Cubic critical, narrow range

Percentage Accurate: 55.1% → 90.7%

Time: 17.0s

Alternatives: 14

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: 55.1% 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: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{-0.375}{t\_0}, a \cdot \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_1\right)}, -0.16666666666666666, \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{t\_1}\right)\right), a, -0.5 \cdot \frac{c}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (fma
    (fma
     c
     (* c (/ -0.375 t_0))
     (*
      a
      (fma
       (/ (* (* c (* c (* c c))) (* a 6.328125)) (* b (* b t_1)))
       -0.16666666666666666
       (/ (* c (* c (* c -0.5625))) t_1))))
    a
    (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return fma(fma(c, (c * (-0.375 / t_0)), (a * fma((((c * (c * (c * c))) * (a * 6.328125)) / (b * (b * t_1))), -0.16666666666666666, ((c * (c * (c * -0.5625))) / t_1)))), a, (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	return fma(fma(c, Float64(c * Float64(-0.375 / t_0)), Float64(a * fma(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_1))), -0.16666666666666666, Float64(Float64(c * Float64(c * Float64(c * -0.5625))) / t_1)))), a, Float64(-0.5 * Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 55.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 90.5%

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

7. Applied egg-rr 90.5%

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

8. Final simplification 90.5%

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

Alternative 2: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(c, \frac{c}{t\_0 \cdot -2.6666666666666665}, a \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot c\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot t\_0}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\left(a \cdot 6.328125\right) \cdot -0.16666666666666666\right)}{b \cdot \left(t\_0 \cdot t\_0\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (/ -0.5 b)
    c
    (*
     a
     (fma
      c
      (/ c (* t_0 -2.6666666666666665))
      (*
       a
       (fma
        c
        (/ (* (* c c) -0.5625) (* (* b b) t_0))
        (/
         (* (* c (* c (* c c))) (* (* a 6.328125) -0.16666666666666666))
         (* b (* t_0 t_0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma((-0.5 / b), c, (a * fma(c, (c / (t_0 * -2.6666666666666665)), (a * fma(c, (((c * c) * -0.5625) / ((b * b) * t_0)), (((c * (c * (c * c))) * ((a * 6.328125) * -0.16666666666666666)) / (b * (t_0 * t_0))))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(Float64(-0.5 / b), c, Float64(a * fma(c, Float64(c / Float64(t_0 * -2.6666666666666665)), Float64(a * fma(c, Float64(Float64(Float64(c * c) * -0.5625) / Float64(Float64(b * b) * t_0)), Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(Float64(a * 6.328125) * -0.16666666666666666)) / Float64(b * Float64(t_0 * t_0))))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.5 / b), $MachinePrecision] * c + N[(a * N[(c * N[(c / N[(t$95$0 * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 6.328125), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 90.5%

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

7. Applied egg-rr 90.5%

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

9. Applied egg-rr 90.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(c, \frac{c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}, a \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot c\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\left(6.328125 \cdot a\right) \cdot -0.16666666666666666\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right)\right)\right)} \]

10. Final simplification 90.4%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(c, \frac{c}{\left(b \cdot \left(b \cdot b\right)\right) \cdot -2.6666666666666665}, a \cdot \mathsf{fma}\left(c, \frac{\left(c \cdot c\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\left(a \cdot 6.328125\right) \cdot -0.16666666666666666\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(c, c \cdot \frac{-0.375}{t\_0}, a \cdot \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_1\right)}, -0.16666666666666666, \frac{c \cdot \left(c \cdot \left(c \cdot -0.5625\right)\right)}{t\_1}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (fma
    (/ -0.5 b)
    c
    (*
     a
     (fma
      c
      (* c (/ -0.375 t_0))
      (*
       a
       (fma
        (/ (* (* c (* c (* c c))) (* a 6.328125)) (* b (* b t_1)))
        -0.16666666666666666
        (/ (* c (* c (* c -0.5625))) t_1))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return fma((-0.5 / b), c, (a * fma(c, (c * (-0.375 / t_0)), (a * fma((((c * (c * (c * c))) * (a * 6.328125)) / (b * (b * t_1))), -0.16666666666666666, ((c * (c * (c * -0.5625))) / t_1))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	return fma(Float64(-0.5 / b), c, Float64(a * fma(c, Float64(c * Float64(-0.375 / t_0)), Float64(a * fma(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_1))), -0.16666666666666666, Float64(Float64(c * Float64(c * Float64(c * -0.5625))) / t_1))))))
end

