Cubic critical, wide range

Percentage Accurate: 18.1% → 97.6%

Time: 14.9s

Alternatives: 10

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: 18.1% 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: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\frac{c}{b}, -0.5, \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(t\_0 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot t\_0\right)}\right), \frac{a \cdot -0.375}{t\_0}\right) \cdot \left(c \cdot c\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (/ c b)
    -0.5
    (*
     (fma
      c
      (fma
       (* a (* a (* a a)))
       (/ (* c -1.0546875) (* (* b b) (* t_0 (* a (* b b)))))
       (/ (* (* a a) -0.5625) (* b (* b t_0))))
      (/ (* a -0.375) t_0))
     (* c c)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma((c / b), -0.5, (fma(c, fma((a * (a * (a * a))), ((c * -1.0546875) / ((b * b) * (t_0 * (a * (b * b))))), (((a * a) * -0.5625) / (b * (b * t_0)))), ((a * -0.375) / t_0)) * (c * c)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(Float64(c / b), -0.5, Float64(fma(c, fma(Float64(a * Float64(a * Float64(a * a))), Float64(Float64(c * -1.0546875) / Float64(Float64(b * b) * Float64(t_0 * Float64(a * Float64(b * b))))), Float64(Float64(Float64(a * a) * -0.5625) / Float64(b * Float64(b * t_0)))), Float64(Float64(a * -0.375) / t_0)) * Float64(c * c)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c / b), $MachinePrecision] * -0.5 + N[(N[(c * N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * -1.0546875), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(t$95$0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(6.328125 \cdot c\right)}{\left(b \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)} \cdot -0.16666666666666666\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \color{blue}{c \cdot c}, \frac{c}{b \cdot -2}\right) \]

9. Applied rewrites 97.4%

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

11. Applied rewrites 97.7%

    \[\leadsto \mathsf{fma}\left(\frac{c}{b}, \color{blue}{-0.5}, \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot c\right)\right) \]
12. Add Preprocessing

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(t\_0 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot t\_0\right)}\right), \frac{a \cdot -0.375}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (/ -0.5 b)
    c
    (*
     (* c c)
     (fma
      c
      (fma
       (/ (* c -1.0546875) (* (* b b) (* t_0 (* a (* b b)))))
       (* a (* a (* a a)))
       (/ (* (* a a) -0.5625) (* b (* b t_0))))
      (/ (* a -0.375) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma((-0.5 / b), c, ((c * c) * fma(c, fma(((c * -1.0546875) / ((b * b) * (t_0 * (a * (b * b))))), (a * (a * (a * a))), (((a * a) * -0.5625) / (b * (b * t_0)))), ((a * -0.375) / t_0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(Float64(-0.5 / b), c, Float64(Float64(c * c) * fma(c, fma(Float64(Float64(c * -1.0546875) / Float64(Float64(b * b) * Float64(t_0 * Float64(a * Float64(b * b))))), Float64(a * Float64(a * Float64(a * a))), Float64(Float64(Float64(a * a) * -0.5625) / Float64(b * Float64(b * t_0)))), Float64(Float64(a * -0.375) / 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[(N[(c * c), $MachinePrecision] * N[(c * N[(N[(N[(c * -1.0546875), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(t$95$0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(6.328125 \cdot c\right)}{\left(b \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)} \cdot -0.16666666666666666\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \color{blue}{c \cdot c}, \frac{c}{b \cdot -2}\right) \]

9. Applied rewrites 97.4%

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

11. Applied rewrites 97.4%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(c \cdot c\right)\right) \]

12. Final simplification 97.4%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{b}, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(\frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right)\right) \]
13. Add Preprocessing

