Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.6%

Time: 11.4s

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: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

   1. flip3-- 28.9%

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

   2. div-inv 28.9%

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

   3. pow2 28.9%

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

   4. pow-pow 28.9%

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

   5. metadata-eval 28.9%

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

   6. associate-*l* 28.9%

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

   7. unpow-prod-down 28.9%

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

   8. metadata-eval 28.9%

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

   9. pow2 28.9%

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

   10. pow2 28.9%

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

   11. pow-prod-up 28.8%

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

   12. metadata-eval 28.8%

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

   13. distribute-rgt-out 28.8%

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

5. Applied egg-rr 28.8%

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

   1. flip-+ 28.8%

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

7. Applied egg-rr 29.7%

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

8. Taylor expanded in b around 0 99.4%

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

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

10. Simplified 99.4%

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

12. Applied egg-rr 37.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}{\mathsf{fma}\left(c \cdot \left(a \cdot 4\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right), {b}^{4}\right)}}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 86.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot \left(a \cdot 4\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\frac{\mathsf{fma}\left(-64, {\left(c \cdot a\right)}^{3}, {b}^{6}\right)}{\mathsf{fma}\left(c \cdot \left(a \cdot 4\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right), {b}^{4}\right)}}\right)}\right)\right)} \]

    2. expm1-log1p 99.3%

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

    3. associate-/r* 99.6%

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

    4. associate-*r* 99.6%

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

    5. *-commutative 99.6%

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

    6. times-frac 99.6%

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

    7. metadata-eval 99.6%

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

    8. *-commutative 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -2e-10) 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 <= -2e-10) {
		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 <= (-2d-10)) 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 <= -2e-10) {
		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 <= -2e-10:
		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 <= -2e-10)
		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 <= -2e-10)
		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, -2e-10], 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)) < -2.00000000000000007e-10

   1. Initial program 67.2%

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

   if -2.00000000000000007e-10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 9.4%

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

      1. *-commutative 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in b around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. distribute-neg-frac 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

   1. flip3-- 28.9%

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

   2. div-inv 28.9%

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

   3. pow2 28.9%

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

   4. pow-pow 28.9%

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

   5. metadata-eval 28.9%

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

   6. associate-*l* 28.9%

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

   7. unpow-prod-down 28.9%

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

   8. metadata-eval 28.9%

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

   9. pow2 28.9%

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

   10. pow2 28.9%

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

   11. pow-prod-up 28.8%

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

   12. metadata-eval 28.8%

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

   13. distribute-rgt-out 28.8%

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

5. Applied egg-rr 28.8%

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

   1. flip-+ 28.8%

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

7. Applied egg-rr 29.7%

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

8. Taylor expanded in b around 0 99.4%

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

   1. associate-*r* 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in b around 0 99.5%

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

12. Final simplification 99.5%

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

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.000225:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.000225)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.000225) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.000225)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.000225], 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[((-c) / b), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2499999999999999e-4

   1. Initial program 81.2%

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

   3. Simplified 81.3%

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

   if 2.2499999999999999e-4 < b

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

   3. Simplified 24.4%

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

   4. Taylor expanded in b around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. unsub-neg 94.5%

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

      3. mul-1-neg 94.5%

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

      4. distribute-neg-frac 94.5%

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

      5. associate-/l* 94.5%

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

   6. Simplified 94.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.000225:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 5: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.000225:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.000225)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.000225) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 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 <= 0.000225d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - (a / ((b ** 3.0d0) / (c ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.000225) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (Math.pow(b, 3.0) / Math.pow(c, 2.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.000225:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - (a / (math.pow(b, 3.0) / math.pow(c, 2.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.000225)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.000225)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - (a / ((b ^ 3.0) / (c ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.000225], 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], N[(N[((-c) / b), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2499999999999999e-4

   1. Initial program 81.2%

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

   if 2.2499999999999999e-4 < b

   1. Initial program 24.4%

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

      1. *-commutative 24.4%

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

   3. Simplified 24.4%

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

   4. Taylor expanded in b around inf 94.5%

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

      1. mul-1-neg 94.5%

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

      2. unsub-neg 94.5%

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

      3. mul-1-neg 94.5%

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

      4. distribute-neg-frac 94.5%

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

      5. associate-/l* 94.5%

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

   6. Simplified 94.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.000225:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 6: 81.2% 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 28.8%

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

   1. *-commutative 28.8%

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

3. Simplified 28.8%

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

4. Taylor expanded in b around inf 82.6%

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

   1. mul-1-neg 82.6%

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

   2. distribute-neg-frac 82.6%

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

6. Simplified 82.6%

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

7. Final simplification 82.6%

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

Reproduce

herbie shell --seed 2023309 
(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)))