Quadratic roots, narrow range

Percentage Accurate: 55.1% → 99.3%

Time: 16.6s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.1% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \frac{\frac{t_0 + \frac{0}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + t_0 \cdot \mathsf{fma}\left(b, b, t_0\right)}}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (/
    (/
     (+ t_0 (/ 0.0 c))
     (-
      (- b)
      (sqrt
       (/
        (+ (pow b 6.0) (* -64.0 (pow (* c a) 3.0)))
        (+ (pow b 4.0) (* t_0 (fma b b t_0)))))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	return ((t_0 + (0.0 / c)) / (-b - sqrt(((pow(b, 6.0) + (-64.0 * pow((c * a), 3.0))) / (pow(b, 4.0) + (t_0 * fma(b, b, t_0))))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	return Float64(Float64(Float64(t_0 + Float64(0.0 / c)) / Float64(Float64(-b) - sqrt(Float64(Float64((b ^ 6.0) + Float64(-64.0 * (Float64(c * a) ^ 3.0))) / Float64((b ^ 4.0) + Float64(t_0 * fma(b, b, t_0))))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 + N[(0.0 / c), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(N[Power[b, 6.0], $MachinePrecision] + N[(-64.0 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 4.0], $MachinePrecision] + N[(t$95$0 * N[(b * b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.0%

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

   1. flip3-- 56.4%

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

   2. sqrt-div 56.1%

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

   3. pow2 56.1%

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

   4. pow-pow 56.2%

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

   5. metadata-eval 56.2%

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

   6. associate-*l* 56.2%

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

   7. pow2 56.2%

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

   8. pow2 56.2%

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

   9. pow-prod-up 56.4%

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

   10. metadata-eval 56.4%

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

   11. distribute-rgt-out 56.4%

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

3. Applied egg-rr 56.3%

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

   1. flip-+ 56.2%

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

5. Applied egg-rr 56.4%

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

7. Simplified 57.2%

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

8. Taylor expanded in c around inf 99.3%

   \[\leadsto \frac{\frac{\color{blue}{-64 \cdot \frac{-0.00390625 \cdot \frac{{b}^{4}}{a} + 0.00390625 \cdot \frac{{b}^{4}}{a}}{c} + 4 \cdot \left(c \cdot a\right)}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]
9. Step-by-step derivation

   1. +-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(c \cdot a\right) + -64 \cdot \frac{-0.00390625 \cdot \frac{{b}^{4}}{a} + 0.00390625 \cdot \frac{{b}^{4}}{a}}{c}}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   2. *-commutative 99.3%

      \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot 4} + -64 \cdot \frac{-0.00390625 \cdot \frac{{b}^{4}}{a} + 0.00390625 \cdot \frac{{b}^{4}}{a}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   3. associate-*r* 99.3%

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)} + -64 \cdot \frac{-0.00390625 \cdot \frac{{b}^{4}}{a} + 0.00390625 \cdot \frac{{b}^{4}}{a}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   4. distribute-rgt-out 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + -64 \cdot \frac{\color{blue}{\frac{{b}^{4}}{a} \cdot \left(-0.00390625 + 0.00390625\right)}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   5. metadata-eval 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + -64 \cdot \frac{\frac{{b}^{4}}{a} \cdot \color{blue}{0}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   6. metadata-eval 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + -64 \cdot \frac{\frac{{b}^{4}}{a} \cdot \color{blue}{\left(-0.25 + 0.25\right)}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   7. distribute-rgt-out 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + -64 \cdot \frac{\color{blue}{-0.25 \cdot \frac{{b}^{4}}{a} + 0.25 \cdot \frac{{b}^{4}}{a}}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   8. associate-*r/ 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \color{blue}{\frac{-64 \cdot \left(-0.25 \cdot \frac{{b}^{4}}{a} + 0.25 \cdot \frac{{b}^{4}}{a}\right)}{c}}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   9. distribute-rgt-out 99.3%

