Cubic critical, medium range

Percentage Accurate: 31.5% → 95.4%

Time: 14.4s

Alternatives: 7

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.5% 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: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{b \cdot t\_0} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (fma
     (* a a)
     (* (/ (* c (* c c)) (* b t_0)) -0.5625)
     (fma
      c
      -0.5
      (fma
       -0.16666666666666666
       (/
        (* (* a a) (* (* a a) (* (* c c) (* (* c c) 6.328125))))
        (* a (* t_0 t_0)))
       (/ (* (* a (* c c)) -0.375) (* b b)))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma((a * a), (((c * (c * c)) / (b * t_0)) * -0.5625), fma(c, -0.5, fma(-0.16666666666666666, (((a * a) * ((a * a) * ((c * c) * ((c * c) * 6.328125)))) / (a * (t_0 * t_0))), (((a * (c * c)) * -0.375) / (b * b))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(fma(Float64(a * a), Float64(Float64(Float64(c * Float64(c * c)) / Float64(b * t_0)) * -0.5625), fma(c, -0.5, fma(-0.16666666666666666, Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * Float64(Float64(c * c) * Float64(Float64(c * c) * 6.328125)))) / Float64(a * Float64(t_0 * t_0))), Float64(Float64(Float64(a * Float64(c * c)) * -0.375) / Float64(b * b))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(c * -0.5 + N[(-0.16666666666666666 * N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{{a}^{4}} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left(\color{blue}{{c}^{4}} \cdot \frac{405}{64}\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \color{blue}{\left({c}^{4} \cdot \frac{405}{64}\right)}}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{a \cdot \color{blue}{{b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{\color{blue}{a \cdot {b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   7. lift-/.f64 94.8

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

7. Applied egg-rr 94.8%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{\color{blue}{\left(3 + 1\right)}}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   2. pow-plus N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{{b}^{3} \cdot b}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   3. cube-unmult N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   6. lower-*.f64 94.8

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot b}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

9. Applied egg-rr 94.8%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot b}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

10. Final simplification 94.8%

    \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]
11. Add Preprocessing

Alternative 2: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, a \cdot -0.5625, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (fma
   (/ (* c (* c c)) (pow b 5.0))
   (* a -0.5625)
   (/ (* (* c c) -0.375) (* b (* b b))))
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(a, fma(((c * (c * c)) / pow(b, 5.0)), (a * -0.5625), (((c * c) * -0.375) / (b * (b * b)))), (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(a, fma(Float64(Float64(c * Float64(c * c)) / (b ^ 5.0)), Float64(a * -0.5625), Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b)))), Float64(-0.5 * Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 30.0%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

5. Simplified 93.3%

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

6. Final simplification 93.3%

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

Alternative 3: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   (* a a)
   (* (/ (* c (* c c)) (* b (* b (* b b)))) -0.5625)
   (fma c -0.5 (/ (* -0.375 (* c (* a c))) (* b b))))
  b))

C

double code(double a, double b, double c) {
	return fma((a * a), (((c * (c * c)) / (b * (b * (b * b)))) * -0.5625), fma(c, -0.5, ((-0.375 * (c * (a * c))) / (b * b)))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(a * a), Float64(Float64(Float64(c * Float64(c * c)) / Float64(b * Float64(b * Float64(b * b)))) * -0.5625), fma(c, -0.5, Float64(Float64(-0.375 * Float64(c * Float64(a * c))) / Float64(b * b)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(c * -0.5 + N[(N[(-0.375 * N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{{a}^{4}} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left(\color{blue}{{c}^{4}} \cdot \frac{405}{64}\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \color{blue}{\left({c}^{4} \cdot \frac{405}{64}\right)}}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   4. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{a \cdot \color{blue}{{b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{{a}^{4} \cdot \left({c}^{4} \cdot \frac{405}{64}\right)}{\color{blue}{a \cdot {b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   7. lift-/.f64 94.8

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

7. Applied egg-rr 94.8%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{\color{blue}{\left(3 + 1\right)}}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   2. pow-plus N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{{b}^{3} \cdot b}} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   3. cube-unmult N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b} \cdot \frac{-9}{16}, \mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(\frac{-1}{6}, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{405}{64}\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \frac{-3}{8}}{b \cdot b}\right)\right)\right)}{b} \]

   6. lower-*.f64 94.8

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot b}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

9. Applied egg-rr 94.8%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{\color{blue}{\left(b \cdot \left(b \cdot b\right)\right) \cdot b}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 6.328125\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot a}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b} \]

10. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. *-commutative N/A

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

    5. unpow2 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 93.2

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

12. Simplified 93.2%

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

13. Final simplification 93.2%

    \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \frac{-0.375 \cdot \left(c \cdot \left(a \cdot c\right)\right)}{b \cdot b}\right)\right)}{b} \]
14. Add Preprocessing

Alternative 4: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 30.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Simplified 90.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 5: 90.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma -0.375 (* a (/ c (* b b))) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma(-0.375, (a * (c / (b * b))), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 30.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

5. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot a, \frac{c \cdot \left(c \cdot c\right)}{{b}^{4}} \cdot -0.5625, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{4} \cdot \left({c}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}\right)\right)\right)}{b}} \]

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 90.5

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

8. Simplified 90.5%

   \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}}{b} \]
9. Add Preprocessing

Alternative 6: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-*.f64 N/A

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

   2. lower-/.f64 82.2

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

5. Simplified 82.2%

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

Alternative 7: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 30.0%

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

4. Applied egg-rr 29.9%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-9, a \cdot \left(c \cdot \left(a \cdot c\right)\right), b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot \left(b \cdot b\right) - \sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot \left(b \cdot \left(b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot \left(b \cdot b\right)}}}{3 \cdot a} \]

6. Applied egg-rr 29.9%

   \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{a}}{\frac{1}{\mathsf{fma}\left(1, \sqrt{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), -9 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\mathsf{fma}\left(c, a \cdot 3, b \cdot b\right)}}, -b \cdot 1\right)}}} \]

8. Applied egg-rr 29.5%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \sqrt{\frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot b\right), \left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -9\right)\right)}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot 3\right)\right)}} + \frac{0.3333333333333333}{a} \cdot \left(-b\right)} \]

9. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{b}{a} + \frac{1}{3} \cdot \frac{b}{a}} \]
10. Step-by-step derivation

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

       \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

    3. mul0-rgt 3.2

       \[\leadsto \color{blue}{0} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))