Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.4%

Time: 12.8s

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: 31.6% 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: 95.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{8}}\\ t_1 := \frac{{c}^{3}}{{b}^{6}}\\ t_2 := t_1 \cdot -256\\ t_3 := \frac{{c}^{2}}{{b}^{4}}\\ t_4 := t_3 \cdot 16\\ 0.5 \cdot \left(\left(a \cdot b\right) \cdot \mathsf{fma}\left(2, t_3, -0.25 \cdot t_4\right) + \mathsf{fma}\left({a}^{2}, \mathsf{fma}\left(b, \mathsf{fma}\left(-1.3333333333333333, t_1, \mathsf{fma}\left(0.5, \frac{c}{\frac{{b}^{2}}{t_4}}, -0.08333333333333333 \cdot t_2\right)\right), -32 \cdot \frac{{c}^{3}}{{b}^{5}}\right), {a}^{3} \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-0.5, \frac{{c}^{2}}{\frac{{b}^{4}}{t_4}}, \mathsf{fma}\left(-0.020833333333333332, t_0 \cdot 1536, \mathsf{fma}\left(0.03125, t_0 \cdot 256, \mathsf{fma}\left(0.16666666666666666, \frac{c \cdot t_2}{{b}^{2}}, t_0 \cdot 0.6666666666666666\right)\right)\right)\right), \frac{{c}^{4} \cdot 64}{{b}^{7}}\right)\right)\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 4.0) (pow b 8.0)))
        (t_1 (/ (pow c 3.0) (pow b 6.0)))
        (t_2 (* t_1 -256.0))
        (t_3 (/ (pow c 2.0) (pow b 4.0)))
        (t_4 (* t_3 16.0)))
   (-
    (*
     0.5
     (+
      (* (* a b) (fma 2.0 t_3 (* -0.25 t_4)))
      (fma
       (pow a 2.0)
       (fma
        b
        (fma
         -1.3333333333333333
         t_1
         (fma 0.5 (/ c (/ (pow b 2.0) t_4)) (* -0.08333333333333333 t_2)))
        (* -32.0 (/ (pow c 3.0) (pow b 5.0))))
       (*
        (pow a 3.0)
        (fma
         b
         (fma
          -0.5
          (/ (pow c 2.0) (/ (pow b 4.0) t_4))
          (fma
           -0.020833333333333332
           (* t_0 1536.0)
           (fma
            0.03125
            (* t_0 256.0)
            (fma
             0.16666666666666666
             (/ (* c t_2) (pow b 2.0))
             (* t_0 0.6666666666666666)))))
         (/ (* (pow c 4.0) 64.0) (pow b 7.0)))))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = pow(c, 4.0) / pow(b, 8.0);
	double t_1 = pow(c, 3.0) / pow(b, 6.0);
	double t_2 = t_1 * -256.0;
	double t_3 = pow(c, 2.0) / pow(b, 4.0);
	double t_4 = t_3 * 16.0;
	return (0.5 * (((a * b) * fma(2.0, t_3, (-0.25 * t_4))) + fma(pow(a, 2.0), fma(b, fma(-1.3333333333333333, t_1, fma(0.5, (c / (pow(b, 2.0) / t_4)), (-0.08333333333333333 * t_2))), (-32.0 * (pow(c, 3.0) / pow(b, 5.0)))), (pow(a, 3.0) * fma(b, fma(-0.5, (pow(c, 2.0) / (pow(b, 4.0) / t_4)), fma(-0.020833333333333332, (t_0 * 1536.0), fma(0.03125, (t_0 * 256.0), fma(0.16666666666666666, ((c * t_2) / pow(b, 2.0)), (t_0 * 0.6666666666666666))))), ((pow(c, 4.0) * 64.0) / pow(b, 7.0))))))) - (c / b);
}