Alternative 3: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(t\_0 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot t\_0\right)}\right), \frac{a \cdot -0.375}{t\_0}\right), \frac{-0.5}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (*
    c
    (fma
     c
     (fma
      c
      (fma
       (* a (* a (* a a)))
       (/ (* c -1.0546875) (* (* b b) (* t_0 (* a (* b b)))))
       (/ (* (* a a) -0.5625) (* b (* b t_0))))
      (/ (* a -0.375) t_0))
     (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return c * fma(c, fma(c, fma((a * (a * (a * a))), ((c * -1.0546875) / ((b * b) * (t_0 * (a * (b * b))))), (((a * a) * -0.5625) / (b * (b * t_0)))), ((a * -0.375) / t_0)), (-0.5 / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(c * fma(c, fma(c, fma(Float64(a * Float64(a * Float64(a * a))), Float64(Float64(c * -1.0546875) / Float64(Float64(b * b) * Float64(t_0 * Float64(a * Float64(b * b))))), Float64(Float64(Float64(a * a) * -0.5625) / Float64(b * Float64(b * t_0)))), Float64(Float64(a * -0.375) / t_0)), Float64(-0.5 / b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(c * N[(c * N[(c * N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * -1.0546875), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(t$95$0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

7. Applied rewrites 97.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(6.328125 \cdot c\right)}{\left(b \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)} \cdot -0.16666666666666666\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \color{blue}{c \cdot c}, \frac{c}{b \cdot -2}\right) \]

9. Applied rewrites 97.4%

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

11. Applied rewrites 97.4%

    \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right), \frac{c \cdot -1.0546875}{\left(b \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}, \frac{\left(a \cdot a\right) \cdot -0.5625}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{-0.5}{b}\right)} \]
12. Add Preprocessing

Alternative 4: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot a}{b}\\ \frac{\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4.5, t\_0, \mathsf{fma}\left(-1.6875, \frac{c \cdot \left(a \cdot \left(a \cdot a\right)\right)}{b \cdot \left(b \cdot b\right)}, t\_0 \cdot 1.125\right)\right), -1.5 \cdot \left(b \cdot a\right)\right)}{b \cdot \sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a a) b)))
   (/
    (/
     (*
      c
      (fma
       c
       (fma
        -4.5
        t_0
        (fma -1.6875 (/ (* c (* a (* a a))) (* b (* b b))) (* t_0 1.125)))
       (* -1.5 (* b a))))
     (* b (sqrt (fma 3.0 (* a c) (* b b)))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) / b;
	return ((c * fma(c, fma(-4.5, t_0, fma(-1.6875, ((c * (a * (a * a))) / (b * (b * b))), (t_0 * 1.125))), (-1.5 * (b * a)))) / (b * sqrt(fma(3.0, (a * c), (b * b))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) / b)
	return Float64(Float64(Float64(c * fma(c, fma(-4.5, t_0, fma(-1.6875, Float64(Float64(c * Float64(a * Float64(a * a))) / Float64(b * Float64(b * b))), Float64(t_0 * 1.125))), Float64(-1.5 * Float64(b * a)))) / Float64(b * sqrt(fma(3.0, Float64(a * c), Float64(b * b))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision]}, N[(N[(N[(c * N[(c * N[(-4.5 * t$95$0 + N[(-1.6875 * N[(N[(c * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 1.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[Sqrt[N[(3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.0%

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

4. Applied rewrites 18.6%

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

5. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(c \cdot \left(\left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{a}^{2}}{b}\right) - \frac{3}{2} \cdot \left(a \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(c \cdot \left(\left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{a}^{2}}{b}\right) - \frac{3}{2} \cdot \left(a \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

   2. sub-neg N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(c \cdot \left(\left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{a}^{2}}{b}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \left(a \cdot b\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{c \cdot \color{blue}{\mathsf{fma}\left(c, \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{a}^{2}}{b}, \mathsf{neg}\left(\frac{3}{2} \cdot \left(a \cdot b\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

7. Applied rewrites 96.1%

   \[\leadsto \frac{\frac{\color{blue}{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4.5, \frac{a \cdot a}{b}, \mathsf{fma}\left(-1.6875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot c}{b \cdot \left(b \cdot b\right)}, 1.125 \cdot \frac{a \cdot a}{b}\right)\right), -1.5 \cdot \left(a \cdot b\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

8. Final simplification 96.1%

   \[\leadsto \frac{\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-4.5, \frac{a \cdot a}{b}, \mathsf{fma}\left(-1.6875, \frac{c \cdot \left(a \cdot \left(a \cdot a\right)\right)}{b \cdot \left(b \cdot b\right)}, \frac{a \cdot a}{b} \cdot 1.125\right)\right), -1.5 \cdot \left(b \cdot a\right)\right)}{b \cdot \sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \]
9. Add Preprocessing

