Cubic critical, wide range

Percentage Accurate: 47.5% → 63.2%

Time: 29.6s

Alternatives: 12

Speedup: 23.2×

• could not determine a ground truth

  1. a = 3.075159361736904e+27
  2. b = 9083801978157616.0
  3. c = 2.1276173242779184e-11

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 47.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18} \lor \neg \left(3 \cdot a \leq 0.02\right):\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (* 3.0 a) 2e-24)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (if (or (<= (* 3.0 a) 2e-18) (not (<= (* 3.0 a) 0.02)))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (pow (* a c) 4.0) (/ 6.328125 (* a (pow b 7.0))))))))
     (log
      (exp
       (/
        (fma -1.125 (* (pow (* a c) 2.0) (pow b -3.0)) (* -1.5 (* a (/ c b))))
        (* 3.0 a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if ((3.0 * a) <= 2e-24) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if (((3.0 * a) <= 2e-18) || !((3.0 * a) <= 0.02)) {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * (pow((a * c), 4.0) * (6.328125 / (a * pow(b, 7.0)))))));
	} else {
		tmp = log(exp((fma(-1.125, (pow((a * c), 2.0) * pow(b, -3.0)), (-1.5 * (a * (c / b)))) / (3.0 * a))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(3.0 * a) <= 2e-24)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif ((Float64(3.0 * a) <= 2e-18) || !(Float64(3.0 * a) <= 0.02))
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / Float64(a * (b ^ 7.0))))))));
	else
		tmp = log(exp(Float64(fma(-1.125, Float64((Float64(a * c) ^ 2.0) * (b ^ -3.0)), Float64(-1.5 * Float64(a * Float64(c / b)))) / Float64(3.0 * a))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18], N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 0.02]], $MachinePrecision]], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[(-1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24

   1. Initial program 73.2%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 0.0200000000000000004 < (*.f64 3 a)

   1. Initial program 42.2%

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

   3. Taylor expanded in b around inf 77.0%

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

   4. Taylor expanded in c around 0 77.0%

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

      1. distribute-rgt-out 77.0%

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

      2. associate-*r* 77.0%

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

      3. *-commutative 77.0%

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

      4. metadata-eval 77.0%

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

      5. pow-sqr 77.0%

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

      6. metadata-eval 77.0%

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

      7. pow-sqr 77.0%

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

      8. unswap-sqr 77.0%

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

      9. unpow2 77.0%

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

      10. unpow2 77.0%

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

      11. swap-sqr 77.0%

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

      12. unpow2 77.0%

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

      13. unpow2 77.0%

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

      14. unpow2 77.0%

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

      15. swap-sqr 77.0%

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

      16. unpow2 77.0%

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

      17. pow-sqr 77.0%

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

      18. metadata-eval 77.0%

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

   6. Simplified 77.0%

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

   if 2.0000000000000001e-18 < (*.f64 3 a) < 0.0200000000000000004

   1. Initial program 63.8%

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

   3. Taylor expanded in b around inf 44.4%

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

      1. add-log-exp 69.0%

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

      2. +-commutative 69.0%

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

      3. fma-define 69.0%

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

      4. div-inv 69.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      5. pow-prod-down 69.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      6. pow-flip 69.0%

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

      7. metadata-eval 69.0%

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

      8. associate-/l* 69.0%

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

      9. *-commutative 69.0%

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

   5. Applied egg-rr 69.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18} \lor \neg \left(3 \cdot a \leq 0.02\right):\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{7}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18} \lor \neg \left(3 \cdot a \leq 0.02\right):\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (* 3.0 a) 2e-24)
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (if (or (<= (* 3.0 a) 2e-18) (not (<= (* 3.0 a) 0.02)))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))
     (log
      (exp
       (/
        (fma -1.125 (* (pow (* a c) 2.0) (pow b -3.0)) (* -1.5 (* a (/ c b))))
        (* 3.0 a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if ((3.0 * a) <= 2e-24) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if (((3.0 * a) <= 2e-18) || !((3.0 * a) <= 0.02)) {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	} else {
		tmp = log(exp((fma(-1.125, (pow((a * c), 2.0) * pow(b, -3.0)), (-1.5 * (a * (c / b)))) / (3.0 * a))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(3.0 * a) <= 2e-24)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif ((Float64(3.0 * a) <= 2e-18) || !(Float64(3.0 * a) <= 0.02))
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	else
		tmp = log(exp(Float64(fma(-1.125, Float64((Float64(a * c) ^ 2.0) * (b ^ -3.0)), Float64(-1.5 * Float64(a * Float64(c / b)))) / Float64(3.0 * a))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18], N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 0.02]], $MachinePrecision]], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[(-1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24