      \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \frac{-64 \cdot \color{blue}{\left(\frac{{b}^{4}}{a} \cdot \left(-0.25 + 0.25\right)\right)}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   10. metadata-eval 99.3%

       \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \frac{-64 \cdot \left(\frac{{b}^{4}}{a} \cdot \color{blue}{0}\right)}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   11. mul0-rgt 99.3%

       \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \frac{-64 \cdot \color{blue}{0}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

   12. metadata-eval 99.3%

       \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \frac{\color{blue}{0}}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

10. Simplified 99.3%

    \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right) + \frac{0}{c}}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{2 \cdot a} \]

11. Final simplification 99.3%

    \[\leadsto \frac{\frac{c \cdot \left(a \cdot 4\right) + \frac{0}{c}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{{b}^{4} + \left(c \cdot \left(a \cdot 4\right)\right) \cdot \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)}}}}{a \cdot 2} \]

Alternative 2: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (fma -2.0 (* (/ (pow c 3.0) (pow b 5.0)) (* a a)) (/ (- c) b))
    (* a (/ c (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), (-c / b)) - (a * (c / (pow(b, 3.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), Float64(Float64(-c) / b)) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $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. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 82.9%

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

   3. Simplified 83.1%

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

   if 2.10000000000000009 < b

   1. Initial program 48.3%

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

   2. Taylor expanded in b around inf 92.1%

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

      1. +-commutative 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. +-commutative 92.1%

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

      5. fma-def 92.1%

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

      6. associate-/l* 92.1%

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

      7. associate-/r/ 92.1%

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

      8. unpow2 92.1%

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

      9. mul-1-neg 92.1%

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

      10. distribute-neg-frac 92.1%

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

      11. associate-/l* 92.1%

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

      12. associate-/r/ 92.1%

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

      13. unpow2 92.1%

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

      14. associate-/l* 92.1%

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

   4. Simplified 92.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3: 85.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-c}{b} - \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 210.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (- c) b) (/ (pow c 3.0) (/ (pow b 5.0) (* a a))))
    (/ (* a (* c c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 210.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-c / b) - (pow(c, 3.0) / (pow(b, 5.0) / (a * a)))) - ((a * (c * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 210.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-c) / b) - Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a)))) - Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 210.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-c) / b), $MachinePrecision] - N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 210

   1. Initial program 79.4%

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

   3. Simplified 79.6%

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

   if 210 < b

   1. Initial program 42.6%

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

   2. Taylor expanded in b around inf 33.5%

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

      1. flip-+ 33.4%

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

      2. associate-/l* 33.4%

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

      3. associate-/r/ 33.4%

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

      4. associate-/l* 33.4%

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

      5. associate-/r/ 33.4%

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

      6. associate-/l* 33.4%

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

      7. associate-/r/ 33.4%

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

   4. Applied egg-rr 33.4%

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

   5. Taylor expanded in b around inf 38.9%

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

      1. unpow2 38.9%

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

      2. fma-def 38.7%

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

   7. Simplified 38.7%

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

   8. Taylor expanded in b around inf 91.9%

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

      1. +-commutative 91.9%

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

      2. mul-1-neg 91.9%

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

      3. unsub-neg 91.9%

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

   10. Simplified 91.9%

       \[\leadsto \color{blue}{\left(\frac{-c}{b} - \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-c}{b} - \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-2 \cdot \left(\frac{c}{b} \cdot \left(-a\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 210.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (/ (* c (* a 4.0)) (- (- (* -2.0 (* (/ c b) (- a))) b) b)) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 210.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 210.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * 4.0)) / Float64(Float64(Float64(-2.0 * Float64(Float64(c / b) * Float64(-a))) - b) - b)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 210.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-2.0 * N[(N[(c / b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 210

