Cubic critical, narrow range

Percentage Accurate: 55.8% → 90.9%

Time: 14.3s

Alternatives: 12

Speedup: 2.9×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.8% 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: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, c \cdot \frac{t\_0}{b \cdot b}, \mathsf{fma}\left(0.5625, \frac{c \cdot c}{{b}^{5}}, \frac{-0.2222222222222222 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c}\right)\right), t\_0\right), \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c (* b (* b b))) -0.375)))
   (/
    1.0
    (fma
     a
     (fma
      a
      (*
       -3.0
       (fma
        a
        (fma
         -0.75
         (* c (/ t_0 (* b b)))
         (fma
          0.5625
          (/ (* c c) (pow b 5.0))
          (/
           (*
            -0.2222222222222222
            (* b (* (/ (pow c 4.0) (pow b 6.0)) 6.328125)))
           (* c c))))
        t_0))
      (/ 1.5 b))
     (/ (* b -2.0) c)))))

C

double code(double a, double b, double c) {
	double t_0 = (c / (b * (b * b))) * -0.375;
	return 1.0 / fma(a, fma(a, (-3.0 * fma(a, fma(-0.75, (c * (t_0 / (b * b))), fma(0.5625, ((c * c) / pow(b, 5.0)), ((-0.2222222222222222 * (b * ((pow(c, 4.0) / pow(b, 6.0)) * 6.328125))) / (c * c)))), t_0)), (1.5 / b)), ((b * -2.0) / c));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / Float64(b * Float64(b * b))) * -0.375)
	return Float64(1.0 / fma(a, fma(a, Float64(-3.0 * fma(a, fma(-0.75, Float64(c * Float64(t_0 / Float64(b * b))), fma(0.5625, Float64(Float64(c * c) / (b ^ 5.0)), Float64(Float64(-0.2222222222222222 * Float64(b * Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125))) / Float64(c * c)))), t_0)), Float64(1.5 / b)), Float64(Float64(b * -2.0) / c)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]}, N[(1.0 / N[(a * N[(a * N[(-3.0 * N[(a * N[(-0.75 * N[(c * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5625 * N[(N[(c * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.2222222222222222 * N[(b * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.6%

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

4. Applied rewrites 51.6%

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

6. Applied rewrites 51.6%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 93.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375}{b \cdot b}, \mathsf{fma}\left(0.5625, \frac{c \cdot c}{{b}^{5}}, \frac{-0.2222222222222222 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c}\right)\right), \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right), \frac{1.5}{b}\right), \frac{-2 \cdot b}{c}\right)}} \]

10. Final simplification 93.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, -3 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-0.75, c \cdot \frac{\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375}{b \cdot b}, \mathsf{fma}\left(0.5625, \frac{c \cdot c}{{b}^{5}}, \frac{-0.2222222222222222 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c}\right)\right), \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right), \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)} \]
11. Add Preprocessing

Alternative 2: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{a \cdot \left(a \cdot t\_0\right)}{t\_1}, -0.5625, c \cdot \mathsf{fma}\left(a, \left(t\_0 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{-1.0546875}{a \cdot \left(\left(b \cdot b\right) \cdot t\_1\right)}, \frac{a \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}\right)\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* b (* b (* b b)))))
   (/
    (fma
     c
     -0.5
     (fma
      (/ (* a (* a t_0)) t_1)
      -0.5625
      (*
       c
       (fma
        a
        (* (* t_0 (* a (* a a))) (/ -1.0546875 (* a (* (* b b) t_1))))
        (/ (* a c) (* (* b b) -2.6666666666666665))))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * (b * b));
	return fma(c, -0.5, fma(((a * (a * t_0)) / t_1), -0.5625, (c * fma(a, ((t_0 * (a * (a * a))) * (-1.0546875 / (a * ((b * b) * t_1)))), ((a * c) / ((b * b) * -2.6666666666666665)))))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * Float64(b * b)))
	return Float64(fma(c, -0.5, fma(Float64(Float64(a * Float64(a * t_0)) / t_1), -0.5625, Float64(c * fma(a, Float64(Float64(t_0 * Float64(a * Float64(a * a))) * Float64(-1.0546875 / Float64(a * Float64(Float64(b * b) * t_1)))), Float64(Float64(a * c) / Float64(Float64(b * b) * -2.6666666666666665)))))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * -0.5 + N[(N[(N[(a * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * -0.5625 + N[(c * N[(a * N[(N[(t$95$0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 55.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 90.6%

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

7. Applied rewrites 90.6%

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

9. Applied rewrites 90.6%

   \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \mathsf{fma}\left(c, \left(c \cdot a\right) \cdot \frac{-0.375}{b \cdot b}, \left(c \cdot \left(\left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \frac{-1.0546875}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right)\right)\right)}{b} \]

11. Applied rewrites 90.6%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(\frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, -0.5625, c \cdot \mathsf{fma}\left(a, \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{-1.0546875}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{a \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}\right)\right)\right)}{b} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))