Julia

function code(a, b, c)
	t_0 = Float64((c ^ 4.0) / (b ^ 8.0))
	t_1 = Float64((c ^ 3.0) / (b ^ 6.0))
	t_2 = Float64(t_1 * -256.0)
	t_3 = Float64((c ^ 2.0) / (b ^ 4.0))
	t_4 = Float64(t_3 * 16.0)
	return Float64(Float64(0.5 * Float64(Float64(Float64(a * b) * fma(2.0, t_3, Float64(-0.25 * t_4))) + fma((a ^ 2.0), fma(b, fma(-1.3333333333333333, t_1, fma(0.5, Float64(c / Float64((b ^ 2.0) / t_4)), Float64(-0.08333333333333333 * t_2))), Float64(-32.0 * Float64((c ^ 3.0) / (b ^ 5.0)))), Float64((a ^ 3.0) * fma(b, fma(-0.5, Float64((c ^ 2.0) / Float64((b ^ 4.0) / t_4)), fma(-0.020833333333333332, Float64(t_0 * 1536.0), fma(0.03125, Float64(t_0 * 256.0), fma(0.16666666666666666, Float64(Float64(c * t_2) / (b ^ 2.0)), Float64(t_0 * 0.6666666666666666))))), Float64(Float64((c ^ 4.0) * 64.0) / (b ^ 7.0))))))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * -256.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * 16.0), $MachinePrecision]}, N[(N[(0.5 * N[(N[(N[(a * b), $MachinePrecision] * N[(2.0 * t$95$3 + N[(-0.25 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[(b * N[(-1.3333333333333333 * t$95$1 + N[(0.5 * N[(c / N[(N[Power[b, 2.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-0.08333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-32.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(b * N[(-0.5 * N[(N[Power[c, 2.0], $MachinePrecision] / N[(N[Power[b, 4.0], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-0.020833333333333332 * N[(t$95$0 * 1536.0), $MachinePrecision] + N[(0.03125 * N[(t$95$0 * 256.0), $MachinePrecision] + N[(0.16666666666666666 * N[(N[(c * t$95$2), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 4.0], $MachinePrecision] * 64.0), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

   1. flip3-- 31.5%

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

   2. sqrt-div 31.2%

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

   3. pow2 31.2%

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

   4. pow-pow 31.2%

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

   5. metadata-eval 31.2%

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

   6. associate-*l* 31.2%

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

   7. unpow-prod-down 31.2%

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

   8. metadata-eval 31.2%

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

5. Applied egg-rr 31.1%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. +-commutative 31.1%

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} + \left(-b\right)}}{a \cdot 2} \]

   2. div-inv 31.1%

      \[\leadsto \frac{\color{blue}{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}} \cdot \frac{1}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}} + \left(-b\right)}{a \cdot 2} \]

   3. fma-def 32.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}, \frac{1}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}, -b\right)}}{a \cdot 2} \]

7. Applied egg-rr 32.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}, {\left(\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right), {b}^{4}\right)\right)}^{-0.5}, -b\right)}}{a \cdot 2} \]

8. Taylor expanded in a around 0 96.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(0.5 \cdot \left(a \cdot \left(b \cdot \left(-0.25 \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{4}} + 32 \cdot \frac{{c}^{2}}{{b}^{4}}\right) + 2 \cdot \frac{{c}^{2}}{{b}^{4}}\right)\right)\right) + \left(0.5 \cdot \left({a}^{2} \cdot \left(-32 \cdot \frac{{c}^{3}}{{b}^{5}} + b \cdot \left(-1.3333333333333333 \cdot \frac{{c}^{3}}{{b}^{6}} + \left(-0.08333333333333333 \cdot \left(-384 \cdot \frac{{c}^{3}}{{b}^{6}} + 128 \cdot \frac{{c}^{3}}{{b}^{6}}\right) + 0.5 \cdot \frac{c \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{4}} + 32 \cdot \frac{{c}^{2}}{{b}^{4}}\right)}{{b}^{2}}\right)\right)\right)\right) + 0.5 \cdot \left({a}^{3} \cdot \left(64 \cdot \frac{{c}^{4}}{{b}^{7}} + b \cdot \left(-0.5 \cdot \frac{{c}^{2} \cdot \left(-16 \cdot \frac{{c}^{2}}{{b}^{4}} + 32 \cdot \frac{{c}^{2}}{{b}^{4}}\right)}{{b}^{4}} + \left(-0.020833333333333332 \cdot \left(-3072 \cdot \frac{{c}^{4}}{{b}^{8}} + \left(-1536 \cdot \frac{{c}^{4}}{{b}^{8}} + 6144 \cdot \frac{{c}^{4}}{{b}^{8}}\right)\right) + \left(0.03125 \cdot {\left(-16 \cdot \frac{{c}^{2}}{{b}^{4}} + 32 \cdot \frac{{c}^{2}}{{b}^{4}}\right)}^{2} + \left(0.16666666666666666 \cdot \frac{c \cdot \left(-384 \cdot \frac{{c}^{3}}{{b}^{6}} + 128 \cdot \frac{{c}^{3}}{{b}^{6}}\right)}{{b}^{2}} + 0.6666666666666666 \cdot \frac{{c}^{4}}{{b}^{8}}\right)\right)\right)\right)\right)\right)\right)\right)} \]

