Cubic critical

Percentage Accurate: 52.6% → 85.4%

Time: 9.6s

Alternatives: 9

Speedup: 2.2×

Specification

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: 52.6% 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: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{-0.5625}{{b}^{5}}, \frac{-0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 8.2e-57)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (*
      c
      (fma
       c
       (* a (fma a (* c (/ -0.5625 (pow b 5.0))) (/ -0.375 (* b (* b b)))))
       (/ -0.5 b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 8.2e-57) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = c * fma(c, (a * fma(a, (c * (-0.5625 / pow(b, 5.0))), (-0.375 / (b * (b * b))))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 8.2e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(c * fma(c, Float64(a * fma(a, Float64(c * Float64(-0.5625 / (b ^ 5.0))), Float64(-0.375 / Float64(b * Float64(b * b))))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.2e-57], 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[(c * N[(c * N[(a * N[(a * N[(c * N[(-0.5625 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

      2. lower-*.f64 92.6

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 92.6%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -1e153 < b < 8.2000000000000003e-57

   1. Initial program 84.7%

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

   if 8.2000000000000003e-57 < b

   1. Initial program 13.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 13.7

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

   5. Simplified 13.7%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{\color{blue}{\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot c}}{{b}^{5}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot {a}^{2}}{{b}^{5}} \cdot c}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   8. Simplified 78.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto c \cdot \mathsf{fma}\left(c, \color{blue}{a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{5}} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{\frac{-1}{2}}{b}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, \color{blue}{a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{5}} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{\frac{-1}{2}}{b}\right) \]

       2. sub-neg N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{5}} + \left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)}, \frac{\frac{-1}{2}}{b}\right) \]

       3. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{5}} \cdot \frac{-9}{16}} + \left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       4. associate-/l* N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \left(\color{blue}{\left(a \cdot \frac{c}{{b}^{5}}\right)} \cdot \frac{-9}{16} + \left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       5. associate-*l* N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \left(\color{blue}{a \cdot \left(\frac{c}{{b}^{5}} \cdot \frac{-9}{16}\right)} + \left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       6. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \left(a \cdot \color{blue}{\left(\frac{-9}{16} \cdot \frac{c}{{b}^{5}}\right)} + \left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{-9}{16} \cdot \frac{c}{{b}^{5}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}, \frac{\frac{-1}{2}}{b}\right) \]

       8. *-commutative N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, \color{blue}{\frac{c}{{b}^{5}} \cdot \frac{-9}{16}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       9. associate-*l/ N/A

          \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, \color{blue}{\frac{c \cdot \frac{-9}{16}}{{b}^{5}}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       10. associate-/l* N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, \color{blue}{c \cdot \frac{\frac{-9}{16}}{{b}^{5}}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, \color{blue}{c \cdot \frac{\frac{-9}{16}}{{b}^{5}}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       12. lower-/.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \color{blue}{\frac{\frac{-9}{16}}{{b}^{5}}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       13. lower-pow.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{\color{blue}{{b}^{5}}}, \mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       14. associate-*r/ N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \mathsf{neg}\left(\color{blue}{\frac{\frac{3}{8} \cdot 1}{{b}^{3}}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       15. metadata-eval N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \mathsf{neg}\left(\frac{\color{blue}{\frac{3}{8}}}{{b}^{3}}\right)\right), \frac{\frac{-1}{2}}{b}\right) \]

       16. distribute-neg-frac N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \color{blue}{\frac{\mathsf{neg}\left(\frac{3}{8}\right)}{{b}^{3}}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       17. metadata-eval N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \frac{\color{blue}{\frac{-3}{8}}}{{b}^{3}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       18. lower-/.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \color{blue}{\frac{\frac{-3}{8}}{{b}^{3}}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       19. cube-mult N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \frac{\frac{-3}{8}}{\color{blue}{b \cdot \left(b \cdot b\right)}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       20. unpow2 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \frac{\frac{-3}{8}}{b \cdot \color{blue}{{b}^{2}}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \frac{\frac{-3}{8}}{\color{blue}{b \cdot {b}^{2}}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       22. unpow2 N/A

           \[\leadsto c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{\frac{-9}{16}}{{b}^{5}}, \frac{\frac{-3}{8}}{b \cdot \color{blue}{\left(b \cdot b\right)}}\right), \frac{\frac{-1}{2}}{b}\right) \]

       23. lower-*.f64 85.6

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

   11. Simplified 85.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, a \cdot \mathsf{fma}\left(a, c \cdot \frac{-0.5625}{{b}^{5}}, \frac{-0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.25e-42)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* (/ c b) (fma -0.375 (* a (/ c (* b b))) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.25e-42) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * fma(-0.375, (a * (c / (b * b))), -0.5);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.25e-42)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.25e-42], 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[(N[(c / b), $MachinePrecision] * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

      2. lower-*.f64 92.6

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 92.6%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -1e153 < b < 2.25e-42

   1. Initial program 84.2%

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

   if 2.25e-42 < b

   1. Initial program 12.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 12.9

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

   5. Simplified 12.9%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{\color{blue}{\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot c}}{{b}^{5}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot {a}^{2}}{{b}^{5}} \cdot c}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   8. Simplified 78.9%

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

   9. Taylor expanded in c around 0

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

       1. sub-neg N/A

          \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       4. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       5. unpow3 N/A

          \[\leadsto \frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{\color{blue}{\left(b \cdot b\right) \cdot b}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       6. unpow2 N/A

          \[\leadsto \frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{\color{blue}{{b}^{2}} \cdot b} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       7. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right)\right) \cdot c \]

