Quadratic roots, narrow range

Percentage Accurate: 55.9% → 90.5%

Time: 15.1s

Alternatives: 11

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ t_2 := c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\\ t_3 := b \cdot t\_1\\ t_4 := c \cdot \left(c \cdot c\right)\\ t_5 := \left(a \cdot \left(c \cdot t\_4\right)\right) \cdot -5\\ t_6 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_7 := t\_0 \cdot t\_1\\ \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_6 - b \cdot b\right)}{b + \sqrt{t\_6}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{t\_5}{t\_7}\right)}^{3} - \frac{\left(t\_4 \cdot \left(t\_4 \cdot t\_4\right)\right) \cdot -8}{t\_0 \cdot \left(t\_1 \cdot \left(b \cdot t\_7\right)\right)}}{\frac{t\_5 \cdot t\_5}{t\_7 \cdot t\_7} + \left(\frac{t\_2 \cdot \left(t\_4 \cdot 2\right)}{t\_3 \cdot t\_3} + \frac{t\_5 \cdot t\_2}{t\_7 \cdot t\_3}\right)}, a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* (* b b) (* b b)))
        (t_2 (* c (* (* c c) -2.0)))
        (t_3 (* b t_1))
        (t_4 (* c (* c c)))
        (t_5 (* (* a (* c t_4)) -5.0))
        (t_6 (fma a (* c -4.0) (* b b)))
        (t_7 (* t_0 t_1)))
   (if (<= b 28.0)
     (/ (* (/ 0.5 a) (- t_6 (* b b))) (+ b (sqrt t_6)))
     (fma
      (/
       (-
        (pow (/ t_5 t_7) 3.0)
        (/ (* (* t_4 (* t_4 t_4)) -8.0) (* t_0 (* t_1 (* b t_7)))))
       (+
        (/ (* t_5 t_5) (* t_7 t_7))
        (+ (/ (* t_2 (* t_4 2.0)) (* t_3 t_3)) (/ (* t_5 t_2) (* t_7 t_3)))))
      (* a a)
      (/ (fma (* c c) (/ a (* b b)) c) (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * (b * b);
	double t_2 = c * ((c * c) * -2.0);
	double t_3 = b * t_1;
	double t_4 = c * (c * c);
	double t_5 = (a * (c * t_4)) * -5.0;
	double t_6 = fma(a, (c * -4.0), (b * b));
	double t_7 = t_0 * t_1;
	double tmp;
	if (b <= 28.0) {
		tmp = ((0.5 / a) * (t_6 - (b * b))) / (b + sqrt(t_6));
	} else {
		tmp = fma(((pow((t_5 / t_7), 3.0) - (((t_4 * (t_4 * t_4)) * -8.0) / (t_0 * (t_1 * (b * t_7))))) / (((t_5 * t_5) / (t_7 * t_7)) + (((t_2 * (t_4 * 2.0)) / (t_3 * t_3)) + ((t_5 * t_2) / (t_7 * t_3))))), (a * a), (fma((c * c), (a / (b * b)), c) / -b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * Float64(b * b))
	t_2 = Float64(c * Float64(Float64(c * c) * -2.0))
	t_3 = Float64(b * t_1)
	t_4 = Float64(c * Float64(c * c))
	t_5 = Float64(Float64(a * Float64(c * t_4)) * -5.0)
	t_6 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_7 = Float64(t_0 * t_1)
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_6 - Float64(b * b))) / Float64(b + sqrt(t_6)));
	else
		tmp = fma(Float64(Float64((Float64(t_5 / t_7) ^ 3.0) - Float64(Float64(Float64(t_4 * Float64(t_4 * t_4)) * -8.0) / Float64(t_0 * Float64(t_1 * Float64(b * t_7))))) / Float64(Float64(Float64(t_5 * t_5) / Float64(t_7 * t_7)) + Float64(Float64(Float64(t_2 * Float64(t_4 * 2.0)) / Float64(t_3 * t_3)) + Float64(Float64(t_5 * t_2) / Float64(t_7 * t_3))))), Float64(a * a), Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), 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[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(N[(c * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(a * N[(c * t$95$4), $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision]}, Block[{t$95$6 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[b, 28.0], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$6 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(t$95$5 / t$95$7), $MachinePrecision], 3.0], $MachinePrecision] - N[(N[(N[(t$95$4 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] / N[(t$95$0 * N[(t$95$1 * N[(b * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$5 * t$95$5), $MachinePrecision] / N[(t$95$7 * t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 * N[(t$95$4 * 2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 * t$95$2), $MachinePrecision] / N[(t$95$7 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   6. Applied rewrites 86.7%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 92.8%

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

   7. Applied rewrites 92.8%

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

   9. Applied rewrites 92.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(\frac{\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}\right)}^{3} - \frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -8}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)}}{\frac{\left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right) \cdot \left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} + \left(\frac{\left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot 2\right)}{\left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right) \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)}\right)}}, a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}\right)}^{3} - \frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -8}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)}}{\frac{\left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right) \cdot \left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} + \left(\frac{\left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right) \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot 2\right)}{\left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(\left(a \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right) \cdot -5\right) \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right)}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)}\right)}, a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)}, a \cdot -5, \frac{\mathsf{fma}\left(a, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right)}{b \cdot b}, a \cdot \left(c \cdot \left(-c\right)\right)\right)}{b \cdot b} - c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 28.0)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ b (sqrt t_0)))
     (/
      (fma
       (/ (* (* c (* c (* c c))) (* a a)) (* b (* b (* (* b b) (* b b)))))
       (* a -5.0)
       (-
        (/
         (fma a (/ (* a (* c (* (* c c) -2.0))) (* b b)) (* a (* c (- c))))
         (* b b))
        c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 28.0) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (b + sqrt(t_0));
	} else {
		tmp = fma((((c * (c * (c * c))) * (a * a)) / (b * (b * ((b * b) * (b * b))))), (a * -5.0), ((fma(a, ((a * (c * ((c * c) * -2.0))) / (b * b)), (a * (c * -c))) / (b * b)) - c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * a)) / Float64(b * Float64(b * Float64(Float64(b * b) * Float64(b * b))))), Float64(a * -5.0), Float64(Float64(fma(a, Float64(Float64(a * Float64(c * Float64(Float64(c * c) * -2.0))) / Float64(b * b)), Float64(a * Float64(c * Float64(-c)))) / Float64(b * b)) - c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 28.0], 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[(N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -5.0), $MachinePrecision] + N[(N[(N[(a * N[(N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   6. Applied rewrites 86.7%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 92.8%

