Quadratic roots, narrow range

Percentage Accurate: 55.3% → 91.5%

Time: 14.9s

Alternatives: 10

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% 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: 91.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(b \cdot \left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)\right)\right) \cdot 0.05}, \left(c \cdot -2\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 4.8)
     (/ (/ (- t_0 (fma b b 0.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (-
       (-
        (fma
         (* -0.25 (* (* a a) (* a a)))
         (/
          (* c (* c (* c c)))
          (* (* a (* b (* (fma b b 0.0) (* b (fma b b 0.0))))) 0.05))
         (* (* c -2.0) (/ (* c (* a (* c a))) (* (* b b) (* b b)))))
        (/ (* a (* c c)) (fma b b 0.0)))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - fma(b, b, 0.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((fma((-0.25 * ((a * a) * (a * a))), ((c * (c * (c * c))) / ((a * (b * (fma(b, b, 0.0) * (b * fma(b, b, 0.0))))) * 0.05)), ((c * -2.0) * ((c * (a * (c * a))) / ((b * b) * (b * b))))) - ((a * (c * c)) / fma(b, b, 0.0))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(fma(Float64(-0.25 * Float64(Float64(a * a) * Float64(a * a))), Float64(Float64(c * Float64(c * Float64(c * c))) / Float64(Float64(a * Float64(b * Float64(fma(b, b, 0.0) * Float64(b * fma(b, b, 0.0))))) * 0.05)), Float64(Float64(c * -2.0) * Float64(Float64(c * Float64(a * Float64(c * a))) / Float64(Float64(b * b) * Float64(b * b))))) - Float64(Float64(a * Float64(c * c)) / fma(b, b, 0.0))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.25 * N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * N[(b * N[(N[(b * b + 0.0), $MachinePrecision] * N[(b * N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.05), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -2.0), $MachinePrecision] * N[(N[(c * N[(a * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   7. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 94.3%

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

   7. Applied egg-rr 94.3%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right), 0\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}}{b} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right) + 0\right)\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right) + 0\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right)\right)}\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(b \cdot b + 0\right)\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      5. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\color{blue}{{b}^{2}} \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      10. cube-unmult N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left({b}^{2} \cdot \color{blue}{{b}^{3}}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      11. pow-prod-up N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{{b}^{\left(2 + 3\right)}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left({b}^{\color{blue}{5}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left({b}^{\color{blue}{\left(3 + 2\right)}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{\left({b}^{3} \cdot {b}^{2}\right)} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      15. cube-unmult N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{2}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      16. pow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

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

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right) \cdot b\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)}\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{{b}^{4}}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{{b}^{\color{blue}{\left(2 \cdot 2\right)}}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

       2. pow-sqr N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{{b}^{2} \cdot {b}^{2}}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

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

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{{b}^{2} \cdot {b}^{2}}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

       4. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

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

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

       6. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

       7. *-lowering-*.f64 94.3

          \[\leadsto \frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

   12. Simplified 94.3%

       \[\leadsto \frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(b \cdot \left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right)\right)\right) \cdot 0.05}, \left(c \cdot -2\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \frac{c \cdot c}{0 - b \cdot b}\right), 0 - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 4.8)
     (/ (/ (- t_0 (fma b b 0.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (fma
       a
       (fma
        -2.0
        (/ (* a (* c (* c c))) (pow b 4.0))
        (/ (* c c) (- 0.0 (* b b))))
       (- 0.0 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - fma(b, b, 0.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = fma(a, fma(-2.0, ((a * (c * (c * c))) / pow(b, 4.0)), ((c * c) / (0.0 - (b * b)))), (0.0 - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, fma(-2.0, Float64(Float64(a * Float64(c * Float64(c * c))) / (b ^ 4.0)), Float64(Float64(c * c) / Float64(0.0 - Float64(b * b)))), Float64(0.0 - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(-2.0 * N[(N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(0.0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   7. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 94.3%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

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

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

   8. Simplified 92.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \frac{-c \cdot c}{b \cdot b}\right), 0 - c\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \frac{c \cdot c}{0 - b \cdot b}\right), 0 - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 4.8)
     (/ (/ (- t_0 (fma b b 0.0)) (+ b (sqrt t_0))) (* a 2.0))
     (/
      (-
       (/
        (- (/ (* (* c (* c c)) (* (* a a) -2.0)) (* b b)) (* a (* c c)))
        (* b b))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 4.8) {
		tmp = ((t_0 - fma(b, b, 0.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((((((c * (c * c)) * ((a * a) * -2.0)) / (b * b)) - (a * (c * c))) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * Float64(c * c)) * Float64(Float64(a * a) * -2.0)) / Float64(b * b)) - Float64(a * Float64(c * c))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $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. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   7. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 94.3%

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

   7. Applied egg-rr 94.3%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right), 0\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}}{b} \]

   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 92.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 4.8)
     (/ (- t_0 (fma b b 0.0)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      (-
       (/
        (- (/ (* (* c (* c c)) (* (* a a) -2.0)) (* b b)) (* a (* c c)))
        (* b b))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 4.8) {
		tmp = (t_0 - fma(b, b, 0.0)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((((((c * (c * c)) * ((a * a) * -2.0)) / (b * b)) - (a * (c * c))) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * Float64(c * c)) * Float64(Float64(a * a) * -2.0)) / Float64(b * b)) - Float64(a * Float64(c * c))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -2.0), $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. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   7. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 94.3%

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

   7. Applied egg-rr 94.3%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right), 0\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}}{b} \]

