Cubic critical, medium range

Percentage Accurate: 32.0% → 96.9%

Time: 27.4s

Alternatives: 11

Speedup: 23.2×

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: 32.0% 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: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\frac{a \cdot -4.9833984375}{{b}^{11}} - \frac{2.21484375}{c \cdot {b}^{9}}\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (*
   a
   (fma
    -0.375
    (/ (pow c 2.0) (pow b 3.0))
    (*
     a
     (fma
      -0.5625
      (/ (pow c 3.0) (pow b 5.0))
      (*
       a
       (fma
        -0.16666666666666666
        (/ (* (/ (pow c 4.0) (pow b 6.0)) 6.328125) b)
        (*
         a
         (*
          (pow c 6.0)
          (-
           (/ (* a -4.9833984375) (pow b 11.0))
           (/ 2.21484375 (* c (pow b 9.0))))))))))))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), (a * fma(-0.375, (pow(c, 2.0) / pow(b, 3.0)), (a * fma(-0.5625, (pow(c, 3.0) / pow(b, 5.0)), (a * fma(-0.16666666666666666, (((pow(c, 4.0) / pow(b, 6.0)) * 6.328125) / b), (a * (pow(c, 6.0) * (((a * -4.9833984375) / pow(b, 11.0)) - (2.21484375 / (c * pow(b, 9.0)))))))))))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), Float64(a * fma(-0.375, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(a * fma(-0.5625, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(a * fma(-0.16666666666666666, Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 6.328125) / b), Float64(a * Float64((c ^ 6.0) * Float64(Float64(Float64(a * -4.9833984375) / (b ^ 11.0)) - Float64(2.21484375 / Float64(c * (b ^ 9.0)))))))))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.16666666666666666 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[Power[c, 6.0], $MachinePrecision] * N[(N[(N[(a * -4.9833984375), $MachinePrecision] / N[Power[b, 11.0], $MachinePrecision]), $MachinePrecision] - N[(2.21484375 / N[(c * N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in a around 0 97.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.16666666666666666 \cdot \frac{a \cdot \left(1.125 \cdot \frac{{c}^{2} \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{c \cdot \left(1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + 2.84765625 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.16666666666666666 \cdot \frac{1.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{c}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

5. Simplified 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left(-0.16666666666666666 \cdot \left(\frac{a \cdot \mathsf{fma}\left(1.125, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{4}}, \mathsf{fma}\left(1.5, c \cdot \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \frac{2.84765625 \cdot {c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(1.5, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{2}}, 3.796875 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right)\right)\right)} \]

6. Taylor expanded in c around inf 97.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \color{blue}{\left({c}^{6} \cdot \left(-4.9833984375 \cdot \frac{a}{{b}^{11}} - 2.21484375 \cdot \frac{1}{{b}^{9} \cdot c}\right)\right)}\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 97.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\color{blue}{\frac{-4.9833984375 \cdot a}{{b}^{11}}} - 2.21484375 \cdot \frac{1}{{b}^{9} \cdot c}\right)\right)\right)\right)\right)\right) \]

   2. associate-*r/ 97.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\frac{-4.9833984375 \cdot a}{{b}^{11}} - \color{blue}{\frac{2.21484375 \cdot 1}{{b}^{9} \cdot c}}\right)\right)\right)\right)\right)\right) \]

   3. metadata-eval 97.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\frac{-4.9833984375 \cdot a}{{b}^{11}} - \frac{\color{blue}{2.21484375}}{{b}^{9} \cdot c}\right)\right)\right)\right)\right)\right) \]

   4. *-commutative 97.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\frac{-4.9833984375 \cdot a}{{b}^{11}} - \frac{2.21484375}{\color{blue}{c \cdot {b}^{9}}}\right)\right)\right)\right)\right)\right) \]

8. Simplified 97.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \color{blue}{\left({c}^{6} \cdot \left(\frac{-4.9833984375 \cdot a}{{b}^{11}} - \frac{2.21484375}{c \cdot {b}^{9}}\right)\right)}\right)\right)\right)\right) \]

