Cubic critical, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 15.2s

Alternatives: 13

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.0% 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: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

5. Simplified 98.7%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

5. Simplified 98.7%

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

7. Applied egg-rr 98.7%

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

9. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{c \cdot \left(\left(a \cdot c\right) \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \cdot -0.5625, \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, \mathsf{fma}\left(-0.16666666666666666, \frac{c \cdot \left(\left(\left(c \cdot c\right) \cdot \left(c \cdot 6.328125\right)\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, c \cdot -0.5\right)\right)\right) \cdot \frac{1}{b}} \]

11. Applied egg-rr 98.7%

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

12. Final simplification 98.7%

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

Alternative 3: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{t\_0}, \frac{c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{\left(b \cdot b\right) \cdot t\_0}\right), c \cdot c, \frac{c}{b \cdot -2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (fma a (/ -0.375 t_0) (/ (* c (* -0.5625 (* a a))) (* (* b b) t_0)))
    (* c c)
    (/ c (* b -2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(fma(a, (-0.375 / t_0), ((c * (-0.5625 * (a * a))) / ((b * b) * t_0))), (c * c), (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(a, Float64(-0.375 / t_0), Float64(Float64(c * Float64(-0.5625 * Float64(a * a))) / Float64(Float64(b * b) * t_0))), Float64(c * c), Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(-0.375 / t$95$0), $MachinePrecision] + N[(N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

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

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

5. Simplified 97.9%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{\left(a \cdot a\right) \cdot -0.5625}{{b}^{5}}, \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}, c \cdot c, c \cdot \frac{\frac{-1}{2}}{b}\right)} \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), c \cdot c, \frac{c}{b \cdot -2}\right)} \]
8. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(a, -0.375 \cdot \frac{c \cdot c}{b \cdot b}, c \cdot -0.5\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -0.5625
   (/ (* a (* a (* c (* c c)))) (* (* b b) (* b b)))
   (fma a (* -0.375 (/ (* c c) (* b b))) (* c -0.5)))
  b))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * (a * (c * (c * c)))) / ((b * b) * (b * b))), fma(a, (-0.375 * ((c * c) / (b * b))), (c * -0.5))) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5625, Float64(Float64(a * Float64(a * Float64(c * Float64(c * c)))) / Float64(Float64(b * b) * Float64(b * b))), fma(a, Float64(-0.375 * Float64(Float64(c * c) / Float64(b * b))), Float64(c * -0.5))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(a * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

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

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

5. Simplified 98.2%

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

Alternative 5: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(c \cdot c\right) \cdot \frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ -0.5 b)
  c
  (*
   (* c c)
   (/ (fma -0.5625 (/ (* c (* a a)) (* b b)) (* a -0.375)) (* b (* b b))))))

C

double code(double a, double b, double c) {
	return fma((-0.5 / b), c, ((c * c) * (fma(-0.5625, ((c * (a * a)) / (b * b)), (a * -0.375)) / (b * (b * b)))));
}

Julia

function code(a, b, c)
	return fma(Float64(-0.5 / b), c, Float64(Float64(c * c) * Float64(fma(-0.5625, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(a * -0.375)) / Float64(b * Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c + N[(N[(c * c), $MachinePrecision] * N[(N[(-0.5625 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

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

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

5. Simplified 97.9%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \frac{\left(a \cdot a\right) \cdot -0.5625}{{b}^{5}}, \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, c \cdot \left(c \cdot \left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2}}{b}}, c, c \cdot \left(c \cdot \left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \color{blue}{\left(c \cdot c\right) \cdot \left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \color{blue}{\left(c \cdot c\right) \cdot \left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \color{blue}{\left(c \cdot c\right)} \cdot \left(c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \left(c \cdot c\right) \cdot \color{blue}{\left(\frac{a \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)} + c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}}\right)}\right) \]

