Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.2%

Time: 16.4s

Alternatives: 7

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in a around 0 96.7%

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

4. Simplified 96.7%

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

5. Final simplification 96.7%

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

Alternative 2: 94.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, c \cdot \frac{a}{b} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -4.0
   (/ (pow (* a c) 3.0) (pow b 5.0))
   (fma
    -2.0
    (+ (* c (/ a b)) (/ a (/ (/ (pow b 3.0) (* c c)) a)))
    (* -0.5 (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return fma(-4.0, (pow((a * c), 3.0) / pow(b, 5.0)), fma(-2.0, ((c * (a / b)) + (a / ((pow(b, 3.0) / (c * c)) / a))), (-0.5 * (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(fma(-4.0, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), fma(-2.0, Float64(Float64(c * Float64(a / b)) + Float64(a / Float64(Float64((b ^ 3.0) / Float64(c * c)) / a))), Float64(-0.5 * Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(-4.0 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in b around inf 96.3%

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

4. Simplified 96.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \frac{-0.5 \cdot \mathsf{fma}\left(16, {a}^{4} \cdot {c}^{4}, 4 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

5. Taylor expanded in a around 0 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{-0.5 \cdot \frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{2 \cdot a} \]
6. Step-by-step derivation

   1. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right) \cdot {a}^{4}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   2. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)} \cdot {a}^{4}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   3. associate-*l* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(\left(4 + 16\right) \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   4. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   5. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   6. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   7. associate-*r* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   8. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   9. associate-/l* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{\frac{{b}^{7}}{4 + 16}}}\right)\right)}{2 \cdot a} \]

7. Simplified 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{-0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}}\right)\right)}{2 \cdot a} \]

8. Taylor expanded in a around 0 96.3%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \color{blue}{\frac{a \cdot c}{b}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]
9. Step-by-step derivation

   1. associate-*l/ 96.3%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \color{blue}{\frac{a}{b} \cdot c} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]

   2. *-commutative 96.3%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]

10. Simplified 96.3%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]

11. Final simplification 96.3%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, c \cdot \frac{a}{b} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \]

Alternative 3: 94.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4, \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}} + \frac{a}{\frac{b}{c}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -4.0
   (/ (* (* a c) (* (* a c) (* a c))) (pow b 5.0))
   (fma
    -2.0
    (+ (/ a (/ (/ (pow b 3.0) (* c c)) a)) (/ a (/ b c)))
    (* -0.5 (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return fma(-4.0, (((a * c) * ((a * c) * (a * c))) / pow(b, 5.0)), fma(-2.0, ((a / ((pow(b, 3.0) / (c * c)) / a)) + (a / (b / c))), (-0.5 * (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(fma(-4.0, Float64(Float64(Float64(a * c) * Float64(Float64(a * c) * Float64(a * c))) / (b ^ 5.0)), fma(-2.0, Float64(Float64(a / Float64(Float64((b ^ 3.0) / Float64(c * c)) / a)) + Float64(a / Float64(b / c))), Float64(-0.5 * Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(-4.0 * N[(N[(N[(a * c), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in b around inf 96.3%

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

4. Simplified 96.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \frac{-0.5 \cdot \mathsf{fma}\left(16, {a}^{4} \cdot {c}^{4}, 4 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{7}}\right)\right)}}{2 \cdot a} \]

5. Taylor expanded in a around 0 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{-0.5 \cdot \frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{2 \cdot a} \]
6. Step-by-step derivation

   1. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right) \cdot {a}^{4}}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   2. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)} \cdot {a}^{4}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   3. associate-*l* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(\left(4 + 16\right) \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   4. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   5. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   6. distribute-rgt-out 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   7. associate-*r* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   8. *-commutative 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{2 \cdot a} \]

   9. associate-/l* 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \color{blue}{\frac{{a}^{4} \cdot {c}^{4}}{\frac{{b}^{7}}{4 + 16}}}\right)\right)}{2 \cdot a} \]

7. Simplified 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, \color{blue}{-0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}}\right)\right)}{2 \cdot a} \]
8. Step-by-step derivation

   1. unpow3 96.2%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]

9. Applied egg-rr 96.2%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}} + \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{2 \cdot a} \]

10. Final simplification 96.2%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\left(a \cdot c\right) \cdot \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{a}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}} + \frac{a}{\frac{b}{c}}, -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \]

Alternative 4: 93.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in b around inf 95.4%

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

   1. associate-+r+ 95.4%

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

   2. mul-1-neg 95.4%

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

   3. unsub-neg 95.4%

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

   4. mul-1-neg 95.4%

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

   5. unsub-neg 95.4%

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

   6. associate-/l* 95.4%

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

   7. associate-*r/ 95.4%

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

   8. unpow2 95.4%

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

   9. associate-/l* 95.4%

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

   10. associate-/r/ 95.4%

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

   11. unpow2 95.4%

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

4. Simplified 95.4%

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

5. Final simplification 95.4%

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

Alternative 5: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in b around inf 92.4%

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

   1. mul-1-neg 92.4%

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

   2. unsub-neg 92.4%

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

   3. mul-1-neg 92.4%

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

   4. distribute-neg-frac 92.4%

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

   5. associate-/l* 92.4%

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

   6. associate-/r/ 92.4%

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

   7. unpow2 92.4%

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

4. Simplified 92.4%

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

5. Final simplification 92.4%

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

Alternative 6: 90.7% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

2. Taylor expanded in b around inf 20.5%

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

   1. flip-+ 20.5%

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

   2. associate-/l* 20.5%

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

   3. associate-/r/ 20.5%

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

   4. associate-/l* 20.5%

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

   5. associate-/r/ 20.5%

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

   6. associate-/l* 20.5%

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

   7. associate-/r/ 20.5%

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

4. Applied egg-rr 20.5%

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

   1. sqr-neg 20.5%

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

   2. *-commutative 20.5%

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

   3. *-commutative 20.5%

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

   4. *-commutative 20.5%

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

6. Simplified 20.5%

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

7. Taylor expanded in b around inf 92.4%

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

8. Final simplification 92.4%

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

Alternative 7: 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 28.7%

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

2. Taylor expanded in b around inf 83.5%

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

   1. mul-1-neg 83.5%

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

   2. distribute-neg-frac 83.5%

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

4. Simplified 83.5%

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

5. Final simplification 83.5%

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

Reproduce

herbie shell --seed 2023275 
(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)))