Quadratic roots, medium range

Percentage Accurate: 31.1% → 98.8%

Time: 15.0s

Alternatives: 6

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ \frac{\frac{\frac{\mathsf{fma}\left(64, \left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right), \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(-48\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot \sqrt{t_0}\right)}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* a 4.0)))))
   (/
    (/
     (/
      (fma
       64.0
       (* (* c a) (* (* c a) (* c a)))
       (fma
        a
        (* (* c (pow b 4.0)) 12.0)
        (* (* a a) (* (* (* b b) (* c c)) (- 48.0)))))
      (- (pow (- b) 3.0) (pow (- (* b b) (* (* c a) 4.0)) 1.5)))
     (+ (pow (- b) 2.0) (+ t_0 (* b (sqrt t_0)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (a * 4.0));
	return ((fma(64.0, ((c * a) * ((c * a) * (c * a))), fma(a, ((c * pow(b, 4.0)) * 12.0), ((a * a) * (((b * b) * (c * c)) * -48.0)))) / (pow(-b, 3.0) - pow(((b * b) - ((c * a) * 4.0)), 1.5))) / (pow(-b, 2.0) + (t_0 + (b * sqrt(t_0))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))
	return Float64(Float64(Float64(fma(64.0, Float64(Float64(c * a) * Float64(Float64(c * a) * Float64(c * a))), fma(a, Float64(Float64(c * (b ^ 4.0)) * 12.0), Float64(Float64(a * a) * Float64(Float64(Float64(b * b) * Float64(c * c)) * Float64(-48.0))))) / Float64((Float64(-b) ^ 3.0) - (Float64(Float64(b * b) - Float64(Float64(c * a) * 4.0)) ^ 1.5))) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(64.0 * N[(N[(c * a), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(c * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 12.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * (-48.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 3.0], $MachinePrecision] - N[Power[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. flip3-+ 31.9%

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

   2. pow1/2 31.9%

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

   3. pow-pow 32.9%

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

   4. *-commutative 32.9%

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

   5. *-commutative 32.9%

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

   6. metadata-eval 32.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   7. pow2 32.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

3. Applied egg-rr 32.9%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]
4. Step-by-step derivation

   1. flip-+ 32.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

5. Applied egg-rr 32.9%

   \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
6. Step-by-step derivation

   1. neg-mul-1 32.9%

      \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(-1 \cdot b\right)}}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   2. neg-mul-1 32.9%

      \[\leadsto \frac{\frac{\frac{{\left(-1 \cdot b\right)}^{3} \cdot {\color{blue}{\left(-1 \cdot b\right)}}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   3. pow-sqr 33.6%

      \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(-1 \cdot b\right)}^{\left(2 \cdot 3\right)}} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   4. neg-mul-1 33.6%

      \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(-b\right)}}^{\left(2 \cdot 3\right)} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. metadata-eval 33.6%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{\color{blue}{6}} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   6. pow-sqr 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - \color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\left(2 \cdot 1.5\right)}}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. associate-*r* 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}^{\left(2 \cdot 1.5\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   8. *-commutative 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - \color{blue}{4 \cdot \left(c \cdot a\right)}\right)}^{\left(2 \cdot 1.5\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   9. metadata-eval 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{\color{blue}{3}}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   10. associate-*r* 33.7%

       \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   11. *-commutative 33.7%

       \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - \color{blue}{4 \cdot \left(c \cdot a\right)}\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

7. Simplified 33.7%

   \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

8. Taylor expanded in a around -inf 98.7%

   \[\leadsto \frac{\frac{\frac{\color{blue}{64 \cdot \left({c}^{3} \cdot {a}^{3}\right) + \left(a \cdot \left(8 \cdot \left(c \cdot {b}^{4}\right) + 4 \cdot \left(c \cdot {b}^{4}\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
9. Step-by-step derivation

