Cubic critical, narrow range

Percentage Accurate: 55.0% → 92.0%

Time: 18.0s

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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.09:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.09)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (pow (/ 0.3333333333333333 a) -1.0))
   (fma
    -0.5625
    (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
    (fma
     -0.5
     (/ c b)
     (fma
      -0.375
      (/ a (/ (pow b 3.0) (* c c)))
      (*
       -0.16666666666666666
       (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.09) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow((0.3333333333333333 / a), -1.0);
	} else {
		tmp = fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.09)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (Float64(0.3333333333333333 / a) ^ -1.0));
	else
		tmp = fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.09], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[(0.3333333333333333 / a), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.089999999999999997

   1. Initial program 88.5%

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

      1. neg-sub0 88.5%

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

      2. sqr-neg 88.5%

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

      3. associate-+l- 88.5%

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

      4. sub0-neg 88.5%

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

      5. neg-mul-1 88.5%

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

   3. Simplified 88.8%

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

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

   5. Applied egg-rr 88.8%

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

   if 0.089999999999999997 < b

   1. Initial program 53.2%

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

      1. sqr-neg 53.2%

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

      2. sqr-neg 53.2%

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

      3. associate-*l* 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in b around inf 94.0%

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

      1. fma-def 94.0%

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

      2. associate-/l* 94.0%

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

      3. unpow2 94.0%

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

      4. fma-def 94.0%

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

      5. fma-def 94.0%

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

   6. Simplified 94.0%

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

   7. Taylor expanded in c around 0 94.0%

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

      1. distribute-rgt-out 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. times-frac 94.0%

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

   9. Simplified 94.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.09:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.215)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (pow (/ 0.3333333333333333 a) -1.0))
   (fma
    -0.5625
    (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
    (fma -0.5 (/ c b) (* -0.375 (/ a (/ (pow b 3.0) (* c c))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.215) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow((0.3333333333333333 / a), -1.0);
	} else {
		tmp = fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), (-0.375 * (a / (pow(b, 3.0) / (c * c))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.215)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (Float64(0.3333333333333333 / a) ^ -1.0));
	else
		tmp = fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), 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_] := If[LessEqual[b, 0.215], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[(0.3333333333333333 / a), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $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 < 0.214999999999999997

   1. Initial program 87.6%

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

      1. neg-sub0 87.6%

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

      2. sqr-neg 87.6%

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

      3. associate-+l- 87.6%

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

      4. sub0-neg 87.6%

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

      5. neg-mul-1 87.6%

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

   3. Simplified 87.7%

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

      1. clear-num 87.7%

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

      2. inv-pow 87.7%

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

   5. Applied egg-rr 87.7%

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

   if 0.214999999999999997 < b

   1. Initial program 52.7%

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

      1. sqr-neg 52.7%

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

      2. sqr-neg 52.7%

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

      3. associate-*l* 52.7%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in b around inf 91.9%

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

      1. fma-def 91.9%

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

      2. associate-/l* 91.9%

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

      3. unpow2 91.9%

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

      4. fma-def 91.9%

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

      5. associate-/l* 91.9%

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

      6. unpow2 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (/
    (- (sqrt (fma b b (* a (* c -3.0)))) b)
    (pow (/ 0.3333333333333333 a) -1.0))
   (fma -0.5 (/ c b) (/ (* a -0.375) (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / pow((0.3333333333333333 / a), -1.0);
	} else {
		tmp = fma(-0.5, (c / b), ((a * -0.375) / (pow(b, 3.0) / (c * c))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / (Float64(0.3333333333333333 / a) ^ -1.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(a * -0.375) / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[(0.3333333333333333 / a), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

      1. clear-num 79.8%

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

      2. inv-pow 79.8%

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

   5. Applied egg-rr 79.8%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. fma-def 90.1%

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

      2. associate-/l* 90.1%

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

      3. associate-*r/ 90.1%

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

      4. unpow2 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\right)\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (fma -0.5 (/ c b) (/ (* a -0.375) (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = fma(-0.5, (c / b), ((a * -0.375) / (pow(b, 3.0) / (c * c))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(a * -0.375) / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. fma-def 90.1%