10. Simplified 96.4%

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(a \cdot b\right) \cdot \mathsf{fma}\left(2, \frac{{c}^{2}}{{b}^{4}}, -0.25 \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot 16\right)\right) + \mathsf{fma}\left({a}^{2}, \mathsf{fma}\left(b, \mathsf{fma}\left(-1.3333333333333333, \frac{{c}^{3}}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{c}{\frac{{b}^{2}}{\frac{{c}^{2}}{{b}^{4}} \cdot 16}}, -0.08333333333333333 \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -256\right)\right)\right), -32 \cdot \frac{{c}^{3}}{{b}^{5}}\right), {a}^{3} \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-0.5, \frac{{c}^{2}}{\frac{{b}^{4}}{\frac{{c}^{2}}{{b}^{4}} \cdot 16}}, \mathsf{fma}\left(-0.020833333333333332, \frac{{c}^{4}}{{b}^{8}} \cdot 1536, \mathsf{fma}\left(0.03125, \frac{{c}^{4}}{{b}^{8}} \cdot 256, \mathsf{fma}\left(0.16666666666666666, \frac{c \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -256\right)}{{b}^{2}}, \frac{{c}^{4}}{{b}^{8}} \cdot 0.6666666666666666\right)\right)\right)\right), \frac{64 \cdot {c}^{4}}{{b}^{7}}\right)\right)\right) - \frac{c}{b}} \]

11. Final simplification 96.4%

    \[\leadsto 0.5 \cdot \left(\left(a \cdot b\right) \cdot \mathsf{fma}\left(2, \frac{{c}^{2}}{{b}^{4}}, -0.25 \cdot \left(\frac{{c}^{2}}{{b}^{4}} \cdot 16\right)\right) + \mathsf{fma}\left({a}^{2}, \mathsf{fma}\left(b, \mathsf{fma}\left(-1.3333333333333333, \frac{{c}^{3}}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{c}{\frac{{b}^{2}}{\frac{{c}^{2}}{{b}^{4}} \cdot 16}}, -0.08333333333333333 \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -256\right)\right)\right), -32 \cdot \frac{{c}^{3}}{{b}^{5}}\right), {a}^{3} \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(-0.5, \frac{{c}^{2}}{\frac{{b}^{4}}{\frac{{c}^{2}}{{b}^{4}} \cdot 16}}, \mathsf{fma}\left(-0.020833333333333332, \frac{{c}^{4}}{{b}^{8}} \cdot 1536, \mathsf{fma}\left(0.03125, \frac{{c}^{4}}{{b}^{8}} \cdot 256, \mathsf{fma}\left(0.16666666666666666, \frac{c \cdot \left(\frac{{c}^{3}}{{b}^{6}} \cdot -256\right)}{{b}^{2}}, \frac{{c}^{4}}{{b}^{8}} \cdot 0.6666666666666666\right)\right)\right)\right), \frac{{c}^{4} \cdot 64}{{b}^{7}}\right)\right)\right) - \frac{c}{b} \]

Alternative 2: 95.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / pow(b, 7.0)) * (20.0 / a))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (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 = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / (b ** 7.0d0)) * (20.0d0 / a))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / Math.pow(b, 7.0)) * (20.0 / a))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / math.pow(b, 7.0)) * (20.0 / a))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / (b ^ 7.0)) * (20.0 / a))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 96.4%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

5. Taylor expanded in c around 0 96.4%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
6. Step-by-step derivation

   1. distribute-rgt-out 96.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 96.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 96.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. *-commutative 96.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

   5. times-frac 96.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{4 + 16}{a}\right)}\right)\right) \]

7. Simplified 96.4%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}\right)}\right)\right) \]

8. Final simplification 96.4%

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

Alternative 3: 93.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 94.9%

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

   1. associate-+r+ 94.9%

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

   2. mul-1-neg 94.9%

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

   3. unsub-neg 94.9%

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

   4. mul-1-neg 94.9%

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

   5. unsub-neg 94.9%

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

   6. associate-*r/ 94.9%

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

   7. *-commutative 94.9%

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

   8. associate-/l* 94.9%

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

6. Simplified 94.9%

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

7. Final simplification 94.9%

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

Alternative 4: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -1.0) t_0 (- (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = t_0;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-1.0d0)) then
        tmp = t_0
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = t_0;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -1.0:
		tmp = t_0
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = t_0;
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = t_0;
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], t$95$0, (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -1

   1. Initial program 70.0%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 23.5%

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

      1. *-commutative 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in b around inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. distribute-neg-frac 87.0%

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

   6. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 5: 90.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 91.7%

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

   1. mul-1-neg 91.7%

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

   2. unsub-neg 91.7%

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

   3. mul-1-neg 91.7%

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

   4. distribute-neg-frac 91.7%

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

   5. associate-/l* 91.7%

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

6. Simplified 91.7%

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

7. Final simplification 91.7%

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

Alternative 6: 81.3% 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 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 81.2%

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

   1. mul-1-neg 81.2%

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

   2. distribute-neg-frac 81.2%

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

6. Simplified 81.2%

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

7. Final simplification 81.2%

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

Reproduce

herbie shell --seed 2023337 
(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)))