   1. Initial program 73.2%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 0.0200000000000000004 < (*.f64 3 a)

   1. Initial program 42.2%

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

   3. Taylor expanded in b around inf 76.1%

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

   if 2.0000000000000001e-18 < (*.f64 3 a) < 0.0200000000000000004

   1. Initial program 63.8%

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

   3. Taylor expanded in b around inf 44.4%

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

      1. add-log-exp 69.0%

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

      2. +-commutative 69.0%

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

      3. fma-define 69.0%

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

      4. div-inv 69.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      5. pow-prod-down 69.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      6. pow-flip 69.0%

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

      7. metadata-eval 69.0%

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

      8. associate-/l* 69.0%

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

      9. *-commutative 69.0%

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

   5. Applied egg-rr 69.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18} \lor \neg \left(3 \cdot a \leq 0.02\right):\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{2}\\ \mathbf{if}\;3 \cdot a \leq 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 0.02:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, t\_0 \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{t\_0}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 2.0)))
   (if (<= (* 3.0 a) 1e-24)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (if (<= (* 3.0 a) 0.02)
       (log
        (exp
         (/
          (fma -1.125 (* t_0 (pow b -3.0)) (* -1.5 (* a (/ c b))))
          (* 3.0 a))))
       (/
        (+
         (* -1.6875 (/ (pow (* a c) 3.0) (pow b 5.0)))
         (+ (* -1.5 (/ (* a c) b)) (* -1.125 (/ t_0 (pow b 3.0)))))
        (* 3.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 2.0);
	double tmp;
	if ((3.0 * a) <= 1e-24) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((3.0 * a) <= 0.02) {
		tmp = log(exp((fma(-1.125, (t_0 * pow(b, -3.0)), (-1.5 * (a * (c / b)))) / (3.0 * a))));
	} else {
		tmp = ((-1.6875 * (pow((a * c), 3.0) / pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * (t_0 / pow(b, 3.0))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 2.0
	tmp = 0.0
	if (Float64(3.0 * a) <= 1e-24)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif (Float64(3.0 * a) <= 0.02)
		tmp = log(exp(Float64(fma(-1.125, Float64(t_0 * (b ^ -3.0)), Float64(-1.5 * Float64(a * Float64(c / b)))) / Float64(3.0 * a))));
	else
		tmp = Float64(Float64(Float64(-1.6875 * Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0))) + Float64(Float64(-1.5 * Float64(Float64(a * c) / b)) + Float64(-1.125 * Float64(t_0 / (b ^ 3.0))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(3.0 * a), $MachinePrecision], 1e-24], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * a), $MachinePrecision], 0.02], N[Log[N[Exp[N[(N[(-1.125 * N[(t$95$0 * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[(N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(t$95$0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 3 a) < 9.99999999999999924e-25

   1. Initial program 72.4%

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

   if 9.99999999999999924e-25 < (*.f64 3 a) < 0.0200000000000000004

   1. Initial program 57.8%

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

   3. Taylor expanded in b around inf 51.4%

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

      1. add-log-exp 65.0%

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

      2. +-commutative 65.0%

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

      3. fma-define 65.0%

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

      4. div-inv 65.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      5. pow-prod-down 65.0%

         \[\leadsto \log \left(e^{\frac{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}{3 \cdot a}}\right) \]

      6. pow-flip 65.0%

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

      7. metadata-eval 65.0%

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

      8. associate-/l* 65.0%

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

      9. *-commutative 65.0%

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

   5. Applied egg-rr 65.0%

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

   if 0.0200000000000000004 < (*.f64 3 a)

