Cubic critical, narrow range

Percentage Accurate: 55.7% → 92.0%

Time: 16.8s

Alternatives: 11

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: 55.7% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.0074:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= b 0.0074)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (+
      (* -0.5 (/ c b))
      (*
       a
       (+
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
        (*
         a
         (+
          (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))
          (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -3.0, pow(b, 2.0));
	double tmp;
	if (b <= 0.0074) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-0.5625 * (pow(c, 3.0) / pow(b, 5.0))) + (-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0)))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.0074)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0074], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0074000000000000003

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. flip-+ 89.4%

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

      3. add-sqr-sqrt 91.1%

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

      4. sub-neg 91.1%

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

      5. +-commutative 91.1%

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

      6. *-commutative 91.1%

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

      7. distribute-rgt-neg-in 91.1%

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

      8. fma-define 91.1%

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

      9. *-commutative 91.1%

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

      10. distribute-rgt-neg-in 91.1%

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

      11. metadata-eval 91.1%

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

      12. pow2 91.1%

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

      13. sqr-neg 91.1%

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

      14. pow2 91.1%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 1.6%

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

   4. Applied egg-rr 91.1%

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

      1. fma-define 91.1%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. fma-define 91.1%

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

      5. fma-define 91.1%

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

      6. associate-*r* 91.0%

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

      7. *-commutative 91.0%

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

      8. fma-define 91.0%

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

   6. Simplified 91.0%

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

   if 0.0074000000000000003 < 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. Add Preprocessing

   3. Taylor expanded in a around 0 92.0%

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

   4. Taylor expanded in c around 0 92.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0074:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + -1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.0085:\\ \;\;\;\;\frac{\frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= b 0.0085)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 3.0))
     (*
      c
      (+
       (*
        c
        (+
         (* -0.375 (/ a (pow b 3.0)))
         (*
          c
          (+
           (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
           (* -1.0546875 (/ (* c (pow a 3.0)) (pow b 7.0)))))))
       (* 0.5 (/ -1.0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -3.0, pow(b, 2.0));
	double tmp;
	if (b <= 0.0085) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 3.0);
	} else {
		tmp = c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + (-1.0546875 * ((c * pow(a, 3.0)) / pow(b, 7.0))))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.0085)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(c * Float64(Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0085], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0085000000000000006

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. flip-+ 89.4%

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

      3. add-sqr-sqrt 91.1%

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

      4. sub-neg 91.1%

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

      5. +-commutative 91.1%

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

      6. *-commutative 91.1%

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

      7. distribute-rgt-neg-in 91.1%

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

      8. fma-define 91.1%

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

      9. *-commutative 91.1%

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

      10. distribute-rgt-neg-in 91.1%

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

      11. metadata-eval 91.1%

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

      12. pow2 91.1%

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

      13. sqr-neg 91.1%

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

      14. pow2 91.1%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 1.6%

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

   4. Applied egg-rr 91.1%

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

      1. fma-define 91.1%

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

      2. associate-*r* 91.1%

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

      3. *-commutative 91.1%

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

      4. fma-define 91.1%

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

      5. fma-define 91.1%

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

      6. associate-*r* 91.0%

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

      7. *-commutative 91.0%

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

      8. fma-define 91.0%

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

   6. Simplified 91.0%

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

   if 0.0085000000000000006 < 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. Add Preprocessing

   3. Taylor expanded in c around 0 91.8%

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

   4. Taylor expanded in a around 0 91.8%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0085:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)\\ \mathbf{if}\;b \leq 0.0072:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \frac{t\_0 - {b}^{2}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (pow b 2.0))))
   (if (<= b 0.0072)
     (* (/ 0.3333333333333333 a) (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))))
     (*
      c
      (+
       (*
        c
        (+
         (* -0.375 (/ a (pow b 3.0)))
         (*
          c
          (+
           (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
           (* -1.0546875 (/ (* c (pow a 3.0)) (pow b 7.0)))))))
       (* 0.5 (/ -1.0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -3.0, pow(b, 2.0));
	double tmp;
	if (b <= 0.0072) {
		tmp = (0.3333333333333333 / a) * ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0)));
	} else {
		tmp = c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + (-1.0546875 * ((c * pow(a, 3.0)) / pow(b, 7.0))))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, (b ^ 2.0))
	tmp = 0.0
	if (b <= 0.0072)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(c * Float64(Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0072], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0071999999999999998

   1. Initial program 88.4%

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

      1. add-cbrt-cube 87.5%

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

      2. pow3 87.6%

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

      3. sub-neg 87.6%

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

      4. +-commutative 87.6%

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

      5. *-commutative 87.6%

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

      6. distribute-rgt-neg-in 87.6%

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

      7. fma-define 87.6%

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

      8. *-commutative 87.6%

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

      9. distribute-rgt-neg-in 87.6%

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

      10. metadata-eval 87.6%

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

      11. pow2 87.6%

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

   4. Applied egg-rr 87.6%

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

      1. div-inv 87.7%

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

      2. unpow3 87.6%

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

      3. add-cbrt-cube 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-/r* 88.4%

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

      3. metadata-eval 88.4%

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

      4. +-commutative 88.4%

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

      5. unsub-neg 88.4%

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

      6. fma-define 88.3%

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

      7. associate-*r* 88.2%

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

      8. *-commutative 88.2%

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

      9. fma-define 88.2%

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

   8. Simplified 88.2%

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

      1. flip-- 89.3%

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

      2. add-sqr-sqrt 90.8%

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

      3. unpow2 90.8%

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

   10. Applied egg-rr 90.8%

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

       1. *-commutative 90.8%

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

       2. +-commutative 90.8%

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

       3. *-commutative 90.8%

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

   12. Simplified 90.8%

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

   if 0.0071999999999999998 < 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. Add Preprocessing