Alternative 5: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot c}{b}\\ \frac{\frac{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-4.5, t\_0, \mathsf{fma}\left(-1.6875, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}, 1.125 \cdot t\_0\right)\right), -1.5 \cdot \left(b \cdot c\right)\right)}{b \cdot \sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c c) b)))
   (/
    (/
     (*
      a
      (fma
       a
       (fma
        -4.5
        t_0
        (fma -1.6875 (/ (* a (* c (* c c))) (* b (* b b))) (* 1.125 t_0)))
       (* -1.5 (* b c))))
     (* b (sqrt (fma 3.0 (* a c) (* b b)))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) / b;
	return ((a * fma(a, fma(-4.5, t_0, fma(-1.6875, ((a * (c * (c * c))) / (b * (b * b))), (1.125 * t_0))), (-1.5 * (b * c)))) / (b * sqrt(fma(3.0, (a * c), (b * b))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) / b)
	return Float64(Float64(Float64(a * fma(a, fma(-4.5, t_0, fma(-1.6875, Float64(Float64(a * Float64(c * Float64(c * c))) / Float64(b * Float64(b * b))), Float64(1.125 * t_0))), Float64(-1.5 * Float64(b * c)))) / Float64(b * sqrt(fma(3.0, Float64(a * c), Float64(b * b))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision]}, N[(N[(N[(a * N[(a * N[(-4.5 * t$95$0 + N[(-1.6875 * N[(N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.125 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[Sqrt[N[(3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.0%

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

4. Applied rewrites 18.6%

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

5. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(a \cdot \left(\left(\frac{-9}{2} \cdot \frac{{c}^{2}}{b} + \frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{c}^{2}}{b}\right) - \frac{3}{2} \cdot \left(b \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(a \cdot \left(\left(\frac{-9}{2} \cdot \frac{{c}^{2}}{b} + \frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{c}^{2}}{b}\right) - \frac{3}{2} \cdot \left(b \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

   2. sub-neg N/A

      \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(a \cdot \left(\left(\frac{-9}{2} \cdot \frac{{c}^{2}}{b} + \frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{c}^{2}}{b}\right) + \left(\mathsf{neg}\left(\frac{3}{2} \cdot \left(b \cdot c\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{a \cdot \color{blue}{\mathsf{fma}\left(a, \left(\frac{-9}{2} \cdot \frac{{c}^{2}}{b} + \frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}}\right) - \frac{-9}{8} \cdot \frac{{c}^{2}}{b}, \mathsf{neg}\left(\frac{3}{2} \cdot \left(b \cdot c\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

7. Applied rewrites 96.1%

   \[\leadsto \frac{\frac{\color{blue}{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-4.5, \frac{c \cdot c}{b}, \mathsf{fma}\left(-1.6875, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}, 1.125 \cdot \frac{c \cdot c}{b}\right)\right), -1.5 \cdot \left(b \cdot c\right)\right)}}{\sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)} \cdot b}}{3 \cdot a} \]

8. Final simplification 96.1%

   \[\leadsto \frac{\frac{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-4.5, \frac{c \cdot c}{b}, \mathsf{fma}\left(-1.6875, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}, 1.125 \cdot \frac{c \cdot c}{b}\right)\right), -1.5 \cdot \left(b \cdot c\right)\right)}{b \cdot \sqrt{\mathsf{fma}\left(3, a \cdot c, b \cdot b\right)}}}{3 \cdot a} \]
9. Add Preprocessing

Alternative 6: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, \frac{c \cdot c}{b \cdot b} \cdot -3.375, -1.6875 \cdot \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right), -1.5 \cdot \left(b \cdot c\right)\right)}{\left(3 \cdot a\right) \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(a, 3 \cdot c, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   a
   (fma
    a
    (fma
     b
     (* (/ (* c c) (* b b)) -3.375)
     (* -1.6875 (/ (* a (* c (* c c))) (* b (* b b)))))
    (* -1.5 (* b c))))
  (* (* 3.0 a) (* b (sqrt (fma a (* 3.0 c) (* b b)))))))