   1. Initial program 41.8%

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

   3. Taylor expanded in b around inf 77.4%

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

      1. pow-prod-down 77.4%

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

   5. Applied egg-rr 77.4%

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

      1. pow1 77.4%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{1}}}{{b}^{3}}\right)}{3 \cdot a} \]

      2. pow-prod-down 77.4%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{1}}{{b}^{3}}\right)}{3 \cdot a} \]

   7. Applied egg-rr 77.4%

      \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left({\left(a \cdot c\right)}^{2}\right)}^{1}}}{{b}^{3}}\right)}{3 \cdot a} \]
   8. Step-by-step derivation

      1. unpow1 77.4%

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

   9. Simplified 77.4%

      \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}\right)}{3 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 0.02:\\ \;\;\;\;\log \left(e^{\frac{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{3 \cdot a}\\ t_1 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (* -1.6875 (/ (pow (* a c) 3.0) (pow b 5.0)))
           (+
            (* -1.5 (/ (* a c) b))
            (* -1.125 (/ (pow (* a c) 2.0) (pow b 3.0)))))
          (* 3.0 a)))
        (t_1 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= (* 3.0 a) 2e-24)
     t_1
     (if (<= (* 3.0 a) 2e-18)
       t_0
       (if (<= (* 3.0 a) 2e-14)
         (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
         (if (<= (* 3.0 a) 3e-10)
           (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))
           (if (<= (* 3.0 a) 2e-6) t_1 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = ((-1.6875 * (pow((a * c), 3.0) / pow(b, 5.0))) + ((-1.5 * ((a * c) / b)) + (-1.125 * (pow((a * c), 2.0) / pow(b, 3.0))))) / (3.0 * a);
	double t_1 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if ((3.0 * a) <= 2e-24) {
		tmp = t_1;
	} else if ((3.0 * a) <= 2e-18) {
		tmp = t_0;
	} else if ((3.0 * a) <= 2e-14) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else if ((3.0 * a) <= 3e-10) {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	} else if ((3.0 * a) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-1.6875 * Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0))) + Float64(Float64(-1.5 * Float64(Float64(a * c) / b)) + Float64(-1.125 * Float64((Float64(a * c) ^ 2.0) / (b ^ 3.0))))) / Float64(3.0 * a))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (Float64(3.0 * a) <= 2e-24)
		tmp = t_1;
	elseif (Float64(3.0 * a) <= 2e-18)
		tmp = t_0;
	elseif (Float64(3.0 * a) <= 2e-14)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	elseif (Float64(3.0 * a) <= 3e-10)
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	elseif (Float64(3.0 * a) <= 2e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(-1.6875 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], t$95$1, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18], t$95$0, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-14], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10], 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], If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 79.2%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 43.3%

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

   3. Taylor expanded in b around inf 73.6%

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

      1. pow-prod-down 73.6%

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

   5. Applied egg-rr 73.6%

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

      1. pow1 73.6%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{1}}}{{b}^{3}}\right)}{3 \cdot a} \]

      2. pow-prod-down 73.6%

         \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{1}}{{b}^{3}}\right)}{3 \cdot a} \]

   7. Applied egg-rr 73.6%

      \[\leadsto \frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{\color{blue}{{\left({\left(a \cdot c\right)}^{2}\right)}^{1}}}{{b}^{3}}\right)}{3 \cdot a} \]
   8. Step-by-step derivation

      1. unpow1 73.6%

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

   9. Simplified 73.6%

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

   if 2.0000000000000001e-18 < (*.f64 3 a) < 2e-14

   1. Initial program 71.3%

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

      1. /-rgt-identity 71.3%

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

      2. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   if 2e-14 < (*.f64 3 a) < 3e-10

