Quadratic roots, narrow range

Percentage Accurate: 55.6% → 92.2%

Time: 14.4s

Alternatives: 12

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: 55.6% 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: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(a \cdot \left(c \cdot c\right)\right)\\ t_2 := c \cdot \left(t\_1 \cdot -5\right)\\ t_3 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ t_4 := -2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\\ t_5 := b \cdot \left(t\_0 \cdot t\_0\right)\\ t_6 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.013:\\ \;\;\;\;\frac{t\_3 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{t\_4 \cdot t\_4}{t\_6 \cdot t\_6} - \frac{\left(\left(c \cdot t\_1\right) \cdot 5\right) \cdot t\_2}{t\_5 \cdot t\_5}}{\frac{t\_4}{t\_6} - \frac{t\_2}{t\_5}}, a, c \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* c (* a (* c c))))
        (t_2 (* c (* t_1 -5.0)))
        (t_3 (fma -4.0 (* c a) (* b b)))
        (t_4 (* -2.0 (* c (* c c))))
        (t_5 (* b (* t_0 t_0)))
        (t_6 (* (* b b) t_0)))
   (if (<= b 0.013)
     (/ (- t_3 (* b b)) (* (* a 2.0) (+ b (sqrt t_3))))
     (fma
      a
      (fma
       (/
        (-
         (/ (* t_4 t_4) (* t_6 t_6))
         (/ (* (* (* c t_1) 5.0) t_2) (* t_5 t_5)))
        (- (/ t_4 t_6) (/ t_2 t_5)))
       a
       (* c (/ c (* b (* b (- b))))))
      (/ c (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (a * (c * c));
	double t_2 = c * (t_1 * -5.0);
	double t_3 = fma(-4.0, (c * a), (b * b));
	double t_4 = -2.0 * (c * (c * c));
	double t_5 = b * (t_0 * t_0);
	double t_6 = (b * b) * t_0;
	double tmp;
	if (b <= 0.013) {
		tmp = (t_3 - (b * b)) / ((a * 2.0) * (b + sqrt(t_3)));
	} else {
		tmp = fma(a, fma(((((t_4 * t_4) / (t_6 * t_6)) - ((((c * t_1) * 5.0) * t_2) / (t_5 * t_5))) / ((t_4 / t_6) - (t_2 / t_5))), a, (c * (c / (b * (b * -b))))), (c / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(a * Float64(c * c)))
	t_2 = Float64(c * Float64(t_1 * -5.0))
	t_3 = fma(-4.0, Float64(c * a), Float64(b * b))
	t_4 = Float64(-2.0 * Float64(c * Float64(c * c)))
	t_5 = Float64(b * Float64(t_0 * t_0))
	t_6 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.013)
		tmp = Float64(Float64(t_3 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_3))));
	else
		tmp = fma(a, fma(Float64(Float64(Float64(Float64(t_4 * t_4) / Float64(t_6 * t_6)) - Float64(Float64(Float64(Float64(c * t_1) * 5.0) * t_2) / Float64(t_5 * t_5))) / Float64(Float64(t_4 / t_6) - Float64(t_2 / t_5))), a, Float64(c * Float64(c / Float64(b * Float64(b * Float64(-b)))))), Float64(c / Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(t$95$1 * -5.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.013], N[(N[(t$95$3 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] / N[(t$95$6 * t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(c * t$95$1), $MachinePrecision] * 5.0), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 / t$95$6), $MachinePrecision] - N[(t$95$2 / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(c * N[(c / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0129999999999999994

   1. Initial program 89.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 89.6

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

   4. Applied egg-rr 89.6%

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

      1. flip-+ N/A

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

      2. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 90.6%

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

   if 0.0129999999999999994 < b

   1. Initial program 50.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 93.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