       9. metadata-eval N/A

          \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \cdot c \]

       10. distribute-neg-frac N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}} \cdot c \]

       11. metadata-eval N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \frac{\color{blue}{\frac{-1}{2}}}{b} \cdot c \]

       12. associate-*l/ N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

       13. associate-*r/ N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

       14. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\frac{c}{b} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right)} \]

       15. lower-*.f64 N/A

           \[\leadsto \color{blue}{\frac{c}{b} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right)} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{c}{b}} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right) \]

       17. associate-/l* N/A

           \[\leadsto \frac{c}{b} \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}} + \frac{-1}{2}\right) \]

   11. Simplified 85.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+153)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.25e-42)
     (/ (- (sqrt (- (* b b) (* a (* 3.0 c)))) b) (* 3.0 a))
     (* (/ c b) (fma -0.375 (* a (/ c (* b b))) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.25e-42) {
		tmp = (sqrt(((b * b) - (a * (3.0 * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * fma(-0.375, (a * (c / (b * b))), -0.5);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+153)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.25e-42)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(3.0 * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+153], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.25e-42], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

      2. lower-*.f64 92.6

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 92.6%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -1e153 < b < 2.25e-42

   1. Initial program 84.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 84.2

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

   5. Simplified 84.2%

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

   if 2.25e-42 < b

   1. Initial program 12.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 12.9

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

   5. Simplified 12.9%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \frac{\color{blue}{\left(\frac{-9}{16} \cdot {a}^{2}\right) \cdot c}}{{b}^{5}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\frac{\frac{-9}{16} \cdot {a}^{2}}{{b}^{5}} \cdot c}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \color{blue}{\left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

   8. Simplified 78.9%

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

   9. Taylor expanded in c around 0

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

       1. sub-neg N/A

          \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

       2. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       4. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       5. unpow3 N/A

          \[\leadsto \frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{\color{blue}{\left(b \cdot b\right) \cdot b}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       6. unpow2 N/A

          \[\leadsto \frac{\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right) \cdot c}{\color{blue}{{b}^{2}} \cdot b} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       7. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \cdot c \]

       8. associate-*r/ N/A

          \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}\right)\right) \cdot c \]

       9. metadata-eval N/A

          \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{2}}}{b}\right)\right) \cdot c \]

       10. distribute-neg-frac N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}} \cdot c \]

       11. metadata-eval N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \frac{\color{blue}{\frac{-1}{2}}}{b} \cdot c \]

       12. associate-*l/ N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}} \]

       13. associate-*r/ N/A

           \[\leadsto \frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} \cdot \frac{c}{b} + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

       14. distribute-rgt-out N/A

           \[\leadsto \color{blue}{\frac{c}{b} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right)} \]

       15. lower-*.f64 N/A

           \[\leadsto \color{blue}{\frac{c}{b} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right)} \]

       16. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{c}{b}} \cdot \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} + \frac{-1}{2}\right) \]

       17. associate-/l* N/A

           \[\leadsto \frac{c}{b} \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}}} + \frac{-1}{2}\right) \]

   11. Simplified 85.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

      2. lower-*.f64 55.6

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 55.6%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 24.6

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

   5. Simplified 24.6%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.0

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

   8. Simplified 72.0%

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

Alternative 5: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;b \cdot \frac{0.6666666666666666}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (* b (/ 0.6666666666666666 (- a))) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (0.6666666666666666 / -a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = b * (0.6666666666666666d0 / -a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = b * (0.6666666666666666 / -a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = b * (0.6666666666666666 / -a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(b * Float64(0.6666666666666666 / Float64(-a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = b * (0.6666666666666666 / -a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(b * N[(0.6666666666666666 / (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 53.4

          \[\leadsto \mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \color{blue}{\left(-b\right)} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{-0.5}{b \cdot b}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{\frac{2}{3}}{a}} \cdot \left(\mathsf{neg}\left(b\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 55.5

         \[\leadsto \color{blue}{\frac{0.6666666666666666}{a}} \cdot \left(-b\right) \]

   8. Simplified 55.5%

      \[\leadsto \color{blue}{\frac{0.6666666666666666}{a}} \cdot \left(-b\right) \]

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 24.6

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

   5. Simplified 24.6%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.0

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

   8. Simplified 72.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;b \cdot \frac{0.6666666666666666}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 78.3

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

   5. Simplified 78.3%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 55.5

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   8. Simplified 55.5%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 24.6

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

   5. Simplified 24.6%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 72.0

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

   8. Simplified 72.0%

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

Alternative 7: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ (* b -0.6666666666666666) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 78.3

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

   5. Simplified 78.3%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot b}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{-2}{3}}}{a} \]

      4. lower-*.f64 55.5

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   8. Simplified 55.5%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.0

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

   5. Simplified 72.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 78.3%

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

   3. Taylor expanded in b around -inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{a} \cdot \frac{-2}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{a} \cdot \frac{-2}{3}} \]

      3. lower-/.f64 55.4

         \[\leadsto \color{blue}{\frac{b}{a}} \cdot -0.6666666666666666 \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if 5.0999999999999998e-272 < b

   1. Initial program 24.6%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.0

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

   5. Simplified 72.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

3. Taylor expanded in b around -inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{b}{a} \cdot \frac{-2}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{a} \cdot \frac{-2}{3}} \]

   3. lower-/.f64 29.1

      \[\leadsto \color{blue}{\frac{b}{a}} \cdot -0.6666666666666666 \]

5. Simplified 29.1%

   \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

6. Final simplification 29.1%

   \[\leadsto -0.6666666666666666 \cdot \frac{b}{a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))