   1. Initial program 79.4%

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

   3. Simplified 79.6%

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

   if 210 < b

   1. Initial program 42.6%

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

   2. Taylor expanded in b around inf 33.5%

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

      1. flip-+ 33.4%

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

      2. associate-/l* 33.4%

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

      3. associate-/r/ 33.4%

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

      4. associate-/l* 33.4%

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

      5. associate-/r/ 33.4%

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

      6. associate-/l* 33.4%

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

      7. associate-/r/ 33.4%

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

   4. Applied egg-rr 33.4%

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

   5. Taylor expanded in b around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-*l* 91.7%

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

   7. Simplified 91.7%

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-2 \cdot \left(\frac{c}{b} \cdot \left(-a\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \]

Alternative 5: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-2 \cdot \left(\frac{c}{b} \cdot \left(-a\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 210.0)
   (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
   (/ (/ (* c (* a 4.0)) (- (- (* -2.0 (* (/ c b) (- a))) b) b)) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 210.0) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0);
	}
	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) :: tmp
    if (b <= 210.0d0) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = ((c * (a * 4.0d0)) / ((((-2.0d0) * ((c / b) * -a)) - b) - b)) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 210.0) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 210.0:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 210.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(a * 4.0)) / Float64(Float64(Float64(-2.0 * Float64(Float64(c / b) * Float64(-a))) - b) - b)) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 210.0)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 210.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-2.0 * N[(N[(c / b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 210

   1. Initial program 79.4%

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

   3. Simplified 79.6%

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

      1. *-commutative 79.6%

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

      2. metadata-eval 79.6%

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

      3. distribute-lft-neg-in 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

      5. *-commutative 79.6%

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

      6. fma-neg 79.4%

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

      7. associate-*l* 79.4%

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

   5. Applied egg-rr 79.4%

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

   if 210 < b

   1. Initial program 42.6%

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

   2. Taylor expanded in b around inf 33.5%

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

      1. flip-+ 33.4%

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

      2. associate-/l* 33.4%

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

      3. associate-/r/ 33.4%

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

      4. associate-/l* 33.4%

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

      5. associate-/r/ 33.4%

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

      6. associate-/l* 33.4%

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

      7. associate-/r/ 33.4%

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

   4. Applied egg-rr 33.4%

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

   5. Taylor expanded in b around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-*l* 91.7%

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

   7. Simplified 91.7%

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a\right)\right)}}{2 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \left(a \cdot 4\right)}{\left(-2 \cdot \left(\frac{c}{b} \cdot \left(-a\right)\right) - b\right) - b}}{a \cdot 2}\\ \end{array} \]

Alternative 6: 82.0% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return ((c * (a * 4.0)) / (((-2.0 * ((c / b) * -a)) - b) - b)) / (a * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in b around inf 35.6%

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

   1. flip-+ 35.5%

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

   2. associate-/l* 35.5%

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

   3. associate-/r/ 35.5%

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

   4. associate-/l* 35.5%

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

   5. associate-/r/ 35.5%

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

   6. associate-/l* 35.5%

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

   7. associate-/r/ 35.5%

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

4. Applied egg-rr 35.5%

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

5. Taylor expanded in b around inf 80.5%

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

   1. *-commutative 80.5%

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

   2. associate-*l* 80.5%

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

7. Simplified 80.5%

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

8. Final simplification 80.5%

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

Alternative 7: 64.6% 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 57.0%

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

2. Taylor expanded in b around inf 63.0%

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

   1. mul-1-neg 63.0%

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

   2. distribute-neg-frac 63.0%

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

4. Simplified 63.0%

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

5. Final simplification 63.0%

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

Alternative 8: 1.6% accurate, 38.7× 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(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 57.0%

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

2. Taylor expanded in b around -inf 11.7%

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

   1. mul-1-neg 11.7%

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

   2. unsub-neg 11.7%

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

4. Simplified 11.7%

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

5. Taylor expanded in c around inf 1.6%

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

6. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023274 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))