Cubic critical, medium range

Percentage Accurate: 31.5% → 95.6%

Time: 15.4s

Alternatives: 6

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: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \frac{\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{t\_0}, a, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot t\_0\right)}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))))
   (/
    (fma
     (/ (* (* c (* c c)) (* a -0.5625)) t_0)
     a
     (fma
      c
      -0.5
      (fma
       a
       (/ (* c c) (* (* b b) -2.6666666666666665))
       (/
        (*
         (* a (* (* a a) -0.16666666666666666))
         (* c (* c (* c (* c 6.328125)))))
        (* b (* b t_0))))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	return fma((((c * (c * c)) * (a * -0.5625)) / t_0), a, fma(c, -0.5, fma(a, ((c * c) / ((b * b) * -2.6666666666666665)), (((a * ((a * a) * -0.16666666666666666)) * (c * (c * (c * (c * 6.328125))))) / (b * (b * t_0)))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	return Float64(fma(Float64(Float64(Float64(c * Float64(c * c)) * Float64(a * -0.5625)) / t_0), a, fma(c, -0.5, fma(a, Float64(Float64(c * c) / Float64(Float64(b * b) * -2.6666666666666665)), Float64(Float64(Float64(a * Float64(Float64(a * a) * -0.16666666666666666)) * Float64(c * Float64(c * Float64(c * Float64(c * 6.328125))))) / Float64(b * Float64(b * t_0)))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * -0.5625), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * a + N[(c * -0.5 + N[(a * N[(N[(c * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(N[(a * a), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * N[(c * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 96.9%

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

7. Applied rewrites 96.9%

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

9. Applied rewrites 96.9%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(-0.5625 \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, a, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)\right)\right)}{b} \]

10. Final simplification 96.9%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, a, \mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)\right)\right)}{b} \]
11. Add Preprocessing

Alternative 2: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{t\_0}, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot t\_0\right)}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))))
   (/
    (fma
     c
     -0.5
     (fma
      a
      (/ (* (* c (* c c)) (* a -0.5625)) t_0)
      (fma
       a
       (/ (* c c) (* (* b b) -2.6666666666666665))
       (/
        (*
         (* a (* (* a a) -0.16666666666666666))
         (* c (* c (* c (* c 6.328125)))))
        (* b (* b t_0))))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	return fma(c, -0.5, fma(a, (((c * (c * c)) * (a * -0.5625)) / t_0), fma(a, ((c * c) / ((b * b) * -2.6666666666666665)), (((a * ((a * a) * -0.16666666666666666)) * (c * (c * (c * (c * 6.328125))))) / (b * (b * t_0)))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	return Float64(fma(c, -0.5, fma(a, Float64(Float64(Float64(c * Float64(c * c)) * Float64(a * -0.5625)) / t_0), fma(a, Float64(Float64(c * c) / Float64(Float64(b * b) * -2.6666666666666665)), Float64(Float64(Float64(a * Float64(Float64(a * a) * -0.16666666666666666)) * Float64(c * Float64(c * Float64(c * Float64(c * 6.328125))))) / Float64(b * Float64(b * t_0)))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * -0.5 + N[(a * N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * -0.5625), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(N[(a * a), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(c * N[(c * N[(c * N[(c * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 96.9%

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

7. Applied rewrites 96.9%

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

9. Applied rewrites 96.9%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(-0.5625 \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)\right)\right)}{b} \]

10. Final simplification 96.9%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(a, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{\left(a \cdot \left(\left(a \cdot a\right) \cdot -0.16666666666666666\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(c \cdot 6.328125\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)\right)\right)}{b} \]
11. Add Preprocessing

Alternative 3: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -3, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   a
   (fma (* a -3.0) (* (/ c (* b (* b b))) -0.375) (/ 1.5 b))
   (* -2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma((a * -3.0), ((c / (b * (b * b))) * -0.375), (1.5 / b)), (-2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(Float64(a * -3.0), Float64(Float64(c / Float64(b * Float64(b * b))) * -0.375), Float64(1.5 / b)), Float64(-2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(N[(a * -3.0), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.0%

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

4. Applied rewrites 31.0%

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

6. Applied rewrites 31.0%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

9. Applied rewrites 95.3%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-3 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)}} \]

10. Final simplification 95.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -3, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{1.5}{b}\right), -2 \cdot \frac{b}{c}\right)} \]
11. Add Preprocessing

Alternative 4: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.0%

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

4. Applied rewrites 31.0%

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

6. Applied rewrites 31.0%

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

7. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 91.4

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

9. Applied rewrites 91.4%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}} \]
10. Add Preprocessing

Alternative 5: 90.9% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 96.9%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 91.1%

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

9. Final simplification 91.1%

   \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \]
10. 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 31.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 81.5

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

5. Applied rewrites 81.5%

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

Reproduce

herbie shell --seed 2024219 
(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)))