   3. Taylor expanded in c around 0 91.8%

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

   4. Taylor expanded in a around 0 91.8%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0072:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \frac{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -3, {b}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0018:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0018)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (+
    (* -0.5 (/ c b))
    (*
     a
     (+
      (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
      (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0018) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0018)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0018], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0018

   1. Initial program 78.9%

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

      1. fma-neg 79.1%

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

      2. *-commutative 79.1%

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

      3. distribute-rgt-neg-in 79.1%

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

      4. *-commutative 79.1%

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

      5. distribute-rgt-neg-in 79.1%

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

      6. metadata-eval 79.1%

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

   4. Applied egg-rr 79.1%

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

   if -0.0018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.9%

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

   3. Taylor expanded in a around 0 93.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0018:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (*
      c
      (+
       (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
       (* -1.0546875 (/ (* c (pow a 3.0)) (pow b 7.0)))))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + (-1.0546875 * ((c * pow(a, 3.0)) / pow(b, 7.0))))))) + (0.5 * (-1.0 / 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 * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + (c * (((-0.5625d0) * ((a ** 2.0d0) / (b ** 5.0d0))) + ((-1.0546875d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (c * ((-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + (-1.0546875 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (c * ((-0.5625 * (math.pow(a, 2.0) / math.pow(b, 5.0))) + (-1.0546875 * ((c * math.pow(a, 3.0)) / math.pow(b, 7.0))))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(c * Float64(Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(-1.0546875 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (c * ((-0.5625 * ((a ^ 2.0) / (b ^ 5.0))) + (-1.0546875 * ((c * (a ^ 3.0)) / (b ^ 7.0))))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in c around 0 90.0%

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

4. Taylor expanded in a around 0 90.0%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

5. Final simplification 90.0%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \]
6. Add Preprocessing

Alternative 6: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0018:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0018)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (*
    c
    (+
     (*
      c
      (+
       (* -0.375 (/ a (pow b 3.0)))
       (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))))
     (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0018) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0018)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0018], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0018

   1. Initial program 78.9%

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

      1. fma-neg 79.1%

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

      2. *-commutative 79.1%

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

      3. distribute-rgt-neg-in 79.1%

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

      4. *-commutative 79.1%

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

      5. distribute-rgt-neg-in 79.1%

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

      6. metadata-eval 79.1%

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

   4. Applied egg-rr 79.1%

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

   if -0.0018 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.9%

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

   3. Taylor expanded in c around 0 93.3%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0018:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0005)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (+ (* -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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0005) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0005)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000001e-4

   1. Initial program 78.2%

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

      1. fma-neg 78.4%

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

      2. *-commutative 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. *-commutative 78.4%

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

      5. distribute-rgt-neg-in 78.4%

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

      6. metadata-eval 78.4%

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

   4. Applied egg-rr 78.4%

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

   if -5.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 43.5%

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

   3. Taylor expanded in a around 0 90.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.0005)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (* c (+ (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.0005) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.0005)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.0005], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000001e-4

   1. Initial program 78.2%

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

      1. fma-neg 78.4%

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

      2. *-commutative 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. *-commutative 78.4%

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

      5. distribute-rgt-neg-in 78.4%

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

      6. metadata-eval 78.4%

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

   4. Applied egg-rr 78.4%

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

   if -5.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 43.5%

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

   3. Taylor expanded in c around 0 90.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.0005)
     t_0
     (* c (+ (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.0005d0)) then
        tmp = t_0
    else
        tmp = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) + (0.5d0 * ((-1.0d0) / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = t_0;
	} else {
		tmp = c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) + (0.5 * (-1.0 / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -0.0005:
		tmp = t_0
	else:
		tmp = c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) + (0.5 * (-1.0 / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -0.0005)
		tmp = t_0;
	else
		tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) + (0.5 * (-1.0 / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0005], t$95$0, N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.0000000000000001e-4

   1. Initial program 78.2%

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

   if -5.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 43.5%

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

   3. Taylor expanded in c around 0 90.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (+ (* -0.375 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * ((a * c) / Math.pow(b, 3.0))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((-0.375 * ((a * c) / math.pow(b, 3.0))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * ((a * c) / (b ^ 3.0))) + (0.5 * (-1.0 / b)));
end

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in c around 0 80.7%

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

4. Final simplification 80.7%

   \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 11: 64.1% 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 55.7%

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

3. Taylor expanded in b around inf 64.1%

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

Reproduce

herbie shell --seed 2024095 
(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)))