Cubic critical, narrow range

Percentage Accurate: 56.1% → 91.7%

Time: 14.4s

Alternatives: 15

Speedup: 2.9×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 56.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_2 := b \cdot \left(b \cdot b\right)\\ t_3 := \left(b \cdot b\right) \cdot t\_2\\ \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_1}{a \cdot \left(b + \sqrt{t\_1}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot t\_0\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_3\right)}, -0.16666666666666666, \frac{t\_0 \cdot -0.5625}{t\_3}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{t\_2}, \frac{c \cdot -0.5}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c)))
        (t_1 (fma c (* a -3.0) (* b b)))
        (t_2 (* b (* b b)))
        (t_3 (* (* b b) t_2)))
   (if (<= b 0.36)
     (/ (/ (- (* b b) t_1) (* a (+ b (sqrt t_1)))) -3.0)
     (fma
      (fma
       (/ (* (* c t_0) (* a 6.328125)) (* b (* b t_3)))
       -0.16666666666666666
       (/ (* t_0 -0.5625) t_3))
      (* a a)
      (fma (* c c) (* a (/ -0.375 t_2)) (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = fma(c, (a * -3.0), (b * b));
	double t_2 = b * (b * b);
	double t_3 = (b * b) * t_2;
	double tmp;
	if (b <= 0.36) {
		tmp = (((b * b) - t_1) / (a * (b + sqrt(t_1)))) / -3.0;
	} else {
		tmp = fma(fma((((c * t_0) * (a * 6.328125)) / (b * (b * t_3))), -0.16666666666666666, ((t_0 * -0.5625) / t_3)), (a * a), fma((c * c), (a * (-0.375 / t_2)), ((c * -0.5) / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = fma(c, Float64(a * -3.0), Float64(b * b))
	t_2 = Float64(b * Float64(b * b))
	t_3 = Float64(Float64(b * b) * t_2)
	tmp = 0.0
	if (b <= 0.36)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_1) / Float64(a * Float64(b + sqrt(t_1)))) / -3.0);
	else
		tmp = fma(fma(Float64(Float64(Float64(c * t_0) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_3))), -0.16666666666666666, Float64(Float64(t_0 * -0.5625) / t_3)), Float64(a * a), fma(Float64(c * c), Float64(a * Float64(-0.375 / t_2)), Float64(Float64(c * -0.5) / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[b, 0.36], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(N[(N[(N[(c * t$95$0), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(t$95$0 * -0.5625), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a * N[(-0.375 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.35999999999999999

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 88.9%

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

   if 0.35999999999999999 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   7. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(6.328125 \cdot a\right)}{\left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, a \cdot \frac{-0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ t_2 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ t_3 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_2}{a \cdot \left(b + \sqrt{t\_2}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot t\_1\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_3\right)}, -0.16666666666666666, \frac{t\_1 \cdot -0.5625}{t\_3}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{t\_0}\right), a, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* c (* c c)))
        (t_2 (fma c (* a -3.0) (* b b)))
        (t_3 (* (* b b) t_0)))
   (if (<= b 0.36)
     (/ (/ (- (* b b) t_2) (* a (+ b (sqrt t_2)))) -3.0)
     (fma
      (fma
       a
       (fma
        (/ (* (* c t_1) (* a 6.328125)) (* b (* b t_3)))
        -0.16666666666666666
        (/ (* t_1 -0.5625) t_3))
       (/ (* c (* c -0.375)) t_0))
      a
      (/ (* c -0.5) 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(c, (a * -3.0), (b * b));
	double t_3 = (b * b) * t_0;
	double tmp;
	if (b <= 0.36) {
		tmp = (((b * b) - t_2) / (a * (b + sqrt(t_2)))) / -3.0;
	} else {
		tmp = fma(fma(a, fma((((c * t_1) * (a * 6.328125)) / (b * (b * t_3))), -0.16666666666666666, ((t_1 * -0.5625) / t_3)), ((c * (c * -0.375)) / t_0)), a, ((c * -0.5) / 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(c, Float64(a * -3.0), Float64(b * b))
	t_3 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.36)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_2) / Float64(a * Float64(b + sqrt(t_2)))) / -3.0);
	else
		tmp = fma(fma(a, fma(Float64(Float64(Float64(c * t_1) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_3))), -0.16666666666666666, Float64(Float64(t_1 * -0.5625) / t_3)), Float64(Float64(c * Float64(c * -0.375)) / t_0)), a, Float64(Float64(c * -0.5) / 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[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.36], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(c * t$95$1), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(t$95$1 * -0.5625), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.35999999999999999

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 88.9%

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

   if 0.35999999999999999 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   7. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(6.328125 \cdot a\right)}{\left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right), a, \frac{c \cdot -0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.36:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right), a, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5625, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 0.38)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (/
      (fma
       -0.5625
       (/ (* (* c (* c c)) (* a a)) (* (* b b) (* b b)))
       (fma -0.375 (/ (* a (* c c)) (* b b)) (* c -0.5)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 0.38) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(-0.5625, (((c * (c * c)) * (a * a)) / ((b * b) * (b * b))), fma(-0.375, ((a * (c * c)) / (b * b)), (c * -0.5))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = Float64(fma(-0.5625, Float64(Float64(Float64(c * Float64(c * c)) * Float64(a * a)) / Float64(Float64(b * b) * Float64(b * b))), fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.38], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(-0.5625 * N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 88.9%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   7. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(6.328125 \cdot a\right)}{\left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), a \cdot a, \mathsf{fma}\left(c \cdot c, \frac{-0.375}{b \cdot \left(b \cdot b\right)} \cdot a, \frac{c \cdot -0.5}{b}\right)\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