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

      2. associate-/l* 90.1%

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

      3. associate-*r/ 90.1%

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

      4. unpow2 90.1%

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

   6. Simplified 90.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}}\right)\\ \end{array} \]

Alternative 5: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (* 0.3333333333333333 (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) a))
   (/
    (+ (* -1.125 (* (* c c) (/ (* a a) (pow b 3.0)))) (* -1.5 (* c (/ a b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = 0.3333333333333333 * ((sqrt(fma(b, b, (a * (c * -3.0)))) - b) / a);
	} else {
		tmp = ((-1.125 * ((c * c) * ((a * a) / pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / a));
	else
		tmp = Float64(Float64(Float64(-1.125 * Float64(Float64(c * c) * Float64(Float64(a * a) / (b ^ 3.0)))) + Float64(-1.5 * Float64(c * Float64(a / b)))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.125 * N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

      1. div-inv 79.7%

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

      2. metadata-eval 79.7%

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

      3. *-commutative 79.7%

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

      4. expm1-log1p-u 79.8%

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

      5. expm1-udef 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. expm1-def 79.8%

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

      2. expm1-log1p-u 79.7%

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

      3. *-commutative 79.7%

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

      4. add-cube-cbrt 79.7%

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

      5. div-sub 79.1%

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

      6. add-cube-cbrt 75.0%

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

      7. add-cube-cbrt 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. div-sub 79.7%

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

      2. *-lft-identity 79.7%

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

      3. *-commutative 79.7%

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

      4. times-frac 79.7%

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

      5. metadata-eval 79.7%

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

   9. Simplified 79.7%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 89.7%

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

      1. fma-def 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 89.8%

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

      4. unpow2 89.8%

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

      5. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. fma-udef 89.7%

         \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}}}{3 \cdot a} \]

      2. associate-/r/ 89.7%

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

      3. associate-/r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 6: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
   (/
    (+ (* -1.125 (* (* c c) (/ (* a a) (pow b 3.0)))) (* -1.5 (* c (/ a b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = ((-1.125 * ((c * c) * ((a * a) / pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(Float64(-1.125 * Float64(Float64(c * c) * Float64(Float64(a * a) / (b ^ 3.0)))) + Float64(-1.5 * Float64(c * Float64(a / b)))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.125 * N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

      1. div-inv 79.7%

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

      2. metadata-eval 79.7%

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

      3. *-commutative 79.7%

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

      4. add-cube-cbrt 79.7%

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

      5. pow3 79.6%

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

   5. Applied egg-rr 79.6%

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

      1. rem-cube-cbrt 79.7%

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

      2. div-sub 78.6%

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

      3. *-commutative 78.6%

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

      4. *-commutative 78.6%

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

   7. Applied egg-rr 78.6%

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

      1. div-sub 79.7%

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

      2. *-lft-identity 79.7%

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

      3. associate-*l/ 79.7%

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

      4. *-commutative 79.7%

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

      5. associate-/r* 79.7%

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

      6. metadata-eval 79.7%

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

   9. Simplified 79.7%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 89.7%

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

      1. fma-def 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 89.8%

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

      4. unpow2 89.8%

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

      5. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. fma-udef 89.7%

         \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}}}{3 \cdot a} \]

      2. associate-/r/ 89.7%

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

      3. associate-/r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 7: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 60:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 60.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (/
    (+ (* -1.125 (* (* c c) (/ (* a a) (pow b 3.0)))) (* -1.5 (* c (/ a b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 60.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = ((-1.125 * ((c * c) * ((a * a) / pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 60.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = Float64(Float64(Float64(-1.125 * Float64(Float64(c * c) * Float64(Float64(a * a) / (b ^ 3.0)))) + Float64(-1.5 * Float64(c * Float64(a / b)))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 60.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.125 * N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 60

