Cubic critical, narrow range

Percentage Accurate: 55.5% → 90.7%

Time: 18.8s

Alternatives: 9

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{-1.0546875 \cdot \frac{{\left(c \cdot a\right)}^{4}}{a}}{{b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (/ a (/ (pow b 3.0) (* c c)))
    (/ (* -1.0546875 (/ (pow (* c a) 4.0) a)) (pow b 7.0))))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), fma(-0.5, (c / b), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), ((-1.0546875 * (pow((c * a), 4.0) / a)) / pow(b, 7.0)))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(Float64(-1.0546875 * Float64((Float64(c * a) ^ 4.0) / a)) / (b ^ 7.0)))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in b around inf 90.0%

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

   1. fma-def 90.0%

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

   2. *-commutative 90.0%

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

   3. unpow2 90.0%

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

   4. fma-def 90.0%

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

   5. fma-def 90.0%

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

4. Simplified 90.0%

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

5. Taylor expanded in c around 0 90.0%

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

7. Simplified 90.0%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}}\right)\right)\right) \]
8. Step-by-step derivation

   1. frac-times 90.0%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\frac{a}{6.328125} \cdot {b}^{7}}}\right)\right)\right) \]

   2. div-inv 90.0%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{\left(a \cdot \frac{1}{6.328125}\right)} \cdot {b}^{7}}\right)\right)\right) \]

   3. metadata-eval 90.0%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\left(a \cdot \color{blue}{0.1580246913580247}\right) \cdot {b}^{7}}\right)\right)\right) \]

9. Applied egg-rr 90.0%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\left(a \cdot 0.1580246913580247\right) \cdot {b}^{7}}}\right)\right)\right) \]
10. Step-by-step derivation

    1. associate-/r* 90.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{a \cdot 0.1580246913580247}}{{b}^{7}}}\right)\right)\right) \]

    2. *-commutative 90.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{\color{blue}{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}}{a \cdot 0.1580246913580247}}{{b}^{7}}\right)\right)\right) \]

    3. *-commutative 90.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{0.1580246913580247 \cdot a}}}{{b}^{7}}\right)\right)\right) \]

    4. times-frac 90.0%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\color{blue}{\frac{-0.16666666666666666}{0.1580246913580247} \cdot \frac{{\left(a \cdot c\right)}^{4}}{a}}}{{b}^{7}}\right)\right)\right) \]

    5. metadata-eval 90.0%

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

11. Simplified 90.0%

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

12. Final simplification 90.0%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{-1.0546875 \cdot \frac{{\left(c \cdot a\right)}^{4}}{a}}{{b}^{7}}\right)\right)\right) \]

Alternative 2: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{{t_1}^{3} - {b}^{3}}{t_0 + b \cdot \left(b + t_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5625 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))) (t_1 (sqrt t_0)))
   (if (<= b 0.16)
     (/ (/ (- (pow t_1 3.0) (pow b 3.0)) (+ t_0 (* b (+ b t_1)))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (fma
       -0.375
       (* (* c c) (/ a (pow b 3.0)))
       (* -0.5625 (* (* a a) (/ (pow c 3.0) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 0.16) {
		tmp = ((pow(t_1, 3.0) - pow(b, 3.0)) / (t_0 + (b * (b + t_1)))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, ((c * c) * (a / pow(b, 3.0))), (-0.5625 * ((a * a) * (pow(c, 3.0) / pow(b, 5.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 0.16)
		tmp = Float64(Float64(Float64((t_1 ^ 3.0) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + t_1)))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64(Float64(c * c) * Float64(a / (b ^ 3.0))), Float64(-0.5625 * Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 0.16], N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * N[(b + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.160000000000000003

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

      1. add-cbrt-cube 84.0%

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

      2. pow3 84.0%

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

      3. pow1/3 81.5%

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

      4. sqrt-pow2 81.4%

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

      5. metadata-eval 81.4%

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

   5. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. flip3-- 81.6%

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

      2. pow-pow 86.5%

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

      3. metadata-eval 86.5%

         \[\leadsto \frac{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}}\right)}^{3} - {b}^{3}}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + \left(b \cdot b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot b\right)}}{3 \cdot a} \]