   10. Applied rewrites 92.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.9999999999999997e-7

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 70.7

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

   4. Applied rewrites 70.7%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 34.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.7

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.9999999999999997e-7

   1. Initial program 70.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 70.5

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

   4. Applied rewrites 70.5%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 34.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.7

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6e-7)
   (* (- b (sqrt (fma a (* c -3.0) (* b b)))) (/ -0.3333333333333333 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6e-7) {
		tmp = (b - sqrt(fma(a, (c * -3.0), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -6e-7)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -6e-7], N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5.9999999999999997e-7

   1. Initial program 70.5%

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

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 34.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.7

         \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

   5. Applied rewrites 82.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.375, c \cdot c, \frac{-0.5625 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 0.38)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (fma
      a
      (/
       (fma -0.375 (* c c) (/ (* -0.5625 (* a (* c (* c c)))) (* b b)))
       (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 0.38) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(a, (fma(-0.375, (c * c), ((-0.5625 * (a * (c * (c * c)))) / (b * b))) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = fma(a, Float64(fma(-0.375, Float64(c * c), Float64(Float64(-0.5625 * Float64(a * Float64(c * Float64(c * c)))) / Float64(b * b))) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.38], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision] + N[(N[(-0.5625 * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 88.9%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2} + \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8}, {c}^{2}, \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{c \cdot c}, \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{c \cdot c}, \frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \color{blue}{\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \color{blue}{\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\color{blue}{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \color{blue}{\left(a \cdot {c}^{3}\right)}}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \color{blue}{\left(c \cdot \left(c \cdot c\right)\right)}\right)}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \color{blue}{{c}^{2}}\right)\right)}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \color{blue}{\left(c \cdot {c}^{2}\right)}\right)}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)}{{b}^{2}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\color{blue}{b \cdot b}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\color{blue}{b \cdot b}}\right)}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-3}{8}, c \cdot c, \frac{\frac{-9}{16} \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      21. lower-*.f64 92.3

          \[\leadsto \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.375, c \cdot c, \frac{-0.5625 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Applied rewrites 92.3%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\mathsf{fma}\left(-0.375, c \cdot c, \frac{-0.5625 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 0.38)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 0.38) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.38], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 88.9%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 86.9

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Applied rewrites 86.9%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 0.38)
     (/ (- (* b b) t_0) (* (* a -3.0) (+ b (sqrt t_0))))
     (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 0.38) {
		tmp = ((b * b) - t_0) / ((a * -3.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.38], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 88.8%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 86.9

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Applied rewrites 86.9%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.38)
   (/ (/ (- b (sqrt (fma b b (* -3.0 (* c a))))) a) -3.0)
   (fma a (/ (* (* c c) -0.375) (* b (* b b))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.38) {
		tmp = ((b - sqrt(fma(b, b, (-3.0 * (c * a))))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / a) / -3.0);
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.38], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\frac{-3}{8} \cdot {c}^{2}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{\frac{-3}{8} \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. lower-*.f64 86.9

          \[\leadsto \mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]

   8. Applied rewrites 86.9%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}}, -0.5 \cdot \frac{c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.38)
   (/ (/ (- b (sqrt (fma b b (* -3.0 (* c a))))) a) -3.0)
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.38) {
		tmp = ((b - sqrt(fma(b, b, (-3.0 * (c * a))))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(b - sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / a) / -3.0);
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.38], N[(N[(N[(b - N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.38)
   (/ (/ (- b (sqrt (fma a (* c -3.0) (* b b)))) a) -3.0)
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.38) {
		tmp = ((b - sqrt(fma(a, (c * -3.0), (b * b)))) / a) / -3.0;
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) / a) / -3.0);
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.38], N[(N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.6%

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

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.38)
   (* (- b (sqrt (fma a (* c -3.0) (* b b)))) (/ -0.3333333333333333 a))
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.38) {
		tmp = (b - sqrt(fma(a, (c * -3.0), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.38], N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.38)
   (* (- b (sqrt (fma a (* c -3.0) (* b b)))) (/ -0.3333333333333333 a))
   (* c (fma a (* -0.375 (/ c (* b (* b b)))) (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.38) {
		tmp = (b - sqrt(fma(a, (c * -3.0), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = c * fma(a, (-0.375 * (c / (b * (b * b)))), (-0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.38)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(c * fma(a, Float64(-0.375 * Float64(c / Float64(b * Float64(b * b)))), Float64(-0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.38], N[(N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * N[(-0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.38

   1. Initial program 87.2%

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

   4. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)} \]

   if 0.38 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in c around 0

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

      1. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)} + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right) + c \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot \frac{-3}{8}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   5. Applied rewrites 86.8%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.38:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 65.8

      \[\leadsto -0.5 \cdot \color{blue}{\frac{c}{b}} \]

5. Applied rewrites 65.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))