seed 2024074 
(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)))