Quadratic roots, medium range

Percentage Accurate: 31.3% → 95.5%

Time: 11.8s

Alternatives: 5

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: 31.3% 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: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{4}\\ \mathsf{fma}\left(-1, \frac{c \cdot c}{\frac{{b}^{3}}{a}}, \mathsf{fma}\left(-0.25, \frac{4 \cdot t_0 + t_0 \cdot 16}{a \cdot {b}^{7}}, \mathsf{fma}\left(-1, \frac{c}{b}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 4.0)))
   (fma
    -1.0
    (/ (* c c) (/ (pow b 3.0) a))
    (fma
     -0.25
     (/ (+ (* 4.0 t_0) (* t_0 16.0)) (* a (pow b 7.0)))
     (fma -1.0 (/ c b) (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 4.0);
	return fma(-1.0, ((c * c) / (pow(b, 3.0) / a)), fma(-0.25, (((4.0 * t_0) + (t_0 * 16.0)) / (a * pow(b, 7.0))), fma(-1.0, (c / b), (-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 4.0
	return fma(-1.0, Float64(Float64(c * c) / Float64((b ^ 3.0) / a)), fma(-0.25, Float64(Float64(Float64(4.0 * t_0) + Float64(t_0 * 16.0)) / Float64(a * (b ^ 7.0))), fma(-1.0, Float64(c / b), Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]}, N[(-1.0 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(4.0 * t$95$0), $MachinePrecision] + N[(t$95$0 * 16.0), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.9%

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

2. Taylor expanded in b around inf 96.4%

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

   1. fma-def 96.4%

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

   2. associate-/l* 96.4%

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

   3. unpow2 96.4%

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

   4. fma-def 96.4%

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

4. Simplified 96.4%

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

   1. pow1 96.4%

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

   2. pow-prod-down 96.4%

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

6. Applied egg-rr 96.4%

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

   1. unpow1 96.4%

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

8. Simplified 96.4%

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

9. Taylor expanded in c around 0 96.4%

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

    1. metadata-eval 96.4%

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

    2. pow-sqr 96.4%

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

    3. unpow2 96.4%

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

    4. unpow2 96.4%

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

    5. metadata-eval 96.4%

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

    6. pow-plus 96.4%

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

    7. unpow3 96.4%

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

    8. associate-*r* 96.4%

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

    9. unswap-sqr 96.4%

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

    10. swap-sqr 96.4%

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

    11. swap-sqr 96.4%

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

    12. unpow2 96.4%

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

    13. unpow2 96.4%

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

    14. pow-sqr 96.4%

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

    15. metadata-eval 96.4%

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

11. Simplified 96.4%

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

12. Final simplification 96.4%

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

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

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 c 3.0) (pow b 5.0)))))
   (/ c b))
  (* a (/ c (/ (pow b 3.0) c)))))

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(c, 3.0) / pow(b, 5.0))))) - (c / b)) - (a * (c / (pow(b, 3.0) / c)));
}

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((c ^ 3.0) / (b ^ 5.0))))) - Float64(c / b)) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))))
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[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.9%

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

2. Taylor expanded in a around 0 96.4%

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

4. Simplified 96.4%

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

5. Taylor expanded in b around 0 96.4%

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

   1. associate-/l* 96.4%

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

   2. distribute-rgt-out 96.4%

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

   3. metadata-eval 96.4%

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

7. Simplified 96.4%

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

8. Final simplification 96.4%

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

Alternative 3: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

2. Taylor expanded in b around inf 95.1%

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

   1. +-commutative 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. +-commutative 95.1%

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

   5. fma-def 95.1%

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

   6. associate-/l* 95.1%

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

   7. associate-/r/ 95.1%

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

   8. unpow2 95.1%

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

   9. mul-1-neg 95.1%

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

   10. distribute-neg-frac 95.1%

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

   11. associate-/l* 95.1%

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

   12. associate-/r/ 95.1%

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

   13. unpow2 95.1%

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

   14. associate-/l* 95.1%

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

4. Simplified 95.1%

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

5. Final simplification 95.1%

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

Alternative 4: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) - (a * (c / ((b ** 3.0d0) / c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

2. Taylor expanded in b around inf 92.7%

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

   1. +-commutative 92.7%

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

   2. mul-1-neg 92.7%

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

   3. unsub-neg 92.7%

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

   4. mul-1-neg 92.7%

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

   5. distribute-neg-frac 92.7%

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

   6. associate-/l* 92.7%

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

   7. associate-/r/ 92.7%

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

   8. unpow2 92.7%

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

   9. associate-/l* 92.7%

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

4. Simplified 92.7%

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

5. Final simplification 92.7%

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

Alternative 5: 81.4% 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 25.9%

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

2. Taylor expanded in b around inf 85.2%

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

   1. mul-1-neg 85.2%

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

   2. distribute-neg-frac 85.2%

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

4. Simplified 85.2%

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

5. Final simplification 85.2%

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

Reproduce

herbie shell --seed 2023274 
(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)))