   1. Initial program 79.6%

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

      1. neg-sub0 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-+l- 79.6%

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

      4. sub0-neg 79.6%

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

      5. neg-mul-1 79.6%

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

   3. Simplified 79.8%

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

   if 60 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 89.7%

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

      1. fma-def 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 89.8%

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

      4. unpow2 89.8%

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

      5. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. fma-udef 89.7%

         \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}}}{3 \cdot a} \]

      2. associate-/r/ 89.7%

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

      3. associate-/r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 60:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 8: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + -1.125 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (/ (- (sqrt (- (* b b) (/ c (/ 0.3333333333333333 a)))) b) (* a 3.0))
   (*
    0.3333333333333333
    (/
     (+ (* -1.5 (* c (/ a b))) (* -1.125 (/ (* (* a a) (* c c)) (pow b 3.0))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} else {
		tmp = 0.3333333333333333 * (((-1.5 * (c * (a / b))) + (-1.125 * (((a * a) * (c * c)) / pow(b, 3.0)))) / a);
	}
	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 <= 59.0d0) then
        tmp = (sqrt(((b * b) - (c / (0.3333333333333333d0 / a)))) - b) / (a * 3.0d0)
    else
        tmp = 0.3333333333333333d0 * ((((-1.5d0) * (c * (a / b))) + ((-1.125d0) * (((a * a) * (c * c)) / (b ** 3.0d0)))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (Math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} else {
		tmp = 0.3333333333333333 * (((-1.5 * (c * (a / b))) + (-1.125 * (((a * a) * (c * c)) / Math.pow(b, 3.0)))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 59.0:
		tmp = (math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0)
	else:
		tmp = 0.3333333333333333 * (((-1.5 * (c * (a / b))) + (-1.125 * (((a * a) * (c * c)) / math.pow(b, 3.0)))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c / Float64(0.3333333333333333 / a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(Float64(-1.5 * Float64(c * Float64(a / b))) + Float64(-1.125 * Float64(Float64(Float64(a * a) * Float64(c * c)) / (b ^ 3.0)))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 59.0)
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	else
		tmp = 0.3333333333333333 * (((-1.5 * (c * (a / b))) + (-1.125 * (((a * a) * (c * c)) / (b ^ 3.0)))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c / N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. sqr-neg 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

      1. associate-*r* 79.6%

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

      2. *-commutative 79.6%

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

      3. *-commutative 79.6%

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

      4. metadata-eval 79.6%

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

      5. div-inv 79.6%

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

      6. clear-num 79.6%

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

      7. un-div-inv 79.6%

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

   5. Applied egg-rr 79.6%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 89.7%

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

      1. fma-def 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 89.8%

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

      4. unpow2 89.8%

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

      5. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. div-inv 89.7%

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

      2. associate-/r/ 89.7%

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

      3. associate-/r/ 89.7%

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

      4. *-commutative 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. associate-*r/ 89.8%

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

      2. *-commutative 89.8%

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

      3. *-commutative 89.8%

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

      4. times-frac 89.7%

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

      5. metadata-eval 89.7%

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

   10. Simplified 89.7%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1.5, c \cdot \frac{a}{b}, \frac{-1.125}{\frac{{b}^{3}}{{\left(a \cdot c\right)}^{2}}}\right)}{a}} \]
   11. Step-by-step derivation

       1. fma-udef 89.6%

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

       2. associate-/r/ 89.6%

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

   12. Applied egg-rr 89.6%

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

   13. Taylor expanded in b around 0 89.6%

       \[\leadsto 0.3333333333333333 \cdot \frac{-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + \color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{a} \]
   14. Step-by-step derivation

       1. *-commutative 89.6%

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

       2. unpow2 89.6%

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

       3. unpow2 89.6%

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

   15. Simplified 89.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{-1.5 \cdot \left(c \cdot \frac{a}{b}\right) + -1.125 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{3}}}{a}\\ \end{array} \]