   10. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \left(c \cdot c\right) \cdot \left(\color{blue}{a \cdot \frac{\frac{-3}{8}}{b \cdot \left(b \cdot b\right)}} + c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}}\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{b}, c, \left(c \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(a, \frac{\frac{-3}{8}}{b \cdot \left(b \cdot b\right)}, c \cdot \frac{\left(a \cdot a\right) \cdot \frac{-9}{16}}{{b}^{5}}\right)}\right) \]

7. Applied egg-rr 97.9%

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

8. Taylor expanded in b around inf

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

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

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

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

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

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

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 97.9

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

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

Alternative 6: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (fma
   c
   (/ (fma -0.5625 (/ (* c (* a a)) (* b b)) (* a -0.375)) (* b (* b b)))
   (/ -0.5 b))))

C

double code(double a, double b, double c) {
	return c * fma(c, (fma(-0.5625, ((c * (a * a)) / (b * b)), (a * -0.375)) / (b * (b * b))), (-0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * fma(c, Float64(fma(-0.5625, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(a * -0.375)) / Float64(b * Float64(b * b))), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(c * N[(N[(-0.5625 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

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

      \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \frac{-3}{8} \cdot \frac{a}{{b}^{3}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

5. Simplified 97.9%

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

6. Taylor expanded in b around inf

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

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

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

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

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

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

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 97.9

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 7: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.375, \frac{c \cdot -0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ (* a (* c c)) (* b (* b b))) -0.375 (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	return fma(((a * (c * c)) / (b * (b * b))), -0.375, ((c * -0.5) / b));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(a * Float64(c * c)) / Float64(b * Float64(b * b))), -0.375, Float64(Float64(c * -0.5) / b))
end

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

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

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

   13. associate-*r/ N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 96.9

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

5. Simplified 96.9%

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

Alternative 8: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, -0.375 \cdot \frac{c \cdot c}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (* -0.375 (/ (* c c) (* b b))) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (-0.375 * ((c * c) / (b * b))), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(-0.375 * Float64(Float64(c * c) / Float64(b * b))), Float64(c * -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 15.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 96.9%

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

Alternative 9: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \mathsf{fma}\left(a \cdot -0.375, \frac{c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (fma (* a -0.375) (/ c (* b b)) -0.5)) b))

C

double code(double a, double b, double c) {
	return (c * fma((a * -0.375), (c / (b * b)), -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * fma(Float64(a * -0.375), Float64(c / Float64(b * b)), -0.5)) / b)
end

Wolfram

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

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in b around inf

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

5. Simplified 98.7%

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

6. Taylor expanded in c around 0

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

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

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 96.8

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 10: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(a \cdot -0.375, \frac{c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma (* a -0.375) (/ c (* b b)) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma((a * -0.375), (c / (b * b)), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(Float64(a * -0.375), Float64(c / Float64(b * b)), -0.5) / b))
end

Wolfram

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

Derivation

1. Initial program 15.8%

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

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. associate-*r/ N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r* N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. associate-*l* N/A

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

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

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

5. Simplified 96.6%

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

6. Taylor expanded in b around inf

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

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

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

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

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

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

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 96.5

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

8. Simplified 96.5%

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

9. Final simplification 96.5%

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

Alternative 11: 90.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 91.8

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

5. Simplified 91.8%

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

Alternative 12: 90.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.8%

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 91.8

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

5. Simplified 91.8%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

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

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

   4. /-lowering-/.f64 91.5

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

7. Applied egg-rr 91.5%

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

8. Final simplification 91.5%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Alternative 13: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 15.8%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

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

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

4. Applied egg-rr 15.7%

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

   1. frac-sub N/A

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

   2. associate-/r* N/A

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

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

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

6. Applied egg-rr 17.2%

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

7. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{9} \cdot \frac{-3 \cdot b + 3 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{9} \cdot \left(-3 \cdot b + 3 \cdot b\right)}{a}} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{\left(b \cdot \left(-3 + 3\right)\right)}}{a} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. /-lowering-/.f64 3.3

      \[\leadsto \color{blue}{\frac{0}{a}} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Step-by-step derivation

    1. div0 3.3

       \[\leadsto \color{blue}{0} \]

11. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024201 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))