Wolfram

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

Derivation

1. Initial program 55.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Simplified 90.5%

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

7. Applied egg-rr 90.4%

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

8. Final simplification 90.4%

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

Alternative 4: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.5e-5)
   (/ (- (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)) <= -1.5e-5) {
		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)) <= -1.5e-5)
		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], -1.5e-5], 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)) < -1.50000000000000004e-5

   1. Initial program 75.7%

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

      1. sub-neg N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval 75.8

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

   4. Applied egg-rr 75.8%

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

   if -1.50000000000000004e-5 < (/.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 33.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 82.5

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

   5. Simplified 82.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.5e-5)
   (* (/ (- b (sqrt (fma c (* a -3.0) (* b b)))) a) -0.3333333333333333)
   (* -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)) <= -1.5e-5) {
		tmp = ((b - sqrt(fma(c, (a * -3.0), (b * b)))) / a) * -0.3333333333333333;
	} 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)) <= -1.5e-5)
		tmp = Float64(Float64(Float64(b - sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))) / a) * -0.3333333333333333);
	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], -1.5e-5], N[(N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.3333333333333333), $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)) < -1.50000000000000004e-5

   1. Initial program 75.7%

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

   4. Applied egg-rr 75.7%

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

      1. div-inv N/A

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

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

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

   6. Applied egg-rr 75.7%

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

   if -1.50000000000000004e-5 < (/.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 33.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 82.5

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

   5. Simplified 82.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.5e-5)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (* -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)) <= -1.5e-5) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} 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)) <= -1.5e-5)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	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], -1.5e-5], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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)) < -1.50000000000000004e-5

   1. Initial program 75.7%

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

   4. Applied egg-rr 75.7%

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

   if -1.50000000000000004e-5 < (/.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 33.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 82.5

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

   5. Simplified 82.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{b + \sqrt{t\_0}} \cdot \frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{-0.5625 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 7.5)
     (/ (* (/ (- (* b b) t_0) (+ b (sqrt t_0))) (/ 1.0 a)) -3.0)
     (fma
      a
      (/
       (fma c (* c -0.375) (/ (* -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 <= 7.5) {
		tmp = ((((b * b) - t_0) / (b + sqrt(t_0))) * (1.0 / a)) / -3.0;
	} else {
		tmp = fma(a, (fma(c, (c * -0.375), ((-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 <= 7.5)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(b + sqrt(t_0))) * Float64(1.0 / a)) / -3.0);
	else
		tmp = fma(a, Float64(fma(c, Float64(c * -0.375), 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, 7.5], N[(N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(a * N[(N[(c * N[(c * -0.375), $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 < 7.5

   1. Initial program 81.7%

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

   4. Applied egg-rr 81.6%

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

      1. flip-- N/A

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

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

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

   6. Applied egg-rr 83.1%

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

   if 7.5 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. Simplified 94.7%

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

   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. /-lowering-/.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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

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

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

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

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

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

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

      19. cube-mult N/A

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

   8. Simplified 92.4%

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

Alternative 8: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{1}{a \cdot -3}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{-0.5625 \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 7.5)
     (/ (* (- (* b b) t_0) (/ 1.0 (* a -3.0))) (+ b (sqrt t_0)))
     (fma
      a
      (/
       (fma c (* c -0.375) (/ (* -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 <= 7.5) {
		tmp = (((b * b) - t_0) * (1.0 / (a * -3.0))) / (b + sqrt(t_0));
	} else {
		tmp = fma(a, (fma(c, (c * -0.375), ((-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 <= 7.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(1.0 / Float64(a * -3.0))) / Float64(b + sqrt(t_0)));
	else
		tmp = fma(a, Float64(fma(c, Float64(c * -0.375), 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, 7.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(c * N[(c * -0.375), $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 < 7.5