   8. Simplified 93.0%

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

   10. Applied egg-rr 93.0%

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

   12. Applied egg-rr 93.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.013:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} - \frac{\left(\left(c \cdot \left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right) \cdot 5\right) \cdot \left(c \cdot \left(\left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot -5\right)\right)}{\left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}}{\frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{c \cdot \left(\left(c \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot -5\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}, a, c \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0135:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \frac{-5 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* c a) (* b b))))
   (if (<= b 0.0135)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (fma
      a
      (fma
       a
       (fma
        -2.0
        (/ (* c (* c c)) (pow b 5.0))
        (/ (* -5.0 (* a (pow c 4.0))) (pow b 7.0)))
       (/ (* c c) (* b (* b (- b)))))
      (/ c (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (c * a), (b * b));
	double tmp;
	if (b <= 0.0135) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, fma(a, fma(-2.0, ((c * (c * c)) / pow(b, 5.0)), ((-5.0 * (a * pow(c, 4.0))) / pow(b, 7.0))), ((c * c) / (b * (b * -b)))), (c / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (b <= 0.0135)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, fma(a, fma(-2.0, Float64(Float64(c * Float64(c * c)) / (b ^ 5.0)), Float64(Float64(-5.0 * Float64(a * (c ^ 4.0))) / (b ^ 7.0))), Float64(Float64(c * c) / Float64(b * Float64(b * Float64(-b))))), Float64(c / Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0135], 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[(a * N[(-2.0 * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0134999999999999998

   1. Initial program 89.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 89.6

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

   4. Applied egg-rr 89.6%

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

      1. flip-+ N/A

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

      2. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 90.6%

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

   if 0.0134999999999999998 < b

   1. Initial program 50.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 93.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

   8. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \frac{-5 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\right), \frac{c \cdot c}{-b \cdot \left(b \cdot b\right)}\right), \frac{c}{-b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0135:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{{b}^{5}}, \frac{-5 \cdot \left(a \cdot {c}^{4}\right)}{{b}^{7}}\right), \frac{c \cdot c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0125:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot t\_1}, \frac{-5 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{t\_1 \cdot \left(b \cdot t\_1\right)}\right), a, c \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* c a) (* b b))) (t_1 (* b (* b b))))
   (if (<= b 0.0125)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (fma
      a
      (fma
       (fma
        -2.0
        (/ (* c (* c c)) (* (* b b) t_1))
        (/ (* -5.0 (* (* c c) (* a (* c c)))) (* t_1 (* b t_1))))
       a
       (* c (/ c (* b (* b (- b))))))
      (/ c (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (c * a), (b * b));
	double t_1 = b * (b * b);
	double tmp;
	if (b <= 0.0125) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, fma(fma(-2.0, ((c * (c * c)) / ((b * b) * t_1)), ((-5.0 * ((c * c) * (a * (c * c)))) / (t_1 * (b * t_1)))), a, (c * (c / (b * (b * -b))))), (c / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(c * a), Float64(b * b))
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 0.0125)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, fma(fma(-2.0, Float64(Float64(c * Float64(c * c)) / Float64(Float64(b * b) * t_1)), Float64(Float64(-5.0 * Float64(Float64(c * c) * Float64(a * Float64(c * c)))) / Float64(t_1 * Float64(b * t_1)))), a, Float64(c * Float64(c / Float64(b * Float64(b * Float64(-b)))))), Float64(c / Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0125], 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[(-2.0 * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-5.0 * N[(N[(c * c), $MachinePrecision] * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(c * N[(c / N[(b * N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.012500000000000001

   1. Initial program 89.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 89.6

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

   4. Applied egg-rr 89.6%

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

      1. flip-+ N/A

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

      2. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 90.6%

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

   if 0.012500000000000001 < b

   1. Initial program 50.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. Simplified 93.0%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

   8. Simplified 93.0%

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

   10. Applied egg-rr 93.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0125:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(\mathsf{fma}\left(-2, \frac{c \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-5 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), a, c \cdot \frac{c}{b \cdot \left(b \cdot \left(-b\right)\right)}\right), \frac{c}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\frac{-2 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b} - c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* c a) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.08)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ b (sqrt t_0)))
     (fma
      a
      (/ (- (/ (* -2.0 (* a (* c (* c c)))) (* b b)) (* c c)) (* b (* b b)))
      (/ c (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (c * a), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.08) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (b + sqrt(t_0));
	} else {
		tmp = fma(a, ((((-2.0 * (a * (c * (c * c)))) / (b * b)) - (c * c)) / (b * (b * b))), (c / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.08)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(t_0)));
	else
		tmp = fma(a, Float64(Float64(Float64(Float64(-2.0 * Float64(a * Float64(c * Float64(c * c)))) / Float64(b * b)) - Float64(c * c)) / Float64(b * Float64(b * b))), Float64(c / Float64(-b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.08], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(-2.0 * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0800000000000000017