   1. Initial program 35.8%

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

   3. Taylor expanded in b around inf 70.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.6875 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}} + \left(-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ t_1 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))
        (t_1 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= (* 3.0 a) 2e-24)
     t_1
     (if (<= (* 3.0 a) 2e-18)
       t_0
       (if (<= (* 3.0 a) 2e-14)
         (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
         (if (<= (* 3.0 a) 3e-10)
           t_0
           (if (<= (* 3.0 a) 2e-6)
             t_1
             (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	double t_1 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if ((3.0 * a) <= 2e-24) {
		tmp = t_1;
	} else if ((3.0 * a) <= 2e-18) {
		tmp = t_0;
	} else if ((3.0 * a) <= 2e-14) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else if ((3.0 * a) <= 3e-10) {
		tmp = t_0;
	} else if ((3.0 * a) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (Float64(3.0 * a) <= 2e-24)
		tmp = t_1;
	elseif (Float64(3.0 * a) <= 2e-18)
		tmp = t_0;
	elseif (Float64(3.0 * a) <= 2e-14)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	elseif (Float64(3.0 * a) <= 3e-10)
		tmp = t_0;
	elseif (Float64(3.0 * a) <= 2e-6)
		tmp = t_1;
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], t$95$1, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18], t$95$0, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-14], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10], t$95$0, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6], t$95$1, N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 79.2%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2e-14 < (*.f64 3 a) < 3e-10

   1. Initial program 41.4%

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

   3. Taylor expanded in b around inf 68.6%

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

   if 2.0000000000000001e-18 < (*.f64 3 a) < 2e-14

   1. Initial program 71.3%

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

      1. /-rgt-identity 71.3%

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

      2. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   if 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 43.2%

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

   3. Taylor expanded in b around inf 72.6%

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

      1. clear-num 72.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 72.5%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 72.5%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-define 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 72.5%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 72.6%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\right)}^{-1} \]

   5. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}^{-1}} \]

   6. Taylor expanded in a around 0 73.0%

      \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}}^{-1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ t_1 := \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0))
        (t_1 (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))))
   (if (<= (* 3.0 a) 2e-24)
     t_1
     (if (<= (* 3.0 a) 2e-18)
       t_0
       (if (<= (* 3.0 a) 2e-13)
         (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* 3.0 a))
         (if (or (<= (* 3.0 a) 3e-10) (not (<= (* 3.0 a) 2e-6))) t_0 t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	double t_1 = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	double tmp;
	if ((3.0 * a) <= 2e-24) {
		tmp = t_1;
	} else if ((3.0 * a) <= 2e-18) {
		tmp = t_0;
	} else if ((3.0 * a) <= 2e-13) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (3.0 * a);
	} else if (((3.0 * a) <= 3e-10) || !((3.0 * a) <= 2e-6)) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (Float64(3.0 * a) <= 2e-24)
		tmp = t_1;
	elseif (Float64(3.0 * a) <= 2e-18)
		tmp = t_0;
	elseif (Float64(3.0 * a) <= 2e-13)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(3.0 * a));
	elseif ((Float64(3.0 * a) <= 3e-10) || !(Float64(3.0 * a) <= 2e-6))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], t$95$1, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18], t$95$0, If[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-13], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10], N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6]], $MachinePrecision]], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 79.2%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2.0000000000000001e-13 < (*.f64 3 a) < 3e-10 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 42.6%

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

   3. Taylor expanded in b around inf 72.0%

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

      1. clear-num 71.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 71.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 71.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-define 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 72.0%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\right)}^{-1} \]

   5. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}^{-1}} \]

   6. Taylor expanded in a around 0 72.4%

      \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}}^{-1} \]

