Cubic critical, medium range

Percentage Accurate: 31.2% → 95.5%

Time: 14.8s

Alternatives: 8

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: 31.2% 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: 95.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 95.3%

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

8. Final simplification 95.3%

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

Alternative 2: 95.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Applied rewrites 95.1%

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

8. Final simplification 95.1%

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

Alternative 3: 94.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. 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. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 93.0%

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

10. Applied rewrites 93.2%

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

Alternative 4: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\mathsf{fma}\left(a, a \cdot \left(0.375 \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right) - \frac{0.5}{b}, \frac{b \cdot 0.6666666666666666}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -0.3333333333333333
  (fma
   a
   (- (* a (* 0.375 (/ c (* b (* b (- b)))))) (/ 0.5 b))
   (/ (* b 0.6666666666666666) c))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / fma(a, ((a * (0.375 * (c / (b * (b * -b))))) - (0.5 / b)), ((b * 0.6666666666666666) / c));
}

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / fma(a, Float64(Float64(a * Float64(0.375 * Float64(c / Float64(b * Float64(b * Float64(-b)))))) - Float64(0.5 / b)), Float64(Float64(b * 0.6666666666666666) / c)))
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(a * N[(N[(a * N[(0.375 * N[(c / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * 0.6666666666666666), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

4. Applied rewrites 36.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 36.4

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. metadata-eval N/A

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

   18. distribute-lft-neg-in N/A

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

   19. lower-fma.f64 N/A

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

6. Applied rewrites 36.4%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

9. Applied rewrites 93.2%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(a, \left(-a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot 0.375\right)\right) - \frac{0.5}{b}, \frac{0.6666666666666666 \cdot b}{c}\right)}} \]

10. Final simplification 93.2%

    \[\leadsto \frac{-0.3333333333333333}{\mathsf{fma}\left(a, a \cdot \left(0.375 \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right) - \frac{0.5}{b}, \frac{b \cdot 0.6666666666666666}{c}\right)} \]
11. Add Preprocessing

Alternative 5: 93.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. 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. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 93.0%

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

Alternative 6: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b \cdot 0.6666666666666666}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -0.3333333333333333 (fma -0.5 (/ a b) (/ (* b 0.6666666666666666) c))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / fma(-0.5, (a / b), ((b * 0.6666666666666666) / c));
}

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / fma(-0.5, Float64(a / b), Float64(Float64(b * 0.6666666666666666) / c)))
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(N[(b * 0.6666666666666666), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.4%

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

4. Applied rewrites 36.4%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 36.4

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. metadata-eval N/A

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

   18. distribute-lft-neg-in N/A

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

   19. lower-fma.f64 N/A

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

6. Applied rewrites 36.4%

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

7. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 89.3

      \[\leadsto \frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{\color{blue}{0.6666666666666666 \cdot b}}{c}\right)} \]

9. Applied rewrites 89.3%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{0.6666666666666666 \cdot b}{c}\right)}} \]

10. Final simplification 89.3%

    \[\leadsto \frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b \cdot 0.6666666666666666}{c}\right)} \]
11. Add Preprocessing

Alternative 7: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.4%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. 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. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

5. Applied rewrites 93.0%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 89.0%

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

Alternative 8: 81.4% 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 36.4%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 77.6

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

5. Applied rewrites 77.6%

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

Reproduce

herbie shell --seed 2024227 
(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)))