   1. Initial program 83.7%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 83.7

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

   4. Applied egg-rr 83.7%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 85.2%

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

   if -0.0800000000000000017 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.4%

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

   3. Taylor expanded in 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. Simplified 94.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) + \color{blue}{-1 \cdot \frac{c}{b}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}, -1 \cdot \frac{c}{b}\right)} \]

   8. Simplified 94.7%

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

   9. Taylor expanded in b around inf

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

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

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

       5. associate-*r/ N/A

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

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

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

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

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

       12. unpow2 N/A

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

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

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

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

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

   11. Simplified 92.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.08:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\frac{-2 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b} - c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* c a) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.032)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ 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(-4.0, (c * a), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.032) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (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(-4.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.032)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / 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[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.032], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 84.7%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.0%

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

   3. Taylor expanded in 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. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. /-lowering-/.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* c a) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.032)
     (/ (- 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(-4.0, (c * a), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.032) {
		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(-4.0, Float64(c * a), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.032)
		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[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.032], 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. flip-+ N/A

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

      2. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 84.7%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.0%

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

   3. Taylor expanded in 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. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. /-lowering-/.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.032)
   (/ 1.0 (/ (* a 2.0) (- (sqrt (fma a (* -4.0 c) (* b b))) b)))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.032) {
		tmp = 1.0 / ((a * 2.0) / (sqrt(fma(a, (-4.0 * c), (b * 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.032)
		tmp = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * 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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.032], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.1%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*r* N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 83.1

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

   4. Applied egg-rr 83.1%

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

      1. flip-+ N/A

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

      2. sqr-neg N/A

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

      3. rem-square-sqrt N/A

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

      4. div-sub N/A

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

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

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

   6. Applied egg-rr 84.1%

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

      1. clear-num N/A

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

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

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

      3. sub-div N/A

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

      4. sqr-neg N/A

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

      5. rem-square-sqrt N/A

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

      6. flip-+ N/A

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

   8. Applied egg-rr 83.2%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.0%

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

   3. Taylor expanded in 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. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. /-lowering-/.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.032)
   (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* -4.0 a))))))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.032) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (-4.0 * a)))));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.032)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(-4.0 * 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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.032], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.1%

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

   4. Applied egg-rr 83.2%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 46.0%

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

   3. Taylor expanded in 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. Simplified 94.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. /-lowering-/.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.032:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 54.3%

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

3. Taylor expanded in a around 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. Simplified 90.6%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

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

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

   5. associate-/l* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. /-lowering-/.f64 82.0

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

8. Simplified 82.0%

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

Alternative 10: 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 54.3%

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

3. Taylor expanded in 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 82.0

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

5. Simplified 82.0%

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

6. Final simplification 82.0%

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

Alternative 11: 64.2% 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 54.3%

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

3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 66.2

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

5. Simplified 66.2%

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

Alternative 12: 1.6% accurate, 4.2× 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 / 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 54.3%

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

3. Taylor expanded in 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. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 66.2

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

5. Simplified 66.2%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{neg}\left(b\right)}{c}}} \]

   2. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(b\right)} \cdot c} \]

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

      \[\leadsto \color{blue}{\frac{1}{\mathsf{neg}\left(b\right)} \cdot c} \]

   4. metadata-eval N/A

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

   5. frac-2neg N/A

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

   6. /-lowering-/.f64 66.1

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

7. Applied egg-rr 66.1%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

      \[\leadsto \frac{c}{\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(b\right)}}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{c}{\frac{1}{\frac{\color{blue}{1}}{\mathsf{neg}\left(b\right)}}} \]

   7. inv-pow N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{c}{{\left(\mathsf{neg}\left(b\right)\right)}^{\color{blue}{1}}} \]

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. pow2 N/A

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

   13. sqr-neg N/A

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

   14. pow2 N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. /-lowering-/.f64 1.6

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

9. Applied egg-rr 1.6%

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

Reproduce

herbie shell --seed 2024204 
(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)))