      4. pow-pow 86.5%

         \[\leadsto \frac{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right)}^{3} - {b}^{3}}{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + \left(b \cdot b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot b\right)}}{3 \cdot a} \]

      5. metadata-eval 86.5%

         \[\leadsto \frac{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right)}^{3} - {b}^{3}}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + \left(b \cdot b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot b\right)}}{3 \cdot a} \]

      6. pow-pow 86.6%

         \[\leadsto \frac{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right)}^{3} - {b}^{3}}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + \left(b \cdot b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot b\right)}}{3 \cdot a} \]

      7. metadata-eval 86.6%

         \[\leadsto \frac{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right)}^{3} - {b}^{3}}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} + \left(b \cdot b + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} \cdot b\right)}}{3 \cdot a} \]

   7. Applied egg-rr 86.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5}\right)}^{3} - {b}^{3}}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} + \left(b \cdot b + {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot b\right)}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. unpow1/2 86.7%

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

      2. associate-*r* 86.7%

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

      3. *-commutative 86.7%

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

      4. rem-square-sqrt 0.0%

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

      5. unpow2 0.0%

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

      6. associate-*r* 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 86.7%

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

   9. Simplified 86.8%

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

   if 0.160000000000000003 < b

   1. Initial program 49.7%

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

   2. Taylor expanded in b around inf 89.1%

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

      1. fma-def 89.1%

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

      2. cube-prod 89.1%

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

      3. fma-def 89.2%

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

      4. associate-/l* 89.1%

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

      5. associate-/l* 89.1%

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

      6. unpow2 89.1%

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

      7. unpow2 89.1%

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

   4. Simplified 89.1%

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

   5. Taylor expanded in a around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. associate-+l+ 89.5%

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

      3. +-commutative 89.5%

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

      4. fma-def 89.5%

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

      5. +-commutative 89.5%

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

      6. fma-def 89.5%

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

      7. unpow2 89.5%

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

      8. associate-/l* 89.5%

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

      9. associate-/r/ 89.5%

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

      10. unpow2 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5625 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \]

Alternative 3: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))))
   (if (<= b 0.16)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5625
      (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
      (fma -0.5 (/ c b) (* -0.375 (/ a (/ (pow b 3.0) (* c c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double tmp;
	if (b <= 0.16) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5625, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), fma(-0.5, (c / b), (-0.375 * (a / (pow(b, 3.0) / (c * c))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (b <= 0.16)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5625, Float64(Float64((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(a / Float64((b ^ 3.0) / Float64(c * c))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.16], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.160000000000000003

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

      1. add-cbrt-cube 84.0%

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

      2. pow3 84.0%

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

      3. pow1/3 81.5%

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

      4. sqrt-pow2 81.4%

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

      5. metadata-eval 81.4%

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

   5. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. flip-- 81.4%

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

      2. pow-pow 82.8%

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

      3. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      4. pow-pow 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      5. metadata-eval 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      6. pow-pow 85.7%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      7. metadata-eval 85.7%

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

   7. Applied egg-rr 85.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} + b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. pow-sqr 86.4%

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

      2. metadata-eval 86.4%

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

      3. unpow1 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-commutative 86.4%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 86.5%

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

      11. +-commutative 86.5%

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

      12. unpow1/2 86.5%

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

      13. associate-*r* 86.4%

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

      14. *-commutative 86.4%

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

      15. rem-square-sqrt 0.0%

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

      16. unpow2 0.0%

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

      17. associate-*r* 0.0%

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

      18. unpow2 0.0%

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

      19. rem-square-sqrt 86.5%

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

   9. Simplified 86.5%

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

   if 0.160000000000000003 < b

   1. Initial program 49.7%

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

   2. Taylor expanded in b around inf 89.5%

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

      1. fma-def 89.5%

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

      2. *-commutative 89.5%

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

      3. unpow2 89.5%

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

      4. fma-def 89.5%

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

      5. associate-/l* 89.5%

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

      6. unpow2 89.5%

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

   4. Simplified 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \left(a \cdot a\right)}{{b}^{5}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \]

