Cubic critical, wide range

Percentage Accurate: 17.5% → 97.7%

Time: 14.5s

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: 17.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathsf{fma}\left(\mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot 6.328125\right)}{\left(b \cdot t\_1\right) \cdot \left(a \cdot b\right)}, -0.16666666666666666, \frac{\left(a \cdot a\right) \cdot -0.5625}{t\_1}\right), \frac{a \cdot -0.375}{t\_0}\right), c \cdot c, \frac{c}{b \cdot -2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (fma
    (fma
     c
     (fma
      (/ (* (* a (* a (* a a))) (* c 6.328125)) (* (* b t_1) (* a b)))
      -0.16666666666666666
      (/ (* (* a a) -0.5625) t_1))
     (/ (* a -0.375) t_0))
    (* c c)
    (/ c (* b -2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return fma(fma(c, fma((((a * (a * (a * a))) * (c * 6.328125)) / ((b * t_1) * (a * b))), -0.16666666666666666, (((a * a) * -0.5625) / t_1)), ((a * -0.375) / t_0)), (c * c), (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	return fma(fma(c, fma(Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(c * 6.328125)) / Float64(Float64(b * t_1) * Float64(a * b))), -0.16666666666666666, Float64(Float64(Float64(a * a) * -0.5625) / t_1)), Float64(Float64(a * -0.375) / t_0)), Float64(c * c), Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(c * N[(N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(b * t$95$1), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 2: 97.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 97.9%

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

8. Final simplification 97.9%

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

Alternative 3: 97.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 97.6

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

10. Applied rewrites 97.6%

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

11. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 97.6

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

13. Applied rewrites 97.6%

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

14. Final simplification 97.6%

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

Alternative 4: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 97.2

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

10. Applied rewrites 97.2%

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

11. Final simplification 97.2%

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

Alternative 5: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.8%

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

6. Final simplification 95.8%

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

Alternative 6: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a \cdot 1.5}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma -2.0 (/ b c) (/ (* a 1.5) b))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), ((a * 1.5) / b));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(Float64(a * 1.5) / b)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 18.1

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

4. Applied rewrites 18.1%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. div-inv N/A

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

   12. lower-*.f64 N/A

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

6. Applied rewrites 18.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}}} \]

7. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 95.5

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{\color{blue}{1.5 \cdot a}}{b}\right)} \]

9. Applied rewrites 95.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{1.5 \cdot a}{b}\right)}} \]

10. Final simplification 95.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a \cdot 1.5}{b}\right)} \]
11. Add Preprocessing

Alternative 7: 95.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.9%

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

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 95.5

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

8. Applied rewrites 95.5%

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

9. Final simplification 95.5%

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

Alternative 8: 90.7% 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 18.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 90.3

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

5. Applied rewrites 90.3%

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

Reproduce

herbie shell --seed 2024219 
(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)))