   1. fma-def 98.7%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(64, {c}^{3} \cdot {a}^{3}, a \cdot \left(8 \cdot \left(c \cdot {b}^{4}\right) + 4 \cdot \left(c \cdot {b}^{4}\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   2. cube-prod 98.7%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, \color{blue}{{\left(c \cdot a\right)}^{3}}, a \cdot \left(8 \cdot \left(c \cdot {b}^{4}\right) + 4 \cdot \left(c \cdot {b}^{4}\right)\right) + -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   3. fma-def 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \color{blue}{\mathsf{fma}\left(a, 8 \cdot \left(c \cdot {b}^{4}\right) + 4 \cdot \left(c \cdot {b}^{4}\right), -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)}\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   4. distribute-rgt-out 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \color{blue}{\left(c \cdot {b}^{4}\right) \cdot \left(8 + 4\right)}, -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. metadata-eval 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot \color{blue}{12}, -1 \cdot \left({a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   6. mul-1-neg 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \color{blue}{-{a}^{2} \cdot \left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. distribute-rgt-neg-in 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \color{blue}{{a}^{2} \cdot \left(-\left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   8. unpow2 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \color{blue}{\left(a \cdot a\right)} \cdot \left(-\left(16 \cdot \left({c}^{2} \cdot {b}^{2}\right) + 32 \cdot \left({c}^{2} \cdot {b}^{2}\right)\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   9. distribute-rgt-out 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(-\color{blue}{\left({c}^{2} \cdot {b}^{2}\right) \cdot \left(16 + 32\right)}\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

10. Simplified 98.8%

    \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(-\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot 48\right)\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
11. Step-by-step derivation

    1. unpow3 98.8%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(-\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot 48\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

12. Applied egg-rr 98.8%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}, \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(-\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot 48\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

13. Final simplification 98.8%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, \left(c \cdot a\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right), \mathsf{fma}\left(a, \left(c \cdot {b}^{4}\right) \cdot 12, \left(a \cdot a\right) \cdot \left(\left(\left(b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(-48\right)\right)\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

Alternative 2: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(a \cdot 4\right)\\ \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(\left(c \cdot a\right) \cdot 12\right) + \left(b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot -48\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot \sqrt{t_0}\right)}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* a 4.0)))))
   (/
    (/
     (/
      (fma
       64.0
       (pow (* c a) 3.0)
       (+
        (* (pow b 4.0) (* (* c a) 12.0))
        (* (* b b) (* (* (* c a) (* c a)) -48.0))))
      (- (pow (- b) 3.0) (pow (- (* b b) (* (* c a) 4.0)) 1.5)))
     (+ (pow (- b) 2.0) (+ t_0 (* b (sqrt t_0)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (a * 4.0));
	return ((fma(64.0, pow((c * a), 3.0), ((pow(b, 4.0) * ((c * a) * 12.0)) + ((b * b) * (((c * a) * (c * a)) * -48.0)))) / (pow(-b, 3.0) - pow(((b * b) - ((c * a) * 4.0)), 1.5))) / (pow(-b, 2.0) + (t_0 + (b * sqrt(t_0))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))
	return Float64(Float64(Float64(fma(64.0, (Float64(c * a) ^ 3.0), Float64(Float64((b ^ 4.0) * Float64(Float64(c * a) * 12.0)) + Float64(Float64(b * b) * Float64(Float64(Float64(c * a) * Float64(c * a)) * -48.0)))) / Float64((Float64(-b) ^ 3.0) - (Float64(Float64(b * b) - Float64(Float64(c * a) * 4.0)) ^ 1.5))) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(64.0 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[(c * a), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision] * -48.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 3.0], $MachinePrecision] - N[Power[N[(N[(b * b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. flip3-+ 31.9%

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

   2. pow1/2 31.9%

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

   3. pow-pow 32.9%

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

   4. *-commutative 32.9%

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

   5. *-commutative 32.9%

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

   6. metadata-eval 32.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   7. pow2 32.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

3. Applied egg-rr 32.9%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]
4. Step-by-step derivation

   1. flip-+ 32.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

5. Applied egg-rr 32.9%

   \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
6. Step-by-step derivation