   if 2.0000000000000001e-18 < (*.f64 3 a) < 2.0000000000000001e-13

   1. Initial program 68.0%

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

      1. /-rgt-identity 68.0%

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

      2. metadata-eval 68.0%

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

   3. Simplified 68.1%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-18}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \mathbf{elif}\;3 \cdot a \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;3 \cdot a \leq 3 \cdot 10^{-10} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-13} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= (* 3.0 a) 2e-24)
         (and (not (<= (* 3.0 a) 2e-18))
              (or (<= (* 3.0 a) 2e-13)
                  (and (not (<= (* 3.0 a) 3e-10)) (<= (* 3.0 a) 2e-6)))))
   (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-13) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	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 (((3.0d0 * a) <= 2d-24) .or. (.not. ((3.0d0 * a) <= 2d-18)) .and. ((3.0d0 * a) <= 2d-13) .or. (.not. ((3.0d0 * a) <= 3d-10)) .and. ((3.0d0 * a) <= 2d-6)) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-13) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((3.0 * a) <= 2e-24) or (not ((3.0 * a) <= 2e-18) and (((3.0 * a) <= 2e-13) or (not ((3.0 * a) <= 3e-10) and ((3.0 * a) <= 2e-6)))):
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((Float64(3.0 * a) <= 2e-24) || (!(Float64(3.0 * a) <= 2e-18) && ((Float64(3.0 * a) <= 2e-13) || (!(Float64(3.0 * a) <= 3e-10) && (Float64(3.0 * a) <= 2e-6)))))
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((3.0 * a) <= 2e-24) || (~(((3.0 * a) <= 2e-18)) && (((3.0 * a) <= 2e-13) || (~(((3.0 * a) <= 3e-10)) && ((3.0 * a) <= 2e-6)))))
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18]], $MachinePrecision], Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-13], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10]], $MachinePrecision], LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6]]]]], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 2.0000000000000001e-18 < (*.f64 3 a) < 2.0000000000000001e-13 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 75.6%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2.0000000000000001e-13 < (*.f64 3 a) < 3e-10 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 42.6%

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

   3. Taylor expanded in b around inf 72.0%

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

      1. clear-num 71.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 71.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 71.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-define 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 72.0%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\right)}^{-1} \]

   5. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}^{-1}} \]

   6. Taylor expanded in a around 0 72.4%

      \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-13} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-13} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\left(b + \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= (* 3.0 a) 2e-24)
         (and (not (<= (* 3.0 a) 2e-18))
              (or (<= (* 3.0 a) 2e-13)
                  (and (not (<= (* 3.0 a) 3e-10)) (<= (* 3.0 a) 2e-6)))))
   (/ (- (+ b (/ (* (* a c) -1.5) b)) b) (* 3.0 a))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-13) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	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 (((3.0d0 * a) <= 2d-24) .or. (.not. ((3.0d0 * a) <= 2d-18)) .and. ((3.0d0 * a) <= 2d-13) .or. (.not. ((3.0d0 * a) <= 3d-10)) .and. ((3.0d0 * a) <= 2d-6)) then
        tmp = ((b + (((a * c) * (-1.5d0)) / b)) - b) / (3.0d0 * a)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-13) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((3.0 * a) <= 2e-24) or (not ((3.0 * a) <= 2e-18) and (((3.0 * a) <= 2e-13) or (not ((3.0 * a) <= 3e-10) and ((3.0 * a) <= 2e-6)))):
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((Float64(3.0 * a) <= 2e-24) || (!(Float64(3.0 * a) <= 2e-18) && ((Float64(3.0 * a) <= 2e-13) || (!(Float64(3.0 * a) <= 3e-10) && (Float64(3.0 * a) <= 2e-6)))))
		tmp = Float64(Float64(Float64(b + Float64(Float64(Float64(a * c) * -1.5) / b)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((3.0 * a) <= 2e-24) || (~(((3.0 * a) <= 2e-18)) && (((3.0 * a) <= 2e-13) || (~(((3.0 * a) <= 3e-10)) && ((3.0 * a) <= 2e-6)))))
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18]], $MachinePrecision], Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-13], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10]], $MachinePrecision], LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6]]]]], N[(N[(N[(b + N[(N[(N[(a * c), $MachinePrecision] * -1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 2.0000000000000001e-18 < (*.f64 3 a) < 2.0000000000000001e-13 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 75.6%

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

      1. /-rgt-identity 75.6%

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

      2. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in b around inf 73.5%

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

      1. associate-*r/ 73.5%

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

   7. Simplified 73.5%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2.0000000000000001e-13 < (*.f64 3 a) < 3e-10 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 42.6%

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

   3. Taylor expanded in b around inf 72.0%

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

      1. clear-num 71.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

      2. inv-pow 71.9%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

      3. *-commutative 71.9%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

      4. +-commutative 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

      5. fma-define 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

      6. div-inv 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      7. pow-prod-down 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 71.9%

