Quadratic roots, medium range

Percentage Accurate: 32.0% → 95.4%

Time: 14.9s

Alternatives: 11

Speedup: 3.3×

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: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(c \cdot c\right)\right)\\ t_1 := c \cdot \left(c \cdot a\right)\\ t_2 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \frac{\mathsf{fma}\left(-2, \frac{t\_1 \cdot \left(c \cdot a\right)}{t\_2}, -5 \cdot \frac{t\_0 \cdot t\_0}{a \cdot \left(\left(b \cdot b\right) \cdot t\_2\right)} - \frac{t\_1}{b \cdot b}\right) - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* a (* c c))))
        (t_1 (* c (* c a)))
        (t_2 (* b (* b (* b b)))))
   (/
    (-
     (fma
      -2.0
      (/ (* t_1 (* c a)) t_2)
      (- (* -5.0 (/ (* t_0 t_0) (* a (* (* b b) t_2)))) (/ t_1 (* b b))))
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = a * (a * (c * c));
	double t_1 = c * (c * a);
	double t_2 = b * (b * (b * b));
	return (fma(-2.0, ((t_1 * (c * a)) / t_2), ((-5.0 * ((t_0 * t_0) / (a * ((b * b) * t_2)))) - (t_1 / (b * b)))) - c) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(a * Float64(c * c)))
	t_1 = Float64(c * Float64(c * a))
	t_2 = Float64(b * Float64(b * Float64(b * b)))
	return Float64(Float64(fma(-2.0, Float64(Float64(t_1 * Float64(c * a)) / t_2), Float64(Float64(-5.0 * Float64(Float64(t_0 * t_0) / Float64(a * Float64(Float64(b * b) * t_2)))) - Float64(t_1 / Float64(b * b)))) - c) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-2.0 * N[(N[(t$95$1 * N[(c * a), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(-5.0 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 95.0%

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

7. Applied egg-rr 95.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\color{blue}{\frac{-1}{4} \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot \frac{1}{20}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\frac{-1}{4} \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\color{blue}{\frac{1}{20} \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   3. times-frac N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{\frac{\frac{-1}{4}}{\frac{1}{20}} \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{\frac{\frac{-1}{4}}{\frac{1}{20}} \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

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

9. Applied egg-rr 95.0%

   \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{-5 \cdot \frac{\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]
10. Add Preprocessing

Alternative 2: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot \left(c \cdot a\right)\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(-5 \cdot t\_0, \frac{t\_0}{t\_1 \cdot \left(b \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(-2 \cdot \left(c \cdot a\right)\right)}{b \cdot t\_1}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a (* c a)))) (t_1 (* b (* b b))))
   (/
    (-
     (fma
      (* -5.0 t_0)
      (/ t_0 (* t_1 (* b (* a (* b b)))))
      (/ (* (* a (* c c)) (* -2.0 (* c a))) (* b t_1)))
     (fma (* c c) (/ a (* b b)) c))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * (c * a));
	double t_1 = b * (b * b);
	return (fma((-5.0 * t_0), (t_0 / (t_1 * (b * (a * (b * b))))), (((a * (c * c)) * (-2.0 * (c * a))) / (b * t_1))) - fma((c * c), (a / (b * b)), c)) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * Float64(c * a)))
	t_1 = Float64(b * Float64(b * b))
	return Float64(Float64(fma(Float64(-5.0 * t_0), Float64(t_0 / Float64(t_1 * Float64(b * Float64(a * Float64(b * b))))), Float64(Float64(Float64(a * Float64(c * c)) * Float64(-2.0 * Float64(c * a))) / Float64(b * t_1))) - fma(Float64(c * c), Float64(a / Float64(b * b)), c)) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(-5.0 * t$95$0), $MachinePrecision] * N[(t$95$0 / N[(t$95$1 * N[(b * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 95.0%

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

7. Applied egg-rr 95.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]
8. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\color{blue}{\frac{-1}{4} \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot \frac{1}{20}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\frac{-1}{4} \cdot \left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\color{blue}{\frac{1}{20} \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   3. times-frac N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{\frac{\frac{-1}{4}}{\frac{1}{20}} \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{\frac{\frac{-1}{4}}{\frac{1}{20}} \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)}} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}{b} \]

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

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

9. Applied egg-rr 95.0%

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

11. Applied egg-rr 95.0%

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

12. Final simplification 95.0%

    \[\leadsto \frac{\mathsf{fma}\left(-5 \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right), \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot \left(-2 \cdot \left(c \cdot a\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]
13. Add Preprocessing

