Cubic critical, medium range

Percentage Accurate: 30.7% → 95.7%

Time: 14.5s

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: 30.7% 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: 95.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in b around inf

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

5. Simplified 94.2%

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

7. Applied egg-rr 94.2%

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

9. Applied egg-rr 94.2%

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

10. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

12. Simplified 94.2%

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

13. Final simplification 94.2%

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

Alternative 2: 94.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in b around inf

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

5. Simplified 94.2%

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

7. Applied egg-rr 94.2%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

10. Simplified 92.2%

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

11. Final simplification 92.2%

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

Alternative 3: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in b around inf

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

5. Simplified 94.2%

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

7. Applied egg-rr 94.2%

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

8. Taylor expanded in b around inf

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

10. Simplified 92.1%

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

11. Final simplification 92.1%

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

Alternative 4: 91.1% 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 33.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
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 88.6%

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

Alternative 5: 91.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.7%

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

3. Taylor expanded in b around inf

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

5. Simplified 94.2%

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

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. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 88.5

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

8. Simplified 88.5%

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

Alternative 6: 81.8% 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 33.7%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 79.3

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

5. Simplified 79.3%

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

Reproduce

herbie shell --seed 2024211 
(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)))