Alternative 4: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5625 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))))
   (if (<= b 0.16)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (fma
       -0.375
       (* (* c c) (/ a (pow b 3.0)))
       (* -0.5625 (* (* a a) (/ (pow c 3.0) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double tmp;
	if (b <= 0.16) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, ((c * c) * (a / pow(b, 3.0))), (-0.5625 * ((a * a) * (pow(c, 3.0) / pow(b, 5.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (b <= 0.16)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64(Float64(c * c) * Float64(a / (b ^ 3.0))), Float64(-0.5625 * Float64(Float64(a * a) * Float64((c ^ 3.0) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.16], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.160000000000000003

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

      1. add-cbrt-cube 84.0%

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

      2. pow3 84.0%

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

      3. pow1/3 81.5%

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

      4. sqrt-pow2 81.4%

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

      5. metadata-eval 81.4%

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

   5. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. flip-- 81.4%

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

      2. pow-pow 82.8%

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

      3. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      4. pow-pow 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      5. metadata-eval 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      6. pow-pow 85.7%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      7. metadata-eval 85.7%

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

   7. Applied egg-rr 85.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} + b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. pow-sqr 86.4%

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

      2. metadata-eval 86.4%

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

      3. unpow1 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-commutative 86.4%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 86.5%

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

      11. +-commutative 86.5%

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

      12. unpow1/2 86.5%

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

      13. associate-*r* 86.4%

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

      14. *-commutative 86.4%

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

      15. rem-square-sqrt 0.0%

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

      16. unpow2 0.0%

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

      17. associate-*r* 0.0%

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

      18. unpow2 0.0%

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

      19. rem-square-sqrt 86.5%

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

   9. Simplified 86.5%

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

   if 0.160000000000000003 < b

   1. Initial program 49.7%

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

   2. Taylor expanded in b around inf 89.1%

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

      1. fma-def 89.1%

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

      2. cube-prod 89.1%

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

      3. fma-def 89.2%

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

      4. associate-/l* 89.1%

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

      5. associate-/l* 89.1%

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

      6. unpow2 89.1%

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

      7. unpow2 89.1%

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

   4. Simplified 89.1%

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

   5. Taylor expanded in a around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. associate-+l+ 89.5%

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

      3. +-commutative 89.5%

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

      4. fma-def 89.5%

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

      5. +-commutative 89.5%

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

      6. fma-def 89.5%

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

      7. unpow2 89.5%

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

      8. associate-/l* 89.5%

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

      9. associate-/r/ 89.5%

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

      10. unpow2 89.5%

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

   7. Simplified 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5625 \cdot \left(\left(a \cdot a\right) \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right)\\ \end{array} \]

Alternative 5: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\\ \mathbf{if}\;b \leq 0.18:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -3.0)))))
   (if (<= b 0.18)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma (/ c b) -0.5 (/ (* -0.375 (* a (* c c))) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -3.0)));
	double tmp;
	if (b <= 0.18) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma((c / b), -0.5, ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (b <= 0.18)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c / b), -0.5, Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.18], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17999999999999999

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

      1. add-cbrt-cube 84.0%

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

      2. pow3 84.0%

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

      3. pow1/3 81.5%

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

      4. sqrt-pow2 81.4%

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

      5. metadata-eval 81.4%

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

   5. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
   6. Step-by-step derivation

      1. flip-- 81.4%

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

      2. pow-pow 82.8%

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

      3. metadata-eval 82.8%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} \cdot {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      4. pow-pow 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      5. metadata-eval 85.5%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{0.5}} - b \cdot b}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333} + b}}{3 \cdot a} \]

      6. pow-pow 85.7%

         \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{\color{blue}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b}}{3 \cdot a} \]

      7. metadata-eval 85.7%

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

   7. Applied egg-rr 85.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} \cdot {\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} - b \cdot b}{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{0.5} + b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. pow-sqr 86.4%

