Cubic critical, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 15.4s

Alternatives: 8

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

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

FPCore

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

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	return Float64(fma(-1.0546875, Float64(a * Float64(Float64(Float64(c * c) * Float64(Float64(a * a) * Float64(c * c))) / Float64(b * Float64(b * t_0)))), fma(c, -0.5, fma(-0.5625, Float64(Float64(a * Float64(a * Float64(c * Float64(c * c)))) / 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 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0546875 * N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5 + N[(-0.5625 * N[(N[(a * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.1%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)\right)}{b}} \]

10. Applied egg-rr 97.4%

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

12. Applied egg-rr 97.4%

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

13. Final simplification 97.4%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.1%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)\right)}{b}} \]

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 3: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.1%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)\right)}{b}} \]

10. Applied egg-rr 97.4%

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

11. Taylor expanded in c around 0

    \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}\right)\right)}{b} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \color{blue}{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}\right)\right)}{b} \]

    2. sub-neg N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}\right)\right)}{b} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-1}{2}}\right)\right)\right)}{b} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}\right)\right)}{b} \]

    5. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \mathsf{fma}\left(\frac{-3}{8}, \color{blue}{a \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2}\right)\right)\right)}{b} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \mathsf{fma}\left(\frac{-3}{8}, \color{blue}{a \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2}\right)\right)\right)}{b} \]

    7. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \mathsf{fma}\left(\frac{-3}{8}, a \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2}\right)\right)\right)}{b} \]

    8. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot \frac{\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot c\right)}{b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)}, \mathsf{fma}\left(\frac{-9}{16}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, c \cdot \mathsf{fma}\left(\frac{-3}{8}, a \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)\right)\right)}{b} \]

    9. lower-*.f64 97.4

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

13. Simplified 97.4%

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

14. Final simplification 97.4%

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

Alternative 4: 96.0% accurate, 0.3× speedup

Math

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

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))) (* t_0 1.125)))
       (* -1.5 (* c b))))
     (* b (sqrt (fma 3.0 (* a c) (* b b)))))
    (* a 3.0))))

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))), (t_0 * 1.125))), (-1.5 * (c * b)))) / (b * sqrt(fma(3.0, (a * c), (b * b))))) / (a * 3.0);
}

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(t_0 * 1.125))), Float64(-1.5 * Float64(c * b)))) / Float64(b * sqrt(fma(3.0, Float64(a * c), Float64(b * b))))) / Float64(a * 3.0))
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[(t$95$0 * 1.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[Sqrt[N[(3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.8%

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

4. Applied egg-rr 19.3%

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

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. Simplified 95.8%

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

8. Final simplification 95.8%

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

Alternative 5: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. 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. Simplified 94.9%

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

Alternative 6: 95.3% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.1%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)\right)\right)}{b}} \]

10. Applied egg-rr 97.4%

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

11. Taylor expanded in c around 0

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

    1. lower-*.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. metadata-eval N/A

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

    4. lower-fma.f64 N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 94.8

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

13. Simplified 94.8%

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

Alternative 7: 95.0% 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 19.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 97.1%

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

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. lower-/.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. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 94.5

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

8. Simplified 94.5%

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

Alternative 8: 90.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.8%

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

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

   2. lower-/.f64 89.1

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

5. Simplified 89.1%

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

Reproduce

herbie shell --seed 2024208 
(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)))