Quadratic roots, medium range

Percentage Accurate: 32.0% → 96.9%

Time: 28.1s

Alternatives: 10

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: 32.0% 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: 96.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}} \cdot 20\\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{t\_0}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{56 \cdot {c}^{5}}{{b}^{10}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{t\_0}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{{c}^{4} \cdot 20}{{b}^{8}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (pow c 4.0) (pow b 6.0)) 20.0)))
   (-
    (*
     a
     (-
      (*
       a
       (fma
        -2.0
        (/ (pow c 3.0) (pow b 5.0))
        (*
         a
         (fma
          -0.25
          (/ t_0 b)
          (*
           a
           (*
            -0.25
            (+
             (*
              a
              (/
               (fma
                2.0
                (* c (/ (* 56.0 (pow c 5.0)) (pow b 10.0)))
                (fma
                 2.0
                 (* (pow c 2.0) (/ t_0 (pow b 4.0)))
                 (* 16.0 (/ (pow c 6.0) (pow b 10.0)))))
               b))
             (/
              (fma
               2.0
               (* c (/ (* (pow c 4.0) 20.0) (pow b 8.0)))
               (* 16.0 (/ (pow c 5.0) (pow b 8.0))))
              b))))))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = (pow(c, 4.0) / pow(b, 6.0)) * 20.0;
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (a * fma(-0.25, (t_0 / b), (a * (-0.25 * ((a * (fma(2.0, (c * ((56.0 * pow(c, 5.0)) / pow(b, 10.0))), fma(2.0, (pow(c, 2.0) * (t_0 / pow(b, 4.0))), (16.0 * (pow(c, 6.0) / pow(b, 10.0))))) / b)) + (fma(2.0, (c * ((pow(c, 4.0) * 20.0) / pow(b, 8.0))), (16.0 * (pow(c, 5.0) / pow(b, 8.0)))) / b)))))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(a * fma(-0.25, Float64(t_0 / b), Float64(a * Float64(-0.25 * Float64(Float64(a * Float64(fma(2.0, Float64(c * Float64(Float64(56.0 * (c ^ 5.0)) / (b ^ 10.0))), fma(2.0, Float64((c ^ 2.0) * Float64(t_0 / (b ^ 4.0))), Float64(16.0 * Float64((c ^ 6.0) / (b ^ 10.0))))) / b)) + Float64(fma(2.0, Float64(c * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 8.0))), Float64(16.0 * Float64((c ^ 5.0) / (b ^ 8.0)))) / b)))))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]}, N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.25 * N[(t$95$0 / b), $MachinePrecision] + N[(a * N[(-0.25 * N[(N[(a * N[(N[(2.0 * N[(c * N[(N[(56.0 * N[Power[c, 5.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[c, 2.0], $MachinePrecision] * N[(t$95$0 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 6.0], $MachinePrecision] / N[Power[b, 10.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(c * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 5.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 97.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.25 \cdot \frac{4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b} + a \cdot \left(-0.25 \cdot \frac{a \cdot \left(2 \cdot \frac{c \cdot \left(2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}} + \left(2 \cdot \frac{{c}^{2} \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{4}} + 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + -0.25 \cdot \frac{2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}}{b}\right)\right)\right)\right)} \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{{b}^{2}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Taylor expanded in c around 0 97.2%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\left(56 \cdot \frac{{c}^{5}}{{b}^{10}}\right)}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
9. Step-by-step derivation

   1. associate-*r/ 97.2%

      \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\frac{56 \cdot {c}^{5}}{{b}^{10}}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

10. Simplified 97.2%

    \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\frac{56 \cdot {c}^{5}}{{b}^{10}}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

11. Taylor expanded in c around 0 97.2%

    \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{56 \cdot {c}^{5}}{{b}^{10}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\left(20 \cdot \frac{{c}^{4}}{{b}^{8}}\right)}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
12. Step-by-step derivation

    1. associate-*r/ 97.2%

       \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{56 \cdot {c}^{5}}{{b}^{10}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{8}}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

13. Simplified 97.2%

    \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{56 \cdot {c}^{5}}{{b}^{10}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \color{blue}{\frac{20 \cdot {c}^{4}}{{b}^{8}}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]

14. Final simplification 97.2%

    \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \mathsf{fma}\left(-0.25, \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, a \cdot \left(-0.25 \cdot \left(a \cdot \frac{\mathsf{fma}\left(2, c \cdot \frac{56 \cdot {c}^{5}}{{b}^{10}}, \mathsf{fma}\left(2, {c}^{2} \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{4}}, 16 \cdot \frac{{c}^{6}}{{b}^{10}}\right)\right)}{b} + \frac{\mathsf{fma}\left(2, c \cdot \frac{{c}^{4} \cdot 20}{{b}^{8}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
15. Add Preprocessing