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

      2. metadata-eval 86.4%

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

      3. unpow1 86.4%

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

      4. associate-*r* 86.4%

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

      5. *-commutative 86.4%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. associate-*r* 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 86.5%

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

      11. +-commutative 86.5%

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

      12. unpow1/2 86.5%

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

      13. associate-*r* 86.4%

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

      14. *-commutative 86.4%

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

      15. rem-square-sqrt 0.0%

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

      16. unpow2 0.0%

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

      17. associate-*r* 0.0%

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

      18. unpow2 0.0%

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

      19. rem-square-sqrt 86.5%

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

   9. Simplified 86.5%

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

   if 0.17999999999999999 < b

   1. Initial program 49.7%

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

      1. neg-sub0 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-+l- 49.7%

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

      4. sub0-neg 49.7%

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

      5. neg-mul-1 49.7%

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

   3. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. add-log-exp 32.0%

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

   5. Applied egg-rr 32.0%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\log \left(e^{\frac{a}{0.3333333333333333}}\right)}} \]
   6. Step-by-step derivation

      1. add-log-exp 49.8%

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

      2. add-cbrt-cube 49.8%

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

      3. div-inv 49.8%

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

      4. metadata-eval 49.8%

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

      5. div-inv 49.8%

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

      6. metadata-eval 49.8%

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

      7. div-inv 49.8%

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

      8. metadata-eval 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in b around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. fma-def 84.7%

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

      3. associate-*r/ 84.7%

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

      4. unpow2 84.7%

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

   10. Simplified 84.7%

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

4. Final simplification 84.9%

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

Alternative 6: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -7.2e-7) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -7.2e-7) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    if (t_0 <= (-7.2d-7)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -7.2e-7) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -7.2e-7:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -7.2e-7)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -7.2e-7)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -7.2e-7], t$95$0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -7.19999999999999989e-7

   1. Initial program 74.1%

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

   if -7.19999999999999989e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.6%

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

   2. Taylor expanded in b around inf 85.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot \left(-0.375 \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.16)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma -0.5 (/ c b) (/ (* a (* -0.375 (* c c))) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.16) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), ((a * (-0.375 * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.16)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(a * Float64(-0.375 * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.16], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(a * N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.160000000000000003

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

   if 0.160000000000000003 < b

   1. Initial program 49.7%

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

   2. Taylor expanded in b around inf 84.7%

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

      1. fma-def 84.7%

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

      2. associate-*r/ 84.7%

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

      3. *-commutative 84.7%

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

      4. associate-*r* 84.7%

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

      5. unpow2 84.7%

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

   4. Simplified 84.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot \left(-0.375 \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.16)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (fma (/ c b) -0.5 (/ (* -0.375 (* a (* c c))) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.16) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma((c / b), -0.5, ((-0.375 * (a * (c * c))) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.16)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(Float64(c / b), -0.5, Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.16], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.160000000000000003

   1. Initial program 86.3%

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

      1. neg-sub0 86.3%

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

      2. sqr-neg 86.3%

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

      3. associate-+l- 86.3%

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

      4. sub0-neg 86.3%

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

   3. Simplified 86.4%

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

   if 0.160000000000000003 < b

   1. Initial program 49.7%

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

      1. neg-sub0 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-+l- 49.7%

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

      4. sub0-neg 49.7%

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

      5. neg-mul-1 49.7%

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

   3. Simplified 49.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. add-log-exp 32.0%

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

   5. Applied egg-rr 32.0%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\log \left(e^{\frac{a}{0.3333333333333333}}\right)}} \]
   6. Step-by-step derivation

      1. add-log-exp 49.8%

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

      2. add-cbrt-cube 49.8%

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

      3. div-inv 49.8%

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

      4. metadata-eval 49.8%

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

      5. div-inv 49.8%

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

      6. metadata-eval 49.8%

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

      7. div-inv 49.8%

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

      8. metadata-eval 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in b around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. fma-def 84.7%

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

      3. associate-*r/ 84.7%

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

      4. unpow2 84.7%

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

   10. Simplified 84.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.16:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b}, -0.5, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9: 64.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in b around inf 65.0%

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

3. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2023285 
(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)))