   1. neg-mul-1 32.9%

      \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(-1 \cdot b\right)}}^{3} \cdot {\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   2. neg-mul-1 32.9%

      \[\leadsto \frac{\frac{\frac{{\left(-1 \cdot b\right)}^{3} \cdot {\color{blue}{\left(-1 \cdot b\right)}}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   3. pow-sqr 33.6%

      \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(-1 \cdot b\right)}^{\left(2 \cdot 3\right)}} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   4. neg-mul-1 33.6%

      \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(-b\right)}}^{\left(2 \cdot 3\right)} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. metadata-eval 33.6%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{\color{blue}{6}} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5} \cdot {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   6. pow-sqr 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - \color{blue}{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\left(2 \cdot 1.5\right)}}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. associate-*r* 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}^{\left(2 \cdot 1.5\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   8. *-commutative 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - \color{blue}{4 \cdot \left(c \cdot a\right)}\right)}^{\left(2 \cdot 1.5\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   9. metadata-eval 33.7%

      \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{\color{blue}{3}}}{{\left(-b\right)}^{3} - {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   10. associate-*r* 33.7%

       \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - \color{blue}{\left(c \cdot a\right) \cdot 4}\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   11. *-commutative 33.7%

       \[\leadsto \frac{\frac{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - \color{blue}{4 \cdot \left(c \cdot a\right)}\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

7. Simplified 33.7%

   \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(-b\right)}^{6} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{3}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

8. Taylor expanded in b around inf 98.8%

   \[\leadsto \frac{\frac{\frac{\color{blue}{64 \cdot \left({c}^{3} \cdot {a}^{3}\right) + \left(-1 \cdot \left(\left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right) \cdot {b}^{4}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]
9. Step-by-step derivation

   1. fma-def 98.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(64, {c}^{3} \cdot {a}^{3}, -1 \cdot \left(\left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right) \cdot {b}^{4}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   2. cube-prod 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, \color{blue}{{\left(c \cdot a\right)}^{3}}, -1 \cdot \left(\left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right) \cdot {b}^{4}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   3. mul-1-neg 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \color{blue}{\left(-\left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right) \cdot {b}^{4}\right)} + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   4. *-commutative 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \left(-\color{blue}{{b}^{4} \cdot \left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right)}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   5. distribute-rgt-neg-in 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, \color{blue}{{b}^{4} \cdot \left(-\left(-8 \cdot \left(c \cdot a\right) + -4 \cdot \left(c \cdot a\right)\right)\right)} + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   6. distribute-rgt-out 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(-\color{blue}{\left(c \cdot a\right) \cdot \left(-8 + -4\right)}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   7. metadata-eval 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(-\left(c \cdot a\right) \cdot \color{blue}{-12}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   8. distribute-rgt-neg-in 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(--12\right)\right)} + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

   9. metadata-eval 98.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(\left(c \cdot a\right) \cdot \color{blue}{12}\right) + -1 \cdot \left(\left(32 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 16 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right) \cdot {b}^{2}\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

10. Simplified 98.8%

    \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(\left(c \cdot a\right) \cdot 12\right) + \left(b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot -48\right)\right)}}{{\left(-b\right)}^{3} - {\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{2 \cdot a} \]

11. Final simplification 98.8%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(64, {\left(c \cdot a\right)}^{3}, {b}^{4} \cdot \left(\left(c \cdot a\right) \cdot 12\right) + \left(b \cdot b\right) \cdot \left(\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot -48\right)\right)}{{\left(-b\right)}^{3} - {\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right)}^{1.5}}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}{a \cdot 2} \]

Alternative 3: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (-
   (fma
    -0.25
    (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
    (* -2.0 (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))))
   (/ c b))
  (/ (* c (* c a)) (pow b 3.0))))

C

double code(double a, double b, double c) {
	return (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 * ((a * a) / (pow(b, 5.0) / pow(c, 3.0))))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 * Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. neg-sub0 31.9%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 31.9%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 31.9%

