Cubic critical, narrow range

Percentage Accurate: 54.9% → 91.3%

Time: 17.8s

Alternatives: 12

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)\\ t_1 := {b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}\\ \mathbf{if}\;b \leq 0.965:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \frac{1}{t_0} \cdot t_1}{\left(-b\right) - {\left(\frac{t_0}{t_1}\right)}^{-0.5}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* 3.0 (* a c)) (fma 3.0 (* a c) (pow b 2.0)) (pow b 4.0)))
        (t_1 (+ (pow b 6.0) (* -27.0 (pow (* a c) 3.0)))))
   (if (<= b 0.965)
     (/
      (/
       (- (pow (- b) 2.0) (* (/ 1.0 t_0) t_1))
       (- (- b) (pow (/ t_0 t_1) -0.5)))
      (* 3.0 a))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (/ (* (pow (* a c) 4.0) 6.328125) (* a (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((3.0 * (a * c)), fma(3.0, (a * c), pow(b, 2.0)), pow(b, 4.0));
	double t_1 = pow(b, 6.0) + (-27.0 * pow((a * c), 3.0));
	double tmp;
	if (b <= 0.965) {
		tmp = ((pow(-b, 2.0) - ((1.0 / t_0) * t_1)) / (-b - pow((t_0 / t_1), -0.5))) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) * 6.328125) / (a * pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(3.0 * Float64(a * c)), fma(3.0, Float64(a * c), (b ^ 2.0)), (b ^ 4.0))
	t_1 = Float64((b ^ 6.0) + Float64(-27.0 * (Float64(a * c) ^ 3.0)))
	tmp = 0.0
	if (b <= 0.965)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - Float64(Float64(1.0 / t_0) * t_1)) / Float64(Float64(-b) - (Float64(t_0 / t_1) ^ -0.5))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) * 6.328125) / Float64(a * (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] * N[(3.0 * N[(a * c), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 6.0], $MachinePrecision] + N[(-27.0 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.965], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Power[N[(t$95$0 / t$95$1), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.964999999999999969

   1. Initial program 84.6%

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

      1. flip3-- 83.8%

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

      2. clear-num 83.7%

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

      3. pow2 83.7%

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

      4. pow2 83.7%

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

      5. pow-prod-up 83.7%

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

      6. metadata-eval 83.7%

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

      7. distribute-rgt-out 83.7%

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

      8. associate-*l* 83.8%

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

      9. associate-*l* 83.8%

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

      10. pow2 83.8%

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

   3. Applied egg-rr 83.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. flip-+ 84.0%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}} \cdot \sqrt{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}{\left(-b\right) - \sqrt{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}}{3 \cdot a} \]

   5. Applied egg-rr 85.4%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)} \cdot \left({b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}}{3 \cdot a} \]

   if 0.964999999999999969 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in b around inf 94.0%

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

   3. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-in 94.0%

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

      2. associate-*r* 94.0%

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

      3. associate-*r* 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.965:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \frac{1}{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)} \cdot \left({b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 2: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.965:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.965)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+
     (* -0.5 (/ c b))
     (+
      (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
      (*
       -0.16666666666666666
       (/ (* (pow (* a c) 4.0) 6.328125) (* a (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.965) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) * 6.328125) / (a * pow(b, 7.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.965)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) * 6.328125) / Float64(a * (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.965], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.964999999999999969

   1. Initial program 84.6%

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

   3. Simplified 84.7%

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

   if 0.964999999999999969 < b

   1. Initial program 51.9%

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

   2. Taylor expanded in b around inf 94.0%

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

   3. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-in 94.0%

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

      2. associate-*r* 94.0%

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

      3. associate-*r* 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.965:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{2} \cdot 2.25\\ \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{-0.5}{b}, -0.16666666666666666 \cdot \left(\frac{t_0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot t_0, \left(a \cdot -3\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow (* a c) 2.0) 2.25)))
   (if (<= b 2.0)
     (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
     (fma
      c
      (/ -0.5 b)
      (*
       -0.16666666666666666
       (+
        (/ t_0 (* a (pow b 3.0)))
        (/
         (fma 1.5 (* (* a c) t_0) (* (* a -3.0) (* c 0.0)))
         (* a (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 2.0) * 2.25;
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = fma(c, (-0.5 / b), (-0.16666666666666666 * ((t_0 / (a * pow(b, 3.0))) + (fma(1.5, ((a * c) * t_0), ((a * -3.0) * (c * 0.0))) / (a * pow(b, 5.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((Float64(a * c) ^ 2.0) * 2.25)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = fma(c, Float64(-0.5 / b), Float64(-0.16666666666666666 * Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(fma(1.5, Float64(Float64(a * c) * t_0), Float64(Float64(a * -3.0) * Float64(c * 0.0))) / Float64(a * (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * 2.25), $MachinePrecision]}, If[LessEqual[b, 2.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(a * -3.0), $MachinePrecision] * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 84.5%