9. Final simplification 97.1%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.375, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125}{b}, a \cdot \left({c}^{6} \cdot \left(\frac{a \cdot -4.9833984375}{{b}^{11}} - \frac{2.21484375}{c \cdot {b}^{9}}\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 96.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(a \cdot \left(-4.9833984375 \cdot \frac{a \cdot {c}^{2}}{{b}^{11}} + -2.21484375 \cdot \frac{c}{{b}^{9}}\right) + 1.0546875 \cdot \frac{-1}{{b}^{7}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (fma
    -0.375
    (/ a (pow b 3.0))
    (*
     c
     (fma
      -0.5625
      (/ (pow a 2.0) (pow b 5.0))
      (*
       c
       (*
        (pow a 3.0)
        (+
         (*
          a
          (+
           (* -4.9833984375 (/ (* a (pow c 2.0)) (pow b 11.0)))
           (* -2.21484375 (/ c (pow b 9.0)))))
         (* 1.0546875 (/ -1.0 (pow b 7.0)))))))))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, fma(-0.375, (a / pow(b, 3.0)), (c * fma(-0.5625, (pow(a, 2.0) / pow(b, 5.0)), (c * (pow(a, 3.0) * ((a * ((-4.9833984375 * ((a * pow(c, 2.0)) / pow(b, 11.0))) + (-2.21484375 * (c / pow(b, 9.0))))) + (1.0546875 * (-1.0 / pow(b, 7.0))))))))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, fma(-0.375, Float64(a / (b ^ 3.0)), Float64(c * fma(-0.5625, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(c * Float64((a ^ 3.0) * Float64(Float64(a * Float64(Float64(-4.9833984375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 11.0))) + Float64(-2.21484375 * Float64(c / (b ^ 9.0))))) + Float64(1.0546875 * Float64(-1.0 / (b ^ 7.0))))))))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(a * N[(N[(-4.9833984375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 11.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.21484375 * N[(c / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0546875 * N[(-1.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + c \cdot \left(-0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}}{a \cdot b} + c \cdot \left(-0.16666666666666666 \cdot \frac{c \cdot \left(1.125 \cdot \frac{{a}^{2} \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a \cdot \left(1.5 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{a}^{5}}{{b}^{8}}\right)}{{b}^{2}} + 2.84765625 \cdot \frac{{a}^{6}}{{b}^{10}}\right)\right)}{a \cdot b} + -0.16666666666666666 \cdot \frac{1.5 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{a}^{5}}{{b}^{8}}}{a \cdot b}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{a}^{4}}{{b}^{6}}}{a} \cdot \frac{6.328125}{b}, c \cdot \left(-0.16666666666666666 \cdot \left(\frac{c}{b} \cdot \frac{\mathsf{fma}\left(1.125, {a}^{2} \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{4}}, \mathsf{fma}\left(1.5, a \cdot \frac{\mathsf{fma}\left(1.5, \frac{a \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)}{{b}^{2}}, \frac{3.796875 \cdot {a}^{5}}{{b}^{8}}\right)}{{b}^{2}}, 2.84765625 \cdot \frac{{a}^{6}}{{b}^{10}}\right)\right)}{a} + \frac{\mathsf{fma}\left(1.5, \frac{a \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)}{{b}^{2}}, \frac{3.796875 \cdot {a}^{5}}{{b}^{8}}\right)}{a \cdot b}\right)\right)\right)\right)\right), \frac{-0.5}{b}\right)} \]

6. Taylor expanded in a around 0 96.8%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \color{blue}{\left({a}^{3} \cdot \left(a \cdot \left(-4.9833984375 \cdot \frac{a \cdot {c}^{2}}{{b}^{11}} + -2.21484375 \cdot \frac{c}{{b}^{9}}\right) - 1.0546875 \cdot \frac{1}{{b}^{7}}\right)\right)}\right)\right), \frac{-0.5}{b}\right) \]

7. Final simplification 96.8%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(a \cdot \left(-4.9833984375 \cdot \frac{a \cdot {c}^{2}}{{b}^{11}} + -2.21484375 \cdot \frac{c}{{b}^{9}}\right) + 1.0546875 \cdot \frac{-1}{{b}^{7}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \]
8. Add Preprocessing