Alternative 9: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 59.0)
   (/ (- (sqrt (- (* b b) (/ c (/ 0.3333333333333333 a)))) b) (* a 3.0))
   (/
    (+ (* -1.125 (* (* c c) (/ (* a a) (pow b 3.0)))) (* -1.5 (* c (/ a b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} else {
		tmp = ((-1.125 * ((c * c) * ((a * a) / pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 59.0d0) then
        tmp = (sqrt(((b * b) - (c / (0.3333333333333333d0 / a)))) - b) / (a * 3.0d0)
    else
        tmp = (((-1.125d0) * ((c * c) * ((a * a) / (b ** 3.0d0)))) + ((-1.5d0) * (c * (a / b)))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 59.0) {
		tmp = (Math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} else {
		tmp = ((-1.125 * ((c * c) * ((a * a) / Math.pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 59.0:
		tmp = (math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0)
	else:
		tmp = ((-1.125 * ((c * c) * ((a * a) / math.pow(b, 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 59.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c / Float64(0.3333333333333333 / a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-1.125 * Float64(Float64(c * c) * Float64(Float64(a * a) / (b ^ 3.0)))) + Float64(-1.5 * Float64(c * Float64(a / b)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 59.0)
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	else
		tmp = ((-1.125 * ((c * c) * ((a * a) / (b ^ 3.0)))) + (-1.5 * (c * (a / b)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 59.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c / N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.125 * N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 59

   1. Initial program 79.6%

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

      1. sqr-neg 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

      1. associate-*r* 79.6%

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

      2. *-commutative 79.6%

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

      3. *-commutative 79.6%

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

      4. metadata-eval 79.6%

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

      5. div-inv 79.6%

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

      6. clear-num 79.6%

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

      7. un-div-inv 79.6%

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

   5. Applied egg-rr 79.6%

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

   if 59 < b

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. associate-*l* 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 89.7%

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

      1. fma-def 89.8%

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

      2. associate-/l* 89.8%

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

      3. associate-/l* 89.8%

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

      4. unpow2 89.8%

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

      5. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. fma-udef 89.7%

         \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.125 \cdot \frac{a \cdot a}{\frac{{b}^{3}}{c \cdot c}}}}{3 \cdot a} \]

      2. associate-/r/ 89.7%

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

      3. associate-/r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 59:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \left(\left(c \cdot c\right) \cdot \frac{a \cdot a}{{b}^{3}}\right) + -1.5 \cdot \left(c \cdot \frac{a}{b}\right)}{a \cdot 3}\\ \end{array} \]

Alternative 10: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 800:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 800.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 800.0) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} 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) :: tmp
    if (b <= 800.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 800.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 800.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 800.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 800.0)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 800.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 800

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

   if 800 < b

   1. Initial program 45.3%

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

      1. sqr-neg 45.3%

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

      2. sqr-neg 45.3%

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

      3. associate-*l* 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in b around inf 74.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 800:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 840:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 840.0)
   (/ (- (sqrt (- (* b b) (/ c (/ 0.3333333333333333 a)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 840.0) {
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} 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) :: tmp
    if (b <= 840.0d0) then
        tmp = (sqrt(((b * b) - (c / (0.3333333333333333d0 / a)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 840.0) {
		tmp = (Math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 840.0:
		tmp = (math.sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 840.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c / Float64(0.3333333333333333 / a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 840.0)
		tmp = (sqrt(((b * b) - (c / (0.3333333333333333 / a)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 840.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c / N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 840

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

      3. *-commutative 75.8%

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

      4. metadata-eval 75.8%

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

      5. div-inv 75.8%

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

      6. clear-num 75.8%

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

      7. un-div-inv 75.8%

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

   5. Applied egg-rr 75.8%

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

   if 840 < b

   1. Initial program 45.3%

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

      1. sqr-neg 45.3%

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

      2. sqr-neg 45.3%

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

      3. associate-*l* 45.3%

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

   3. Simplified 45.3%

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

   4. Taylor expanded in b around inf 74.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 840:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \frac{c}{\frac{0.3333333333333333}{a}}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 64.7% 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 56.4%

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

   1. sqr-neg 56.4%

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

   2. sqr-neg 56.4%

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

   3. associate-*l* 56.4%

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

3. Simplified 56.4%

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

4. Taylor expanded in b around inf 64.3%

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

5. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2023293 
(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)))