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

   10. Applied rewrites 92.8%

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

   12. Applied rewrites 92.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)\right)}, a \cdot -5, \frac{\mathsf{fma}\left(a, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -2\right)\right)}{b \cdot b}, a \cdot \left(c \cdot \left(-c\right)\right)\right)}{b \cdot b} - c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ t_2 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_2 - b \cdot b\right)}{b + \sqrt{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, a \cdot \frac{\left(c \cdot t\_1\right) \cdot \left(a \cdot a\right)}{t\_0 \cdot t\_0}, \frac{\frac{\left(a \cdot -2\right) \cdot \left(a \cdot t\_1\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* c (* c c)))
        (t_2 (fma a (* c -4.0) (* b b))))
   (if (<= b 28.0)
     (/ (* (/ 0.5 a) (- t_2 (* b b))) (+ b (sqrt t_2)))
     (/
      (-
       (fma
        -5.0
        (* a (/ (* (* c t_1) (* a a)) (* t_0 t_0)))
        (/ (- (/ (* (* a -2.0) (* a t_1)) (* b b)) (* a (* c c))) (* b b)))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	double t_2 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 28.0) {
		tmp = ((0.5 / a) * (t_2 - (b * b))) / (b + sqrt(t_2));
	} else {
		tmp = (fma(-5.0, (a * (((c * t_1) * (a * a)) / (t_0 * t_0))), (((((a * -2.0) * (a * t_1)) / (b * b)) - (a * (c * c))) / (b * b))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(c * c))
	t_2 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_2 - Float64(b * b))) / Float64(b + sqrt(t_2)));
	else
		tmp = Float64(Float64(fma(-5.0, Float64(a * Float64(Float64(Float64(c * t_1) * Float64(a * a)) / Float64(t_0 * t_0))), Float64(Float64(Float64(Float64(Float64(a * -2.0) * Float64(a * t_1)) / Float64(b * b)) - Float64(a * Float64(c * c))) / Float64(b * b))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 28.0], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$2 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-5.0 * N[(a * N[(N[(N[(c * t$95$1), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(a * -2.0), $MachinePrecision] * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   6. Applied rewrites 86.7%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 92.8%

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

   10. Applied rewrites 92.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, a \cdot \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\frac{\left(a \cdot -2\right) \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b} - a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 28.0)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ b (sqrt t_0)))
     (/
      (- (/ (* (* c c) (fma -2.0 (/ (* c (* a a)) (* b b)) (- a))) (* b b)) c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 28.0) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (b + sqrt(t_0));
	} else {
		tmp = ((((c * c) * fma(-2.0, ((c * (a * a)) / (b * b)), -a)) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * c) * fma(-2.0, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(-a))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 28.0], 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[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   6. Applied rewrites 86.7%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lower-neg.f64 90.5

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

   10. Applied rewrites 90.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))))
   (if (<= b 28.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      (- (/ (* (* c c) (fma -2.0 (/ (* c (* a a)) (* b b)) (- a))) (* b b)) c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double tmp;
	if (b <= 28.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = ((((c * c) * fma(-2.0, ((c * (a * a)) / (b * b)), -a)) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * c) * fma(-2.0, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(-a))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 28.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   6. Applied rewrites 86.5%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lower-neg.f64 90.5

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

   10. Applied rewrites 90.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 28.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/
    (- (/ (* (* c c) (fma -2.0 (/ (* c (* a a)) (* b b)) (- a))) (* b b)) c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 28.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((((c * c) * fma(-2.0, ((c * (a * a)) / (b * b)), -a)) / (b * b)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * c) * fma(-2.0, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(-a))) / Float64(b * b)) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 28.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 85.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 85.7

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

   6. Applied rewrites 85.7%

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

   if 28 < b

   1. Initial program 48.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lower-neg.f64 90.5

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

   10. Applied rewrites 90.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 82.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.0

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

   4. Applied rewrites 83.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 83.0

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

   6. Applied rewrites 83.0%

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

   if 115 < b

   1. Initial program 45.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 82.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if 115 < b

   1. Initial program 45.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 115.0)
   (* (/ -0.5 a) (- b (sqrt (fma a (* c -4.0) (* b b)))))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 115.0) {
		tmp = (-0.5 / a) * (b - sqrt(fma(a, (c * -4.0), (b * b))));
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 115.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 115.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 115

   1. Initial program 82.7%

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

   4. Applied rewrites 82.7%

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

   if 115 < b

   1. Initial program 45.9%

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

   3. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-/l* N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 115:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% 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 55.8%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Final simplification 79.2%

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

Alternative 11: 63.9% 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 55.8%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 63.7

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

5. Applied rewrites 63.7%

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

Reproduce

herbie shell --seed 2024214 
(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)))