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

   3. Simplified 84.7%

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

   if 2 < b

   1. Initial program 51.4%

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

      1. flip3-- 51.1%

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

      2. clear-num 51.0%

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

      3. pow2 51.0%

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

      4. pow2 51.0%

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

      5. pow-prod-up 50.3%

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

      6. metadata-eval 50.3%

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

      7. distribute-rgt-out 50.4%

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

      8. associate-*l* 50.4%

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

      9. associate-*l* 50.4%

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

      10. pow2 50.4%

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

   3. Applied egg-rr 51.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]

   4. Taylor expanded in b around inf 92.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

   6. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25 + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25 + 0\right), \left(-3 \cdot a\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{-0.5}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right), \left(a \cdot -3\right) \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right)\right)\\ \end{array} \]

Alternative 4: 89.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 84.5%

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

   3. Simplified 84.7%

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

   if 2 < b

   1. Initial program 51.4%

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

   2. Taylor expanded in b around inf 92.5%

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

4. Final simplification 91.2%

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

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 34:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 34.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 34.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 34.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 34.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 34

   1. Initial program 80.6%

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

   3. Simplified 80.8%

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

   if 34 < b

   1. Initial program 48.6%

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

   2. Taylor expanded in b around inf 90.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 34:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 6: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 430.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 430.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 430.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 430.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 430

   1. Initial program 76.6%

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

      1. neg-sub0 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l- 76.6%

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

      4. sub0-neg 76.6%

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

   3. Simplified 76.8%

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

   if 430 < b

   1. Initial program 45.8%

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

   2. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 430.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 430.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 430.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 430.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 430

   1. Initial program 76.6%

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

   3. Simplified 76.9%

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

   if 430 < b

   1. Initial program 45.8%

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

   2. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 420:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 420.0)
   (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 420.0) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 420.0d0) then
        tmp = (sqrt(((b * b) - (a * (3.0d0 * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 420.0) {
		tmp = (Math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 420.0:
		tmp = (math.sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 420.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 420.0)
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 420.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 420

   1. Initial program 76.6%

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

   2. Taylor expanded in a around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l* 76.6%

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

   4. Simplified 76.6%

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

   if 420 < b

   1. Initial program 45.8%

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

   2. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 420:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 430.0)
   (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 430.0) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 430.0d0) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 430.0) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 430.0:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 430.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 430.0)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 430.0], 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], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 430

   1. Initial program 76.6%

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

   if 430 < b

   1. Initial program 45.8%

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

   2. Taylor expanded in b around inf 74.2%

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

      1. associate-*r/ 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 11.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.3333333333333333 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.3333333333333333d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 * (b / a);
}

Python

def code(a, b, c):
	return -0.3333333333333333 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

   1. flip3-- 56.7%

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

   2. clear-num 56.6%

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

   3. pow2 56.6%

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

   4. pow2 56.6%

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

   5. pow-prod-up 56.0%

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

   6. metadata-eval 56.0%

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

   7. distribute-rgt-out 56.1%

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

   8. associate-*l* 56.1%

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

   9. associate-*l* 56.1%

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

   10. pow2 56.1%

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

3. Applied egg-rr 56.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. expm1-log1p-u 45.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}{3 \cdot a}\right)\right)} \]

   2. expm1-udef 43.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + {b}^{2}\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}{3 \cdot a}\right)} - 1} \]

5. Applied egg-rr 43.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{a \cdot 3}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 45.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{a \cdot 3}\right)\right)} \]

   2. expm1-log1p 56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{a \cdot 3}} \]

   3. *-lft-identity 56.7%

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}}{a \cdot 3} \]

   4. *-commutative 56.7%

      \[\leadsto \frac{1 \cdot \mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{\color{blue}{3 \cdot a}} \]

   5. times-frac 56.7%

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{a}} \]

   6. metadata-eval 56.7%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(-1, b, {\left(\frac{\mathsf{fma}\left(3 \cdot \left(a \cdot c\right), \mathsf{fma}\left(3, a \cdot c, {b}^{2}\right), {b}^{4}\right)}{{b}^{6} + -27 \cdot {\left(a \cdot c\right)}^{3}}\right)}^{-0.5}\right)}{a} \]

7. Simplified 56.7%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(b, -1, {\left(\frac{\mathsf{fma}\left(\left(a \cdot 3\right) \cdot c, \mathsf{fma}\left(a \cdot 3, c, {b}^{2}\right), {b}^{4}\right)}{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, {b}^{6}\right)}\right)}^{-0.5}\right)}{a}} \]

8. Taylor expanded in b around inf 11.8%

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

9. Final simplification 11.8%

   \[\leadsto -0.3333333333333333 \cdot \frac{b}{a} \]

Alternative 11: 64.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

2. Taylor expanded in b around inf 63.9%

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

   1. associate-*r/ 63.9%

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

   2. associate-/l* 63.9%

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

4. Simplified 63.9%

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

   1. associate-/r/ 63.9%

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

6. Applied egg-rr 63.9%

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

7. Final simplification 63.9%

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

Alternative 12: 64.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

2. Taylor expanded in b around inf 63.9%

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

   1. associate-*r/ 63.9%

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

4. Simplified 63.9%

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

5. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2023306 
(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)))