C

double code(double a, double b, double c) {
	return (a * fma(a, fma(b, (((c * c) / (b * b)) * -3.375), (-1.6875 * ((a * (c * (c * c))) / (b * (b * b))))), (-1.5 * (b * c)))) / ((3.0 * a) * (b * sqrt(fma(a, (3.0 * c), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(a * fma(a, fma(b, Float64(Float64(Float64(c * c) / Float64(b * b)) * -3.375), Float64(-1.6875 * Float64(Float64(a * Float64(c * Float64(c * c))) / Float64(b * Float64(b * b))))), Float64(-1.5 * Float64(b * c)))) / Float64(Float64(3.0 * a) * Float64(b * sqrt(fma(a, Float64(3.0 * c), Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(a * N[(b * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -3.375), $MachinePrecision] + N[(-1.6875 * N[(N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 * a), $MachinePrecision] * N[(b * N[Sqrt[N[(a * N[(3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.0%

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

4. Applied rewrites 18.6%

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

6. Applied rewrites 19.0%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{a \cdot \left(\frac{-3}{2} \cdot \left(b \cdot c\right) + a \cdot \left(\frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}} + b \cdot \left(\frac{-9}{2} \cdot \frac{{c}^{2}}{{b}^{2}} - \frac{-9}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}\right)}{\left(a \cdot 3\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) \cdot \sqrt{\mathsf{fma}\left(a, c \cdot 3, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{a \cdot \left(\frac{-3}{2} \cdot \left(b \cdot c\right) + a \cdot \left(\frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}} + b \cdot \left(\frac{-9}{2} \cdot \frac{{c}^{2}}{{b}^{2}} - \frac{-9}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}\right)}{\left(a \cdot 3\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) \cdot \sqrt{\mathsf{fma}\left(a, c \cdot 3, b \cdot b\right)}\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(a \cdot \color{blue}{\left(a \cdot \left(\frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}} + b \cdot \left(\frac{-9}{2} \cdot \frac{{c}^{2}}{{b}^{2}} - \frac{-9}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right) + \frac{-3}{2} \cdot \left(b \cdot c\right)\right)}\right)}{\left(a \cdot 3\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) \cdot \sqrt{\mathsf{fma}\left(a, c \cdot 3, b \cdot b\right)}\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(a, \frac{-27}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{3}} + b \cdot \left(\frac{-9}{2} \cdot \frac{{c}^{2}}{{b}^{2}} - \frac{-9}{8} \cdot \frac{{c}^{2}}{{b}^{2}}\right), \frac{-3}{2} \cdot \left(b \cdot c\right)\right)}\right)}{\left(a \cdot 3\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) \cdot \sqrt{\mathsf{fma}\left(a, c \cdot 3, b \cdot b\right)}\right)} \]

9. Applied rewrites 96.1%

   \[\leadsto \frac{-\color{blue}{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, \frac{c \cdot c}{b \cdot b} \cdot -3.375, -1.6875 \cdot \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right), -1.5 \cdot \left(b \cdot c\right)\right)}}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) \cdot \sqrt{\mathsf{fma}\left(a, c \cdot 3, b \cdot b\right)}\right)} \]

10. Final simplification 96.1%

    \[\leadsto \frac{a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(b, \frac{c \cdot c}{b \cdot b} \cdot -3.375, -1.6875 \cdot \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right), -1.5 \cdot \left(b \cdot c\right)\right)}{\left(3 \cdot a\right) \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(a, 3 \cdot c, b \cdot b\right)}\right)} \]
11. Add Preprocessing

Alternative 7: 95.2% accurate, 1.0× speedup

Math

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

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(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := 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. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.4%

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

6. Final simplification 95.4%

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

Alternative 8: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

4. Applied rewrites 19.2%

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

5. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 95.4

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

7. Applied rewrites 95.4%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 95.4%

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

Alternative 9: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 95.1%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 95.1%

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

Alternative 10: 90.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 89.7

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

5. Applied rewrites 89.7%

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

6. Final simplification 89.7%

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

Reproduce

herbie shell --seed 2024221 
(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)))