Alternative 2: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}} \cdot 20\\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.25 \cdot \left(\frac{t\_0}{b} + \frac{a \cdot \mathsf{fma}\left(2, c \cdot \frac{t\_0}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (pow c 4.0) (pow b 6.0)) 20.0)))
   (-
    (*
     a
     (-
      (*
       a
       (fma
        -2.0
        (/ (pow c 3.0) (pow b 5.0))
        (*
         a
         (*
          -0.25
          (+
           (/ t_0 b)
           (/
            (*
             a
             (fma
              2.0
              (* c (/ t_0 (pow b 2.0)))
              (* 16.0 (/ (pow c 5.0) (pow b 8.0)))))
            b))))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = (pow(c, 4.0) / pow(b, 6.0)) * 20.0;
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (a * (-0.25 * ((t_0 / b) + ((a * fma(2.0, (c * (t_0 / pow(b, 2.0))), (16.0 * (pow(c, 5.0) / pow(b, 8.0))))) / b)))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(a * Float64(-0.25 * Float64(Float64(t_0 / b) + Float64(Float64(a * fma(2.0, Float64(c * Float64(t_0 / (b ^ 2.0))), Float64(16.0 * Float64((c ^ 5.0) / (b ^ 8.0))))) / b)))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]}, N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.25 * N[(N[(t$95$0 / b), $MachinePrecision] + N[(N[(a * N[(2.0 * N[(c * N[(t$95$0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 5.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 96.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + a \cdot \left(-0.25 \cdot \frac{a \cdot \left(2 \cdot \frac{c \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{b}^{2}} + 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b} + -0.25 \cdot \frac{4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}}{b}\right)\right)\right)} \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.25 \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b} + \frac{a \cdot \mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Final simplification 96.7%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, a \cdot \left(-0.25 \cdot \left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b} + \frac{a \cdot \mathsf{fma}\left(2, c \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{{b}^{2}}, 16 \cdot \frac{{c}^{5}}{{b}^{8}}\right)}{b}\right)\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
9. Add Preprocessing

Alternative 3: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, -0.25 \cdot \left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (fma
      -2.0
      (/ (pow c 3.0) (pow b 5.0))
      (* -0.25 (* a (/ (* (/ (pow c 4.0) (pow b 6.0)) 20.0) b)))))
    (/ (pow c 2.0) (pow b 3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (-0.25 * (a * (((pow(c, 4.0) / pow(b, 6.0)) * 20.0) / b))))) - (pow(c, 2.0) / pow(b, 3.0)))) - (c / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(-0.25 * Float64(a * Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0) / b))))) - Float64((c ^ 2.0) / (b ^ 3.0)))) - Float64(c / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(a * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 96.0%

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

   1. +-commutative 96.0%

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

   2. mul-1-neg 96.0%

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

   3. unsub-neg 96.0%

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

7. Simplified 96.0%

   \[\leadsto \color{blue}{a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, \left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}\right) \cdot -0.25\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}} \]

8. Final simplification 96.0%

   \[\leadsto a \cdot \left(a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, -0.25 \cdot \left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}\right)\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b} \]
9. Add Preprocessing

Alternative 4: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -4 + -0.5 \cdot \frac{a \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (+
    (* -2.0 (/ c b))
    (*
     a
     (+
      (* -2.0 (/ (pow c 2.0) (pow b 3.0)))
      (*
       a
       (+
        (* (/ (pow c 3.0) (pow b 5.0)) -4.0)
        (* -0.5 (/ (* a (/ (* (pow c 4.0) 20.0) (pow b 6.0))) b))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (pow(c, 2.0) / pow(b, 3.0))) + (a * (((pow(c, 3.0) / pow(b, 5.0)) * -4.0) + (-0.5 * ((a * ((pow(c, 4.0) * 20.0) / pow(b, 6.0))) / 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 = (a * (((-2.0d0) * (c / b)) + (a * (((-2.0d0) * ((c ** 2.0d0) / (b ** 3.0d0))) + (a * ((((c ** 3.0d0) / (b ** 5.0d0)) * (-4.0d0)) + ((-0.5d0) * ((a * (((c ** 4.0d0) * 20.0d0) / (b ** 6.0d0))) / b)))))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (Math.pow(c, 2.0) / Math.pow(b, 3.0))) + (a * (((Math.pow(c, 3.0) / Math.pow(b, 5.0)) * -4.0) + (-0.5 * ((a * ((Math.pow(c, 4.0) * 20.0) / Math.pow(b, 6.0))) / b)))))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (a * ((-2.0 * (c / b)) + (a * ((-2.0 * (math.pow(c, 2.0) / math.pow(b, 3.0))) + (a * (((math.pow(c, 3.0) / math.pow(b, 5.0)) * -4.0) + (-0.5 * ((a * ((math.pow(c, 4.0) * 20.0) / math.pow(b, 6.0))) / b)))))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(-2.0 * Float64(c / b)) + Float64(a * Float64(Float64(-2.0 * Float64((c ^ 2.0) / (b ^ 3.0))) + Float64(a * Float64(Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * -4.0) + Float64(-0.5 * Float64(Float64(a * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 6.0))) / b)))))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((-2.0 * (c / b)) + (a * ((-2.0 * ((c ^ 2.0) / (b ^ 3.0))) + (a * ((((c ^ 3.0) / (b ^ 5.0)) * -4.0) + (-0.5 * ((a * (((c ^ 4.0) * 20.0) / (b ^ 6.0))) / b)))))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-2.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] + N[(-0.5 * N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 95.9%

   \[\leadsto \frac{\color{blue}{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)\right)}}{a \cdot 2} \]
6. Step-by-step derivation