         \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

      10. associate-/l* 72.0%

          \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\right)}^{-1} \]

   5. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}^{-1}} \]

   6. Taylor expanded in a around 0 72.4%

      \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}}^{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-13} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\left(b + \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-14} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\left(b + \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= (* 3.0 a) 2e-24)
         (and (not (<= (* 3.0 a) 2e-18))
              (or (<= (* 3.0 a) 2e-14)
                  (and (not (<= (* 3.0 a) 3e-10)) (<= (* 3.0 a) 2e-6)))))
   (/ (- (+ b (/ (* (* a c) -1.5) b)) b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-14) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((3.0d0 * a) <= 2d-24) .or. (.not. ((3.0d0 * a) <= 2d-18)) .and. ((3.0d0 * a) <= 2d-14) .or. (.not. ((3.0d0 * a) <= 3d-10)) .and. ((3.0d0 * a) <= 2d-6)) then
        tmp = ((b + (((a * c) * (-1.5d0)) / b)) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-14) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((3.0 * a) <= 2e-24) or (not ((3.0 * a) <= 2e-18) and (((3.0 * a) <= 2e-14) or (not ((3.0 * a) <= 3e-10) and ((3.0 * a) <= 2e-6)))):
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((Float64(3.0 * a) <= 2e-24) || (!(Float64(3.0 * a) <= 2e-18) && ((Float64(3.0 * a) <= 2e-14) || (!(Float64(3.0 * a) <= 3e-10) && (Float64(3.0 * a) <= 2e-6)))))
		tmp = Float64(Float64(Float64(b + Float64(Float64(Float64(a * c) * -1.5) / b)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((3.0 * a) <= 2e-24) || (~(((3.0 * a) <= 2e-18)) && (((3.0 * a) <= 2e-14) || (~(((3.0 * a) <= 3e-10)) && ((3.0 * a) <= 2e-6)))))
		tmp = ((b + (((a * c) * -1.5) / b)) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18]], $MachinePrecision], Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-14], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10]], $MachinePrecision], LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6]]]]], N[(N[(N[(b + N[(N[(N[(a * c), $MachinePrecision] * -1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 2.0000000000000001e-18 < (*.f64 3 a) < 2e-14 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 77.0%

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

      1. /-rgt-identity 77.0%

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

      2. metadata-eval 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in b around inf 74.7%

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

      1. associate-*r/ 74.7%

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

   7. Simplified 74.7%

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

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2e-14 < (*.f64 3 a) < 3e-10 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 42.8%

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

   3. Taylor expanded in b around inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.3%

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

   5. Simplified 66.3%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-14} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\left(b + \frac{\left(a \cdot c\right) \cdot -1.5}{b}\right) - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-14} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{b - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= (* 3.0 a) 2e-24)
         (and (not (<= (* 3.0 a) 2e-18))
              (or (<= (* 3.0 a) 2e-14)
                  (and (not (<= (* 3.0 a) 3e-10)) (<= (* 3.0 a) 2e-6)))))
   (/ (- b b) (* 3.0 a))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-14) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = (b - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((3.0d0 * a) <= 2d-24) .or. (.not. ((3.0d0 * a) <= 2d-18)) .and. ((3.0d0 * a) <= 2d-14) .or. (.not. ((3.0d0 * a) <= 3d-10)) .and. ((3.0d0 * a) <= 2d-6)) then
        tmp = (b - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((3.0 * a) <= 2e-24) || (!((3.0 * a) <= 2e-18) && (((3.0 * a) <= 2e-14) || (!((3.0 * a) <= 3e-10) && ((3.0 * a) <= 2e-6))))) {
		tmp = (b - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((3.0 * a) <= 2e-24) or (not ((3.0 * a) <= 2e-18) and (((3.0 * a) <= 2e-14) or (not ((3.0 * a) <= 3e-10) and ((3.0 * a) <= 2e-6)))):
		tmp = (b - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if ((Float64(3.0 * a) <= 2e-24) || (!(Float64(3.0 * a) <= 2e-18) && ((Float64(3.0 * a) <= 2e-14) || (!(Float64(3.0 * a) <= 3e-10) && (Float64(3.0 * a) <= 2e-6)))))
		tmp = Float64(Float64(b - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((3.0 * a) <= 2e-24) || (~(((3.0 * a) <= 2e-18)) && (((3.0 * a) <= 2e-14) || (~(((3.0 * a) <= 3e-10)) && ((3.0 * a) <= 2e-6)))))
		tmp = (b - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-24], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-18]], $MachinePrecision], Or[LessEqual[N[(3.0 * a), $MachinePrecision], 2e-14], And[N[Not[LessEqual[N[(3.0 * a), $MachinePrecision], 3e-10]], $MachinePrecision], LessEqual[N[(3.0 * a), $MachinePrecision], 2e-6]]]]], N[(N[(b - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 3 a) < 1.99999999999999985e-24 or 2.0000000000000001e-18 < (*.f64 3 a) < 2e-14 or 3e-10 < (*.f64 3 a) < 1.99999999999999991e-6