   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 92.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \frac{c \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{a}{b \cdot b}, -1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 4.8)
     (/ (- t_0 (fma b b 0.0)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      (*
       c
       (fma
        c
        (- (/ (* c (* (* a a) -2.0)) (* (* b b) (* b b))) (/ a (* b b)))
        -1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 4.8) {
		tmp = (t_0 - fma(b, b, 0.0)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = (c * fma(c, (((c * ((a * a) * -2.0)) / ((b * b) * (b * b))) - (a / (b * b))), -1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(c * fma(c, Float64(Float64(Float64(c * Float64(Float64(a * a) * -2.0)) / Float64(Float64(b * b) * Float64(b * b))) - Float64(a / Float64(b * b))), -1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.8], N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(N[(N[(c * N[(N[(a * a), $MachinePrecision] * -2.0), $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. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 84.1%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   7. Applied egg-rr 85.2%

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

   if 4.79999999999999982 < b

   1. Initial program 49.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 94.3%

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

   7. Applied egg-rr 94.3%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right), 0\right)\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}}{b} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right) + 0\right)\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right) + 0\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(b \cdot b + 0\right) \cdot \left(b \cdot b + 0\right)\right)}\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(b \cdot b + 0\right)\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      5. +-rgt-identity N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(b \cdot b\right) \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot b\right)\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\color{blue}{{b}^{2}} \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      10. cube-unmult N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left({b}^{2} \cdot \color{blue}{{b}^{3}}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      11. pow-prod-up N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{{b}^{\left(2 + 3\right)}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left({b}^{\color{blue}{5}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left({b}^{\color{blue}{\left(3 + 2\right)}} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      14. pow-prod-up N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\color{blue}{\left({b}^{3} \cdot {b}^{2}\right)} \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      15. cube-unmult N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot {b}^{2}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

      16. pow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot b\right)\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

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

          \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{-1}{4} \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right) \cdot b\right)}\right) \cdot \frac{1}{20}}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

   9. Applied egg-rr 94.3%

      \[\leadsto \frac{\left(\mathsf{fma}\left(-0.25 \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right), \frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(a \cdot \color{blue}{\left(\left(\mathsf{fma}\left(b, b, 0\right) \cdot \left(b \cdot \mathsf{fma}\left(b, b, 0\right)\right)\right) \cdot b\right)}\right) \cdot 0.05}, \left(-2 \cdot c\right) \cdot \frac{c \cdot \left(a \cdot \left(c \cdot a\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 0\right), \mathsf{fma}\left(b, b, 0\right), 0\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{\mathsf{fma}\left(b, b, 0\right)}\right) - c}{b} \]

   10. Taylor expanded in b around inf

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

   12. Simplified 92.0%

       \[\leadsto \color{blue}{\frac{c \cdot \mathsf{fma}\left(c, \frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{a}{b \cdot b}, -1\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \frac{c \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{a}{b \cdot b}, -1\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)\\ \mathbf{if}\;b \leq 27:\\ \;\;\;\;\frac{t\_0 - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (fma b b 0.0))))
   (if (<= b 27.0)
     (/ (- t_0 (fma b b 0.0)) (* (* a 2.0) (+ b (sqrt t_0))))
     (- (- 0.0 (/ c b)) (/ (* a (* c c)) (* b (* b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), fma(b, b, 0.0));
	double tmp;
	if (b <= 27.0) {
		tmp = (t_0 - fma(b, b, 0.0)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = (0.0 - (c / b)) - ((a * (c * c)) / (b * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), fma(b, b, 0.0))
	tmp = 0.0
	if (b <= 27.0)
		tmp = Float64(Float64(t_0 - fma(b, b, 0.0)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(0.0 - Float64(c / b)) - Float64(Float64(a * Float64(c * c)) / Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 27.0], N[(N[(t$95$0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 27

   1. Initial program 82.4%

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

   3. Taylor expanded in b around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 82.4

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

   5. Simplified 82.4%

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

   7. Applied egg-rr 83.7%

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

   if 27 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \left(0 - \color{blue}{\frac{c}{b}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \left(0 - \frac{c}{b}\right) - \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}}} \]

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

         \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      10. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}} \]

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

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}} \]

      12. cube-mult N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}} \]

      13. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}} \]

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

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

      16. *-lowering-*.f64 87.9

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

   8. Simplified 87.9%

      \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 27:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right) - \mathsf{fma}\left(b, b, 0\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(b, b, 0\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 27.5:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 27.5)
   (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
   (- (- 0.0 (/ c b)) (/ (* a (* c c)) (* b (* b b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 27.5) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = (0.0 - (c / b)) - ((a * (c * c)) / (b * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 27.5)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(0.0 - Float64(c / b)) - Float64(Float64(a * Float64(c * c)) / Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 27.5], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 27.5

   1. Initial program 82.4%

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

   4. Applied egg-rr 82.6%

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

   if 27.5 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \left(0 - \color{blue}{\frac{c}{b}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

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

         \[\leadsto \left(0 - \frac{c}{b}\right) - \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}}} \]

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

         \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} \]

      10. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}} \]

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

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}} \]

      12. cube-mult N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}} \]

      13. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}} \]

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

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

      16. *-lowering-*.f64 87.9

          \[\leadsto \left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

   8. Simplified 87.9%

      \[\leadsto \color{blue}{\left(0 - \frac{c}{b}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 27:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 27.0)
   (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
   (/ (fma (* c c) (/ a (* b b)) c) (- 0.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 27.0) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / (0.0 - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 27.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	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[b, 27.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $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 b < 27

   1. Initial program 82.4%

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

   4. Applied egg-rr 82.6%

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

   if 27 < b

   1. Initial program 47.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 27:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% 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 57.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 79.3

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

5. Simplified 79.3%

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

6. Final simplification 79.3%

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

Alternative 10: 64.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 62.1

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

5. Simplified 62.1%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
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 62.1

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

7. Applied egg-rr 62.1%

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

8. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))