Cubic critical, medium range

Percentage Accurate: 31.6% → 95.5%

Time: 13.5s

Alternatives: 7

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.6% 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: 95.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Taylor expanded in 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. Simplified 95.8%

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

7. Applied egg-rr 95.8%

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

8. Final simplification 95.8%

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

Alternative 2: 95.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Taylor expanded in 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. Simplified 95.8%

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

7. Applied egg-rr 95.5%

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

8. Final simplification 95.5%

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

Alternative 3: 93.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(a, Float64(fma(c, Float64(c * -0.375), Float64(Float64(a * Float64(c * Float64(Float64(c * c) * -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[(c * N[(c * -0.375), $MachinePrecision] + N[(N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $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 32.0%

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

3. Taylor expanded in 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. Simplified 95.8%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

8. Simplified 94.1%

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

Alternative 4: 93.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 94.1%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

8. Simplified 94.0%

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

9. Final simplification 94.0%

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

Alternative 5: 90.6% 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 32.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 90.8%

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

6. Final simplification 90.8%

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

Alternative 6: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 90.8%

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

6. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. associate-*r* N/A

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

   11. associate-*r/ N/A

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

   12. associate-/l* N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

8. Simplified 90.7%

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

9. Final simplification 90.7%

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

Alternative 7: 81.1% 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 32.0%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 80.7

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

5. Simplified 80.7%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))