Alternative 3: 96.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{c \cdot a}{{b}^{9}} - \frac{1.0546875}{{b}^{7}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (fma
    -0.375
    (/ a (pow b 3.0))
    (*
     c
     (fma
      -0.5625
      (/ (pow a 2.0) (pow b 5.0))
      (*
       c
       (*
        (pow a 3.0)
        (-
         (* -2.21484375 (/ (* c a) (pow b 9.0)))
         (/ 1.0546875 (pow b 7.0))))))))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, fma(-0.375, (a / pow(b, 3.0)), (c * fma(-0.5625, (pow(a, 2.0) / pow(b, 5.0)), (c * (pow(a, 3.0) * ((-2.21484375 * ((c * a) / pow(b, 9.0))) - (1.0546875 / pow(b, 7.0)))))))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, fma(-0.375, Float64(a / (b ^ 3.0)), Float64(c * fma(-0.5625, Float64((a ^ 2.0) / (b ^ 5.0)), Float64(c * Float64((a ^ 3.0) * Float64(Float64(-2.21484375 * Float64(Float64(c * a) / (b ^ 9.0))) - Float64(1.0546875 / (b ^ 7.0)))))))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-2.21484375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0546875 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 96.8%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + c \cdot \left(-0.16666666666666666 \cdot \frac{1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}}{a \cdot b} + c \cdot \left(-0.16666666666666666 \cdot \frac{c \cdot \left(1.125 \cdot \frac{{a}^{2} \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{4}} + \left(1.5 \cdot \frac{a \cdot \left(1.5 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{a}^{5}}{{b}^{8}}\right)}{{b}^{2}} + 2.84765625 \cdot \frac{{a}^{6}}{{b}^{10}}\right)\right)}{a \cdot b} + -0.16666666666666666 \cdot \frac{1.5 \cdot \frac{a \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 3.796875 \cdot \frac{{a}^{5}}{{b}^{8}}}{a \cdot b}\right)\right)\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

5. Simplified 96.8%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{\frac{{a}^{4}}{{b}^{6}}}{a} \cdot \frac{6.328125}{b}, c \cdot \left(-0.16666666666666666 \cdot \left(\frac{c}{b} \cdot \frac{\mathsf{fma}\left(1.125, {a}^{2} \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{{b}^{4}}, \mathsf{fma}\left(1.5, a \cdot \frac{\mathsf{fma}\left(1.5, \frac{a \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)}{{b}^{2}}, \frac{3.796875 \cdot {a}^{5}}{{b}^{8}}\right)}{{b}^{2}}, 2.84765625 \cdot \frac{{a}^{6}}{{b}^{10}}\right)\right)}{a} + \frac{\mathsf{fma}\left(1.5, \frac{a \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)}{{b}^{2}}, \frac{3.796875 \cdot {a}^{5}}{{b}^{8}}\right)}{a \cdot b}\right)\right)\right)\right)\right), \frac{-0.5}{b}\right)} \]

6. Taylor expanded in a around 0 96.2%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \color{blue}{\left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{a \cdot c}{{b}^{9}} - 1.0546875 \cdot \frac{1}{{b}^{7}}\right)\right)}\right)\right), \frac{-0.5}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 96.2%

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{a \cdot c}{{b}^{9}} - \color{blue}{\frac{1.0546875 \cdot 1}{{b}^{7}}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \]

   2. metadata-eval 96.2%

      \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{a \cdot c}{{b}^{9}} - \frac{\color{blue}{1.0546875}}{{b}^{7}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \]

8. Simplified 96.2%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \color{blue}{\left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{a \cdot c}{{b}^{9}} - \frac{1.0546875}{{b}^{7}}\right)\right)}\right)\right), \frac{-0.5}{b}\right) \]

9. Final simplification 96.2%

   \[\leadsto c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, c \cdot \left({a}^{3} \cdot \left(-2.21484375 \cdot \frac{c \cdot a}{{b}^{9}} - \frac{1.0546875}{{b}^{7}}\right)\right)\right)\right), \frac{-0.5}{b}\right) \]
10. Add Preprocessing

Alternative 4: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (*
     a
     (+
      (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 7.0)))
      (* -0.5625 (/ (pow c 3.0) (pow b 5.0)))))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (a * ((-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 7.0))) + (-0.5625 * (pow(c, 3.0) / pow(b, 5.0)))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * (((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))) + ((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0)))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * ((-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))) + (-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * ((-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))) + (-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0)))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))) + Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 7.0))) + (-0.5625 * ((c ^ 3.0) / (b ^ 5.0)))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 95.2%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + c \cdot \left(-1.125 \cdot \frac{{a}^{2}}{{b}^{3}} + c \cdot \left(-1.6875 \cdot \frac{{a}^{3}}{{b}^{5}} + -0.5 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{3 \cdot a} \]

