Cubic critical, wide range

Percentage Accurate: 17.9% → 97.7%

Time: 13.7s

Alternatives: 9

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: 17.9% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(a \cdot 6.328125\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (fma
   a
   (fma
    (/ (* (/ (pow c 4.0) (pow b 6.0)) (* a 6.328125)) b)
    -0.16666666666666666
    (/ (* (* c (* c c)) -0.5625) (pow b 5.0)))
   (/ (* (* c c) -0.375) (* b (* b b))))
  (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(a, fma(a, fma((((pow(c, 4.0) / pow(b, 6.0)) * (a * 6.328125)) / b), -0.16666666666666666, (((c * (c * c)) * -0.5625) / pow(b, 5.0))), (((c * c) * -0.375) / (b * (b * b)))), (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(a, fma(a, fma(Float64(Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * Float64(a * 6.328125)) / b), -0.16666666666666666, Float64(Float64(Float64(c * Float64(c * c)) * -0.5625) / (b ^ 5.0))), Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b)))), Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(a * N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

6. Final simplification 97.2%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(a, fma(fma(c, Float64(Float64(c * c) * Float64(-0.5625 / Float64(Float64(b * b) * t_0))), Float64(-0.16666666666666666 * Float64(Float64(Float64(a * 6.328125) * Float64(c * Float64(c * Float64(c * c)))) / Float64(b * Float64(Float64(b * b) * Float64(b * t_0)))))), 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]}, N[(a * N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-0.5625 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(a * 6.328125), $MachinePrecision] * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

7. Applied rewrites 97.2%

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

8. Final simplification 97.2%

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

Alternative 3: 97.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 97.2%

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

12. Applied rewrites 97.2%

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

13. Final simplification 97.2%

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

Alternative 4: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 97.2%

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

12. Applied rewrites 97.2%

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

13. Final simplification 97.2%

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

Alternative 5: 97.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

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 96.4%

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

9. Final simplification 96.4%

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

Alternative 6: 95.3% 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 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 94.8%

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

Alternative 7: 95.3% 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 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 97.2%

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

7. Applied rewrites 97.2%

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

8. Taylor expanded in b around inf

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

10. Applied rewrites 97.2%

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

11. Taylor expanded in c around 0

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

13. Applied rewrites 94.8%

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

14. Final simplification 94.8%

    \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \]
15. 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 15.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 91.6

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

5. Applied rewrites 91.6%

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

Alternative 9: 3.3% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 15.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 15.4%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 17.1%

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

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. mul0-rgt N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 3.3

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

9. Applied rewrites 3.3%

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

10. Taylor expanded in a around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))