   1. Initial program 77.0%

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

      1. /-rgt-identity 77.0%

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

      2. metadata-eval 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in b around inf 69.6%

      \[\leadsto \frac{\color{blue}{b} - b}{3 \cdot a} \]

   if 1.99999999999999985e-24 < (*.f64 3 a) < 2.0000000000000001e-18 or 2e-14 < (*.f64 3 a) < 3e-10 or 1.99999999999999991e-6 < (*.f64 3 a)

   1. Initial program 42.8%

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

   3. Taylor expanded in b around inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.3%

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

   5. Simplified 66.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot a \leq 2 \cdot 10^{-24} \lor \neg \left(3 \cdot a \leq 2 \cdot 10^{-18}\right) \land \left(3 \cdot a \leq 2 \cdot 10^{-14} \lor \neg \left(3 \cdot a \leq 3 \cdot 10^{-10}\right) \land 3 \cdot a \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{b - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in b around inf 60.7%

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

   1. clear-num 60.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}} \]

   2. inv-pow 60.6%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1}} \]

   3. *-commutative 60.6%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}\right)}^{-1} \]

   4. +-commutative 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{a \cdot c}{b}}}\right)}^{-1} \]

   5. fma-define 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}}\right)}^{-1} \]

   6. div-inv 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   7. pow-prod-down 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, \color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   8. pow-flip 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   9. metadata-eval 60.6%

      \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{\color{blue}{-3}}, -1.5 \cdot \frac{a \cdot c}{b}\right)}\right)}^{-1} \]

   10. associate-/l* 60.7%

       \[\leadsto {\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\right)}^{-1} \]

5. Applied egg-rr 60.7%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1.125, {\left(a \cdot c\right)}^{2} \cdot {b}^{-3}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}^{-1}} \]

6. Taylor expanded in a around 0 55.9%

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

   1. metadata-eval 55.9%

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

   2. pow-prod-up 0.0%

      \[\leadsto \color{blue}{{\left(-2 \cdot \frac{b}{c}\right)}^{-0.5} \cdot {\left(-2 \cdot \frac{b}{c}\right)}^{-0.5}} \]

8. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{{\left(-2 \cdot \frac{b}{c}\right)}^{-0.5} \cdot {\left(-2 \cdot \frac{b}{c}\right)}^{-0.5}} \]
9. Step-by-step derivation

   1. pow-sqr 55.9%

      \[\leadsto \color{blue}{{\left(-2 \cdot \frac{b}{c}\right)}^{\left(2 \cdot -0.5\right)}} \]

   2. metadata-eval 55.9%

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

   3. unpow-1 55.9%

      \[\leadsto \color{blue}{\frac{1}{-2 \cdot \frac{b}{c}}} \]

   4. associate-/r* 55.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{-2}}{\frac{b}{c}}} \]

   5. metadata-eval 55.9%

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

   6. associate-/r/ 55.9%

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

10. Simplified 55.9%

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

11. Final simplification 55.9%

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

Alternative 12: 60.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in b around inf 56.1%

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

   1. *-commutative 56.1%

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

   2. associate-*l/ 56.1%

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

5. Simplified 56.1%

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

6. Final simplification 56.1%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))