   1. distribute-rgt-out 95.9%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \color{blue}{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(4 + 16\right)\right)}}{b}\right)\right)\right)}{a \cdot 2} \]

   2. metadata-eval 95.9%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot \color{blue}{20}\right)}{b}\right)\right)\right)}{a \cdot 2} \]

   3. associate-*l/ 95.9%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \color{blue}{\frac{{c}^{4} \cdot 20}{{b}^{6}}}}{b}\right)\right)\right)}{a \cdot 2} \]

7. Applied egg-rr 95.9%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.5 \cdot \frac{a \cdot \color{blue}{\frac{{c}^{4} \cdot 20}{{b}^{6}}}}{b}\right)\right)\right)}{a \cdot 2} \]

8. Final simplification 95.9%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -4 + -0.5 \cdot \frac{a \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right)\right)\right)}{a \cdot 2} \]
9. Add Preprocessing

Alternative 5: 93.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 94.6%

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

   1. +-commutative 94.6%

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

   2. mul-1-neg 94.6%

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

   3. unsub-neg 94.6%

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

   4. mul-1-neg 94.6%

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

   5. unsub-neg 94.6%

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

7. Simplified 94.6%

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

8. Final simplification 94.6%

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

Alternative 6: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / 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 * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-2.0 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * ((-2.0 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) - (a / math.pow(b, 3.0)))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-2.0 * ((c * (a ^ 2.0)) / (b ^ 5.0))) - (a / (b ^ 3.0)))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in c around 0 94.3%

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

6. Final simplification 94.3%

   \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right) \]
7. Add Preprocessing

Alternative 7: 90.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 / b) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in a around 0 91.7%

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

   1. mul-1-neg 91.7%

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

   2. unsub-neg 91.7%

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

   3. mul-1-neg 91.7%

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

   4. distribute-neg-frac2 91.7%

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

7. Simplified 91.7%

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

8. Final simplification 91.7%

   \[\leadsto \frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 8: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* a (/ c (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / pow(b, 3.0))));
}

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 * (((-1.0d0) / b) - (a * (c / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * (c / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * (c / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(c / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * (c / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in b around inf 91.6%

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

   1. mul-1-neg 91.6%

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

   2. unsub-neg 91.6%

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

   3. mul-1-neg 91.6%

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

7. Simplified 91.6%

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

8. Taylor expanded in a around inf 91.5%

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

   1. sub-neg 91.5%

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

   2. sub-neg 91.5%

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

   3. associate-*r/ 91.5%

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

   4. neg-mul-1 91.5%

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

10. Simplified 91.5%

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

11. Taylor expanded in c around 0 91.4%

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

    1. sub-neg 91.4%

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

    2. distribute-neg-frac 91.4%

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

    3. metadata-eval 91.4%

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

    4. +-commutative 91.4%

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

    5. mul-1-neg 91.4%

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

    6. unsub-neg 91.4%

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

    7. associate-/l* 91.4%

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

13. Simplified 91.4%

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

14. Final simplification 91.4%

    \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \frac{c}{{b}^{3}}\right) \]
15. Add Preprocessing

Alternative 9: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- -1.0 (* a (/ c (pow b 2.0))))) b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 - (a * (c / pow(b, 2.0))))) / 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 * ((-1.0d0) - (a * (c / (b ** 2.0d0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 - (a * (c / Math.pow(b, 2.0))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 - (a * (c / math.pow(b, 2.0))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 - Float64(a * Float64(c / (b ^ 2.0))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 - (a * (c / (b ^ 2.0))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 - N[(a * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in b around inf 91.6%

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

   1. mul-1-neg 91.6%

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

   2. unsub-neg 91.6%

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

   3. mul-1-neg 91.6%

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

7. Simplified 91.6%

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

8. Taylor expanded in c around 0 91.6%

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

   1. sub-neg 91.6%

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

   2. metadata-eval 91.6%

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

   3. +-commutative 91.6%

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

   4. mul-1-neg 91.6%

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

   5. unsub-neg 91.6%

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

   6. associate-/l* 91.6%

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

10. Simplified 91.6%

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

11. Final simplification 91.6%

    \[\leadsto \frac{c \cdot \left(-1 - a \cdot \frac{c}{{b}^{2}}\right)}{b} \]
12. Add Preprocessing

Alternative 10: 80.8% 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 30.8%

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

   1. *-commutative 30.8%

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

3. Simplified 30.8%

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

5. Taylor expanded in b around inf 82.0%

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

   1. associate-*r/ 82.0%

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

   2. mul-1-neg 82.0%

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

7. Simplified 82.0%

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

8. Final simplification 82.0%

   \[\leadsto \frac{-c}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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)))