   1. Initial program 81.7%

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

   4. Applied egg-rr 81.6%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 83.1%

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

   if 7.5 < b

   1. Initial program 46.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. Simplified 94.7%

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

   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. /-lowering-/.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. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. associate-*r/ N/A

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

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

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

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

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

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

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

      19. cube-mult N/A

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

   8. Simplified 92.4%

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

Alternative 9: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{1}{a \cdot -3}}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 8.5)
     (/ (* (- (* b b) t_0) (/ 1.0 (* a -3.0))) (+ b (sqrt t_0)))
     (/ (fma a (/ (* -0.375 (* 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 <= 8.5) {
		tmp = (((b * b) - t_0) * (1.0 / (a * -3.0))) / (b + sqrt(t_0));
	} else {
		tmp = fma(a, ((-0.375 * (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 <= 8.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(1.0 / Float64(a * -3.0))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * 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, 8.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5

   1. Initial program 81.6%

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

   4. Applied egg-rr 81.5%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 83.0%

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

   if 8.5 < b

   1. Initial program 46.5%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.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. Simplified 87.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{a \cdot -3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 8.6:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{1}{a}}{-3 \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 8.6)
     (/ (* (- (* b b) t_0) (/ 1.0 a)) (* -3.0 (+ b (sqrt t_0))))
     (/ (fma a (/ (* -0.375 (* 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 <= 8.6) {
		tmp = (((b * b) - t_0) * (1.0 / a)) / (-3.0 * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, ((-0.375 * (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 <= 8.6)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(1.0 / a)) / Float64(-3.0 * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * 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, 8.6], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / N[(-3.0 * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.59999999999999964

   1. Initial program 81.6%

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

   4. Applied egg-rr 81.5%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. frac-times N/A

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

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

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

   6. Applied egg-rr 83.0%

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

   if 8.59999999999999964 < b

   1. Initial program 46.5%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.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. Simplified 87.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot \frac{1}{a}}{-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{a \cdot \left(b + \sqrt{t\_0}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 8.5)
     (/ (/ (- (* b b) t_0) (* a (+ b (sqrt t_0)))) -3.0)
     (/ (fma a (/ (* -0.375 (* 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 <= 8.5) {
		tmp = (((b * b) - t_0) / (a * (b + sqrt(t_0)))) / -3.0;
	} else {
		tmp = fma(a, ((-0.375 * (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 <= 8.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(a * Float64(b + sqrt(t_0)))) / -3.0);
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * 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, 8.5], 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[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5

   1. Initial program 81.6%

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

   4. Applied egg-rr 81.5%

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

      1. un-div-inv N/A

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

      2. flip-- N/A

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

      3. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 83.0%

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

   if 8.5 < b

   1. Initial program 46.5%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.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. Simplified 87.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (fma a (/ (* -0.375 (* c c)) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5], 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[(N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5

   1. Initial program 81.6%

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

      1. sub-neg N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval 81.7

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

   4. Applied egg-rr 81.7%

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

   if 8.5 < b

   1. Initial program 46.5%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.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. Simplified 87.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* 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 <= 8.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} 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 <= 8.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	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, 8.5], 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[(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 < 8.5

   1. Initial program 81.6%

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

      1. sub-neg N/A

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

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

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

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

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

      9. metadata-eval 81.7

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

   4. Applied egg-rr 81.7%

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

   if 8.5 < b

   1. Initial program 46.5%

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

   3. Taylor expanded in 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. *-lowering-*.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. Simplified 87.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.7% 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 55.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-lowering-*.f64 N/A

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

   2. /-lowering-/.f64 64.1

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

5. Simplified 64.1%

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

Reproduce

herbie shell --seed 2024199 
(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)))