Cubic critical, medium range

Percentage Accurate: 30.4% → 95.6%

Time: 16.2s

Alternatives: 9

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: 30.4% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -0.5 c)
   (*
    a
    (+
     (* -0.375 (/ (pow c 2.0) (pow b 2.0)))
     (*
      a
      (+
       (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 6.0)))
       (* -0.5625 (/ (pow c 3.0) (pow b 4.0))))))))
  b))

C

double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * (pow(c, 2.0) / pow(b, 2.0))) + (a * ((-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-0.5625 * (pow(c, 3.0) / pow(b, 4.0)))))))) / 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 = (((-0.5d0) * c) + (a * (((-0.375d0) * ((c ** 2.0d0) / (b ** 2.0d0))) + (a * (((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * ((c ** 3.0d0) / (b ** 4.0d0)))))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 2.0))) + (a * ((-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 4.0)))))))) / b;
}

Python

def code(a, b, c):
	return ((-0.5 * c) + (a * ((-0.375 * (math.pow(c, 2.0) / math.pow(b, 2.0))) + (a * ((-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(c, 3.0) / math.pow(b, 4.0)))))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5 * c) + Float64(a * Float64(Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 2.0))) + Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 4.0)))))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5 * c) + (a * ((-0.375 * ((c ^ 2.0) / (b ^ 2.0))) + (a * ((-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-0.5625 * ((c ^ 3.0) / (b ^ 4.0)))))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * c), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in a around -inf 96.1%

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

8. Simplified 96.1%

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

9. Taylor expanded in a around 0 96.6%

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

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (+
      (* -0.375 (/ a (pow b 2.0)))
      (*
       c
       (+
        (* -1.0546875 (/ (* c (pow a 3.0)) (pow b 6.0)))
        (* -0.5625 (/ (pow a 2.0) (pow b 4.0)))))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / pow(b, 2.0))) + (c * ((-1.0546875 * ((c * pow(a, 3.0)) / pow(b, 6.0))) + (-0.5625 * (pow(a, 2.0) / pow(b, 4.0))))))) - 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 * ((c * (((-0.375d0) * (a / (b ** 2.0d0))) + (c * (((-1.0546875d0) * ((c * (a ** 3.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * ((a ** 2.0d0) / (b ** 4.0d0))))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / Math.pow(b, 2.0))) + (c * ((-1.0546875 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * ((-0.375 * (a / math.pow(b, 2.0))) + (c * ((-1.0546875 * ((c * math.pow(a, 3.0)) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(a, 2.0) / math.pow(b, 4.0))))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 2.0))) + Float64(c * Float64(Float64(-1.0546875 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * ((-0.375 * (a / (b ^ 2.0))) + (c * ((-1.0546875 * ((c * (a ^ 3.0)) / (b ^ 6.0))) + (-0.5625 * ((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-1.0546875 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in a around -inf 96.1%

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

8. Simplified 96.1%

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

9. Taylor expanded in c around 0 96.4%

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}}{b} \]

10. Final simplification 96.4%

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

Alternative 3: 95.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.375 (/ a (pow b 3.0)))
     (*
      c
      (*
       (pow a 3.0)
       (- (/ (* c -1.0546875) (pow b 7.0)) (/ 0.5625 (* a (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))) + (c * (pow(a, 3.0) * (((c * -1.0546875) / pow(b, 7.0)) - (0.5625 / (a * 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))) + (c * ((a ** 3.0d0) * (((c * (-1.0546875d0)) / (b ** 7.0d0)) - (0.5625d0 / (a * (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))) + (c * (Math.pow(a, 3.0) * (((c * -1.0546875) / Math.pow(b, 7.0)) - (0.5625 / (a * 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))) + (c * (math.pow(a, 3.0) * (((c * -1.0546875) / math.pow(b, 7.0)) - (0.5625 / (a * 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(c * Float64((a ^ 3.0) * Float64(Float64(Float64(c * -1.0546875) / (b ^ 7.0)) - Float64(0.5625 / Float64(a * (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))) + (c * ((a ^ 3.0) * (((c * -1.0546875) / (b ^ 7.0)) - (0.5625 / (a * (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[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(N[(c * -1.0546875), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] - N[(0.5625 / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in c around 0 96.3%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

7. Taylor expanded in a around 0 96.3%

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

8. Taylor expanded in a around inf 96.3%

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

   1. associate-*r/ 96.3%

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

   2. associate-*r/ 96.3%

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

   3. metadata-eval 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 4: 94.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Simplified 30.4%

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

5. Taylor expanded in a around 0 94.9%

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

6. Taylor expanded in c around inf 94.9%

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

   1. associate-*r/ 94.9%

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

   2. metadata-eval 94.9%

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

   3. *-commutative 94.9%

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

8. Simplified 94.9%

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

Alternative 5: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in c around 0 96.3%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]

7. Taylor expanded in a around 0 96.3%

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

8. Taylor expanded in a around 0 94.6%

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

   1. associate-*r/ 94.6%

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

   2. associate-*r/ 94.6%

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

   3. metadata-eval 94.6%

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

10. Simplified 94.6%

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

11. Final simplification 94.6%

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

Alternative 6: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.5, c, \left(a \cdot -0.375\right) \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in b around inf 91.7%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
7. Step-by-step derivation

   1. fma-define 91.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}}{b} \]

   2. associate-*r/ 91.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}{b} \]

   3. associate-*r* 91.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, \color{blue}{\left(-0.375 \cdot a\right) \cdot \frac{{c}^{2}}{{b}^{2}}}\right)}{b} \]

   4. unpow2 91.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, \left(-0.375 \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b} \]

   5. unpow2 91.7%

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

   6. times-frac 91.7%

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

   7. unpow1 91.7%

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

   8. pow-plus 91.7%

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

   9. metadata-eval 91.7%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, \left(-0.375 \cdot a\right) \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}\right)}{b} \]

8. Simplified 91.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, c, \left(-0.375 \cdot a\right) \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b}} \]

9. Final simplification 91.7%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, \left(a \cdot -0.375\right) \cdot {\left(\frac{c}{b}\right)}^{2}\right)}{b} \]
10. Add Preprocessing

Alternative 7: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((c * a) / pow(b, 2.0))) - 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 ** 2.0d0))) - 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, 2.0))) - 0.5)) / b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 96.6%

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

6. Taylor expanded in c around 0 91.6%

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

7. Final simplification 91.6%

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

Alternative 8: 90.7% accurate, 1.0× speedup

Math

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

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(-0.375 * Float64(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[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $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. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in c around 0 91.4%

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

   1. associate-*r/ 91.4%

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

   2. metadata-eval 91.4%

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

7. Simplified 91.4%

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

8. Final simplification 91.4%

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

Alternative 9: 81.9% accurate, 23.2× 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 30.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

3. Simplified 30.4%

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

5. Taylor expanded in b around inf 82.4%

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

Reproduce

herbie shell --seed 2024170 
(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)))