      \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. neg-mul-1 31.9%

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

   5. associate-*l/ 31.9%

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

   6. *-commutative 31.9%

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

   7. associate-/r* 31.9%

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

   8. /-rgt-identity 31.9%

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

   9. metadata-eval 31.9%

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

3. Simplified 31.8%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in a around 0 96.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + \left(-0.25 \cdot \frac{\left({\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right) \cdot {a}^{3}}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

6. Simplified 96.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b} \cdot {a}^{3}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]

7. Taylor expanded in b around 0 96.1%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
8. Step-by-step derivation

   1. associate-/l* 96.1%

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

   2. distribute-rgt-out 96.1%

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

   3. metadata-eval 96.1%

      \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

9. Simplified 96.1%

   \[\leadsto \left(\mathsf{fma}\left(-0.25, \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

10. Final simplification 96.1%

    \[\leadsto \left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right) - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

Alternative 4: 93.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.9%

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

   2. +-commutative 31.9%

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

   3. unsub-neg 31.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{a \cdot 2} \]

   4. fma-neg 31.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2} \]

   5. associate-*l* 31.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{4 \cdot \left(a \cdot c\right)}\right)} - b}{a \cdot 2} \]

   6. *-commutative 31.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(a \cdot c\right) \cdot 4}\right)} - b}{a \cdot 2} \]

   7. distribute-rgt-neg-in 31.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)} - b}{a \cdot 2} \]

   8. metadata-eval 31.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-4}\right)} - b}{a \cdot 2} \]

3. Simplified 31.9%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}{a \cdot 2}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 31.3%

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

   2. pow3 31.3%

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

   3. pow1/2 31.3%

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

   4. pow-pow 31.5%

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

   5. *-commutative 31.5%

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

   6. metadata-eval 31.5%

      \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{1.5}}} - b}{a \cdot 2} \]

5. Applied egg-rr 31.5%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5}}} - b}{a \cdot 2} \]

6. Taylor expanded in b around inf 94.5%

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

   1. +-commutative 94.5%

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

   2. mul-1-neg 94.5%

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

   3. unsub-neg 94.5%

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

8. Simplified 94.5%

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

9. Final simplification 94.5%

   \[\leadsto \left(-2 \cdot \left({c}^{3} \cdot \frac{a}{\frac{{b}^{5}}{a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternative 5: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. neg-sub0 31.9%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 31.9%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 31.9%

      \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. neg-mul-1 31.9%

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

   5. associate-*l/ 31.9%

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

   6. *-commutative 31.9%

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

   7. associate-/r* 31.9%

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

   8. /-rgt-identity 31.9%

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

   9. metadata-eval 31.9%

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

3. Simplified 31.8%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 91.0%

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

   1. +-commutative 91.0%

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

   2. mul-1-neg 91.0%

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

   3. unsub-neg 91.0%

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

   4. associate-*r/ 91.0%

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

   5. neg-mul-1 91.0%

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

   6. unpow2 91.0%

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

   7. associate-*l* 91.0%

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

6. Simplified 91.0%

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

7. Taylor expanded in c around 0 91.0%

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

   1. associate-/l* 91.0%

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

   2. unpow2 91.0%

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

9. Simplified 91.0%

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

10. Final simplification 91.0%

    \[\leadsto \frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]

Alternative 6: 81.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. neg-sub0 31.9%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 31.9%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 31.9%

      \[\leadsto \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. neg-mul-1 31.9%

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

   5. associate-*l/ 31.9%

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

   6. *-commutative 31.9%

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

   7. associate-/r* 31.9%

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

   8. /-rgt-identity 31.9%

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

   9. metadata-eval 31.9%

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

3. Simplified 31.8%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 81.0%

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

   1. associate-*r/ 81.0%

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

   2. neg-mul-1 81.0%

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

6. Simplified 81.0%

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

7. Final simplification 81.0%

   \[\leadsto \frac{-c}{b} \]

Reproduce

herbie shell --seed 2023214 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))