4. Taylor expanded in a around 0 95.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)} \]

5. Final simplification 95.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right) \]
6. Add Preprocessing

Alternative 5: 95.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1.0546875 \cdot \left(c \cdot {a}^{3}\right)}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (*
      c
      (+
       (* -0.5625 (/ (pow a 2.0) (pow b 5.0)))
       (/ (* -1.0546875 (* c (pow a 3.0))) (pow b 7.0))))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (c * ((-0.5625 * (pow(a, 2.0) / pow(b, 5.0))) + ((-1.0546875 * (c * pow(a, 3.0))) / pow(b, 7.0)))))) + (0.5 * (-1.0 / 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 = c * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + (c * (((-0.5625d0) * ((a ** 2.0d0) / (b ** 5.0d0))) + (((-1.0546875d0) * (c * (a ** 3.0d0))) / (b ** 7.0d0)))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (c * ((-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))) + ((-1.0546875 * (c * Math.pow(a, 3.0))) / Math.pow(b, 7.0)))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (c * ((-0.5625 * (math.pow(a, 2.0) / math.pow(b, 5.0))) + ((-1.0546875 * (c * math.pow(a, 3.0))) / math.pow(b, 7.0)))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(c * Float64(Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 5.0))) + Float64(Float64(-1.0546875 * Float64(c * (a ^ 3.0))) / (b ^ 7.0)))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (c * ((-0.5625 * ((a ^ 2.0) / (b ^ 5.0))) + ((-1.0546875 * (c * (a ^ 3.0))) / (b ^ 7.0)))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 95.5%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in a around 0 95.5%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*r/ 95.5%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

   2. *-commutative 95.5%

      \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1.0546875 \cdot \color{blue}{\left(c \cdot {a}^{3}\right)}}{{b}^{7}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

6. Simplified 95.5%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \color{blue}{\frac{-1.0546875 \cdot \left(c \cdot {a}^{3}\right)}{{b}^{7}}}\right)\right) - 0.5 \cdot \frac{1}{b}\right) \]

7. Final simplification 95.5%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{-1.0546875 \cdot \left(c \cdot {a}^{3}\right)}{{b}^{7}}\right)\right) + 0.5 \cdot \frac{-1}{b}\right) \]
8. Add Preprocessing

Alternative 6: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))
    (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 3.0))) + (-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + ((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0)))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 3.0))) + (-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in a around 0 94.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

4. Final simplification 94.2%

   \[\leadsto -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right) \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / pow(b, 3.0))) + (-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))))) + (0.5 * (-1.0 / 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 = c * ((c * (((-0.375d0) * (a / (b ** 3.0d0))) + ((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.375 * (a / Math.pow(b, 3.0))) + (-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.375 * (a / math.pow(b, 3.0))) + (-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.375 * (a / (b ^ 3.0))) + (-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 93.8%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Final simplification 93.8%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + -0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}}\right) + 0.5 \cdot \frac{-1}{b}\right) \]
5. Add Preprocessing

Alternative 8: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in a around 0 91.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 91.0%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ (* -0.375 (* c a)) (pow b 3.0)) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * (((-0.375 * (c * a)) / pow(b, 3.0)) - (0.5 / 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 = c * ((((-0.375d0) * (c * a)) / (b ** 3.0d0)) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * (((-0.375 * (c * a)) / Math.pow(b, 3.0)) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * (((-0.375 * (c * a)) / math.pow(b, 3.0)) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(-0.375 * Float64(c * a)) / (b ^ 3.0)) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((-0.375 * (c * a)) / (b ^ 3.0)) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in c around 0 90.7%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 90.7%

      \[\leadsto c \cdot \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r/ 90.7%

      \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   3. metadata-eval 90.7%

      \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{\color{blue}{0.5}}{b}\right) \]

5. Simplified 90.7%

   \[\leadsto \color{blue}{c \cdot \left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{0.5}{b}\right)} \]

6. Final simplification 90.7%

   \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} - \frac{0.5}{b}\right) \]
7. Add Preprocessing

Alternative 10: 80.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in a around 0 91.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Taylor expanded in c around 0 82.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
5. Step-by-step derivation

   1. associate-*r/ 82.0%

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

   2. *-commutative 82.0%

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

   3. associate-/l* 81.7%

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

6. Simplified 81.7%

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

7. Final simplification 81.7%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
8. Add Preprocessing

Alternative 11: 80.8% accurate, 23.2× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

3. Taylor expanded in b around inf 82.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ 82.0%

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

   2. *-commutative 82.0%

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

5. Simplified 82.0%

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

6. Final simplification 82.0%

   \[\leadsto \frac{-0.5 \cdot c}{b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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)))