Cubic critical, wide range

Percentage Accurate: 17.7% → 97.5%

Time: 14.3s

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: 17.7% 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: 97.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

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

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot \left(b \cdot t\_0\right)\\ \mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(\mathsf{fma}\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right), \frac{a \cdot -1.0546875}{b \cdot \left(b \cdot t\_1\right)}, \frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{t\_1}\right), a, \frac{c \cdot \left(c \cdot -0.375\right)}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b (* b t_0))))
   (fma
    (/ -0.5 b)
    c
    (*
     a
     (fma
      (fma
       (* c (* c (* c c)))
       (/ (* a -1.0546875) (* b (* b t_1)))
       (/ (* (* c c) (* c -0.5625)) t_1))
      a
      (/ (* c (* c -0.375)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * (b * t_0);
	return fma((-0.5 / b), c, (a * fma(fma((c * (c * (c * c))), ((a * -1.0546875) / (b * (b * t_1))), (((c * c) * (c * -0.5625)) / t_1)), a, ((c * (c * -0.375)) / t_0))));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * Float64(b * t_0))
	return fma(Float64(-0.5 / b), c, Float64(a * fma(fma(Float64(c * Float64(c * Float64(c * c))), Float64(Float64(a * -1.0546875) / Float64(b * Float64(b * t_1))), Float64(Float64(Float64(c * c) * Float64(c * -0.5625)) / t_1)), a, Float64(Float64(c * Float64(c * -0.375)) / t_0))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.5 / b), $MachinePrecision] * c + N[(a * N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a * -1.0546875), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * N[(c * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.6%

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

9. Applied rewrites 97.2%

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

11. Applied rewrites 97.2%

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

Alternative 3: 96.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in c around 0

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

11. Applied rewrites 96.9%

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

Alternative 4: 96.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in c around 0

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

   1. lower-*.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. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. lower-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. Applied rewrites 96.6%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 96.6%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 96.6%

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

Alternative 5: 95.2% 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 17.6%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.3%

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

Alternative 6: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 95.0%

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

6. Final simplification 95.0%

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

Alternative 7: 94.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. div-sub N/A

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

   9. lower-/.f64 N/A

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

8. Applied rewrites 95.0%

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

Alternative 8: 90.4% 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 17.6%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 90.5

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

5. Applied rewrites 90.5%

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

Reproduce

herbie shell --seed 2024234 
(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)))