Quadratic roots, narrow range

Percentage Accurate: 55.9% → 92.1%

Time: 12.9s

Alternatives: 10

Speedup: 3.6×

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: 55.9% 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: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, -0.25, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 0.025)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (-
      (*
       (* a a)
       (fma
        (* (/ (* (pow c 4.0) 20.0) (pow b 6.0)) (/ a b))
        -0.25
        (/ (* (* c (* c c)) -2.0) (pow b 5.0))))
      (/ (fma (* c c) (/ a (* b b)) c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 0.025) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((a * a) * fma((((pow(c, 4.0) * 20.0) / pow(b, 6.0)) * (a / b)), -0.25, (((c * (c * c)) * -2.0) / pow(b, 5.0)))) - (fma((c * c), (a / (b * b)), c) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.025)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(a * a) * fma(Float64(Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 6.0)) * Float64(a / b)), -0.25, Float64(Float64(Float64(c * Float64(c * c)) * -2.0) / (b ^ 5.0)))) - Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.025], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.025000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 89.7%

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

   if 0.025000000000000001 < 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 a around 0

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

   5. Applied rewrites 96.3%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{{c}^{4} \cdot 20}{{b}^{6}} \cdot \frac{a}{b}, -0.25, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -2}{{b}^{5}}\right) - \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{t\_1}, \frac{-0.25 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot t\_1\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))) (t_1 (* (* b b) (* b (* b b)))))
   (if (<= b 0.025)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (fma
      (fma
       c
       (* (* c c) (/ -2.0 t_1))
       (/ (* -0.25 (* a (* (* c c) (* (* c c) 20.0)))) (* b (* b t_1))))
      (* a a)
      (/ (fma c (* c (/ a (* b b))) c) (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double t_1 = (b * b) * (b * (b * b));
	double tmp;
	if (b <= 0.025) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(fma(c, ((c * c) * (-2.0 / t_1)), ((-0.25 * (a * ((c * c) * ((c * c) * 20.0)))) / (b * (b * t_1)))), (a * a), (fma(c, (c * (a / (b * b))), c) / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_1 = Float64(Float64(b * b) * Float64(b * Float64(b * b)))
	tmp = 0.0
	if (b <= 0.025)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(fma(c, Float64(Float64(c * c) * Float64(-2.0 / t_1)), Float64(Float64(-0.25 * Float64(a * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(b * Float64(b * t_1)))), Float64(a * a), Float64(fma(c, Float64(c * Float64(a / Float64(b * b))), c) / Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.025], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(c * N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.025000000000000001

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 89.7%

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

   if 0.025000000000000001 < 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 a around 0

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 96.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \color{blue}{a \cdot a}, \frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c, c \cdot \frac{a}{b \cdot b}, c\right)}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot a\right) \cdot -2\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{b \cdot b} - c \cdot \left(a \cdot c\right)}{b \cdot b}}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 0.027)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (-
      (/
       (/
        (- (* (* (* a a) -2.0) (/ (* c (* c c)) (* b b))) (* c (* a c)))
        (* b b))
       b)
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 0.027) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((((((a * a) * -2.0) * ((c * (c * c)) / (b * b))) - (c * (a * c))) / (b * b)) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * -2.0) * Float64(Float64(c * Float64(c * c)) / Float64(b * b))) - Float64(c * Float64(a * c))) / Float64(b * b)) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.027], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 89.7%

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

   if 0.0269999999999999997 < 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

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.1%

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

   7. Applied rewrites 94.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot a\right) \cdot -2\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{b \cdot b} - c \cdot \left(a \cdot c\right)}{b \cdot b}}{b} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot a\right) \cdot -2\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{b \cdot b} - c \cdot \left(a \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 0.027)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      (-
       (/
        (- (* (* (* a a) -2.0) (/ (* c (* c c)) (* b b))) (* c (* a c)))
        (* b b))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 0.027) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((((((a * a) * -2.0) * ((c * (c * c)) / (b * b))) - (c * (a * c))) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * -2.0) * Float64(Float64(c * Float64(c * c)) / Float64(b * b))) - Float64(c * Float64(a * c))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.027], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 87.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 89.7%

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

   if 0.0269999999999999997 < 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

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.1%

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

   7. Applied rewrites 94.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot a\right) \cdot -2\right) \cdot \frac{c \cdot \left(c \cdot c\right)}{b \cdot b} - c \cdot \left(a \cdot c\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.35:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 1.35)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (- (fma a (/ (* c c) (* b (* b b))) (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 1.35) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 1.35)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.35], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3500000000000001

   1. Initial program 84.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-prod N/A

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

   7. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 86.4%

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

   if 1.3500000000000001 < b

   1. Initial program 45.9%

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

   4. Applied rewrites 45.0%

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

   6. Applied rewrites 45.5%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2} - b \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2)
   (- (/ (sqrt (fma a (* c -4.0) (* b b))) (* a 2.0)) (* b (/ 0.5 a)))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.2) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) / (a * 2.0)) - (b * (0.5 / a));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.2)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) / Float64(a * 2.0)) - Float64(b * Float64(0.5 / a)));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.2], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision] - N[(b * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 1.19999999999999996

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 84.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 84.2

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 84.6

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2} - b \cdot \color{blue}{\frac{0.5}{a}} \]

   6. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2} - b \cdot \frac{0.5}{a}} \]

   if 1.19999999999999996 < b

   1. Initial program 45.9%

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

   4. Applied rewrites 45.0%

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

   6. Applied rewrites 45.5%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

Alternative 7: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2)
   (* (/ -0.5 a) (- b (sqrt (fma c (* a -4.0) (* b b)))))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.2) {
		tmp = (-0.5 / a) * (b - sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.2)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.2], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 1.19999999999999996

   1. Initial program 84.3%

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

   4. Applied rewrites 84.4%

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

   if 1.19999999999999996 < b

   1. Initial program 45.9%

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

   4. Applied rewrites 45.0%

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

   6. Applied rewrites 45.5%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

Alternative 8: 81.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

4. Applied rewrites 49.7%

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

6. Applied rewrites 50.1%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 85.7

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

9. Applied rewrites 85.7%

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

Alternative 9: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.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. lower-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. lower-*.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. lower-/.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. lower-*.f64 85.7

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

5. Applied rewrites 85.7%

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

6. Final simplification 85.7%

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

Alternative 10: 64.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 68.6

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

5. Applied rewrites 68.6%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))