Alternative 3: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_1 := a \cdot \left(a \cdot \left(c \cdot c\right)\right)\\ \frac{\mathsf{fma}\left(-5, \frac{t\_1 \cdot t\_1}{a \cdot \left(\left(b \cdot b\right) \cdot t\_0\right)}, \frac{-2 \cdot \left(c \cdot t\_1\right)}{t\_0}\right) - c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b)))) (t_1 (* a (* a (* c c)))))
   (/
    (-
     (fma
      -5.0
      (/ (* t_1 t_1) (* a (* (* b b) t_0)))
      (/ (* -2.0 (* c t_1)) t_0))
     (* c (fma a (/ c (* b b)) 1.0)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	double t_1 = a * (a * (c * c));
	return (fma(-5.0, ((t_1 * t_1) / (a * ((b * b) * t_0))), ((-2.0 * (c * t_1)) / t_0)) - (c * fma(a, (c / (b * b)), 1.0))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	t_1 = Float64(a * Float64(a * Float64(c * c)))
	return Float64(Float64(fma(-5.0, Float64(Float64(t_1 * t_1) / Float64(a * Float64(Float64(b * b) * t_0))), Float64(Float64(-2.0 * Float64(c * t_1)) / t_0)) - Float64(c * fma(a, Float64(c / Float64(b * b)), 1.0))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-5.0 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(a * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(c * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 95.0%

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

7. Applied egg-rr 95.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

9. Applied egg-rr 94.9%

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

10. Final simplification 94.9%

    \[\leadsto \frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{a \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{-2 \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)}{b} \]
11. Add Preprocessing

Alternative 4: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{a \cdot 2}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c a) -4.0 (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -500.0)
     (/ (/ (- t_0 (* b b)) (* a 2.0)) (+ b (sqrt t_0)))
     (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * a), -4.0, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -500.0) {
		tmp = ((t_0 - (b * b)) / (a * 2.0)) / (b + sqrt(t_0));
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / (0.0 - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * a), -4.0, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -500.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(a * 2.0)) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(0.0 - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -500.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -500

   1. Initial program 78.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval 78.4

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

   4. Applied egg-rr 78.4%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. rem-square-sqrt N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

   6. Applied egg-rr 78.0%

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

      1. sub-div N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. associate-*r* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-lowering-+.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

   8. Applied egg-rr 79.9%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right) - b \cdot b}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -500.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ b (sqrt t_0)))
     (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -500.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (b + sqrt(t_0));
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / (0.0 - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -500.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(0.0 - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -500.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -500

   1. Initial program 78.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval 78.4

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

   4. Applied egg-rr 78.4%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -500.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -500.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / (0.0 - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -500.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(0.0 - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -500.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -500

   1. Initial program 78.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval 78.4

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

   4. Applied egg-rr 78.4%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. --lowering--.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 79.9%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -500.0)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (* a 2.0))
   (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -500.0) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a * 2.0);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / (0.0 - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -500.0)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(0.0 - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -500.0], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -500

   1. Initial program 78.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval 78.4

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

   4. Applied egg-rr 78.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 78.5

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

   6. Applied egg-rr 78.5%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 27.2%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 93.9

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

   5. Simplified 93.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -500:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 95.0%

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

7. Applied egg-rr 95.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

8. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

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

10. Simplified 93.6%

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

Alternative 9: 93.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * fma(c, (((-2.0 * (a * (c * a))) / ((b * b) * (b * b))) - (a / (b * b))), -1.0)) / b;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

5. Simplified 95.0%

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

7. Applied egg-rr 95.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot a\right)\right) \cdot 0.05} - \frac{c \cdot \left(c \cdot a\right)}{b \cdot b}\right) - c}}{b} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}}{b} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

10. Simplified 93.5%

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

11. Final simplification 93.5%

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

Alternative 10: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b)))

C

double code(double a, double b, double c) {
	return fma((c * c), (a / (b * b)), c) / (0.0 - b);
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(0.0 - b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 90.9

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

5. Simplified 90.9%

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

6. Final simplification 90.9%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b} \]
7. Add Preprocessing

Alternative 11: 80.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{0 - b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- 0.0 b)))

C

double code(double a, double b, double c) {
	return c / (0.0 - 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 / (0.0d0 - b)
end function

Java

public static double code(double a, double b, double c) {
	return c / (0.0 - b);
}

Python

def code(a, b, c):
	return c / (0.0 - b)

Julia

function code(a, b, c)
	return Float64(c / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   4. /-lowering-/.f64 81.6

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

5. Simplified 81.6%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   3. /-lowering-/.f64 81.6

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

7. Applied egg-rr 81.6%

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

8. Final simplification 81.6%

   \[\leadsto \frac{c}{0 - b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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)))