Cubic critical, wide range

Percentage Accurate: 18.3% → 97.5%

Time: 15.1s

Alternatives: 5

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{2}\\ \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{t_0}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(c \cdot a\right) \cdot \left(t_0 \cdot 2.25\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(c \cdot a\right) \cdot \left(2.25 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 2.0)))
   (fma
    -0.5
    (/ c b)
    (*
     -0.16666666666666666
     (+
      (* (/ t_0 (pow b 3.0)) (/ 2.25 a))
      (+
       (/ (* 1.5 (* (* c a) (* t_0 2.25))) (* a (pow b 5.0)))
       (/
        (fma
         (pow (* c (* a -1.5)) 4.0)
         0.25
         (* (* a 1.5) (* c (* 1.5 (* (* c a) (* 2.25 (* (* c a) (* c a))))))))
        (* a (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 2.0);
	return fma(-0.5, (c / b), (-0.16666666666666666 * (((t_0 / pow(b, 3.0)) * (2.25 / a)) + (((1.5 * ((c * a) * (t_0 * 2.25))) / (a * pow(b, 5.0))) + (fma(pow((c * (a * -1.5)), 4.0), 0.25, ((a * 1.5) * (c * (1.5 * ((c * a) * (2.25 * ((c * a) * (c * a)))))))) / (a * pow(b, 7.0)))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 2.0
	return fma(-0.5, Float64(c / b), Float64(-0.16666666666666666 * Float64(Float64(Float64(t_0 / (b ^ 3.0)) * Float64(2.25 / a)) + Float64(Float64(Float64(1.5 * Float64(Float64(c * a) * Float64(t_0 * 2.25))) / Float64(a * (b ^ 5.0))) + Float64(fma((Float64(c * Float64(a * -1.5)) ^ 4.0), 0.25, Float64(Float64(a * 1.5) * Float64(c * Float64(1.5 * Float64(Float64(c * a) * Float64(2.25 * Float64(Float64(c * a) * Float64(c * a)))))))) / Float64(a * (b ^ 7.0)))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]}, N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(2.25 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] * N[(t$95$0 * 2.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(c * N[(a * -1.5), $MachinePrecision]), $MachinePrecision], 4.0], $MachinePrecision] * 0.25 + N[(N[(a * 1.5), $MachinePrecision] * N[(c * N[(1.5 * N[(N[(c * a), $MachinePrecision] * N[(2.25 * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.2%

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

   1. flip3-- 17.5%

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

   2. clear-num 17.3%

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

   3. pow2 17.3%

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

   4. pow2 17.3%

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

   5. pow-prod-up 17.6%

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

   6. metadata-eval 17.6%

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

   7. distribute-rgt-out 17.6%

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

   8. associate-*l* 17.6%

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

   9. +-commutative 17.6%

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

   10. fma-def 17.6%

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

   11. associate-*l* 17.6%

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

3. Applied egg-rr 17.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]

4. Taylor expanded in b around inf 97.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

6. Simplified 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\left(\frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0\right), a \cdot \left(c \cdot 0\right)\right)}{a \cdot {b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot {\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2}\right)}^{2} + 0\right) + \mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{2} + 0\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{a \cdot {b}^{7}}\right)\right)} \]

8. Applied egg-rr 97.8%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right)}{a \cdot {b}^{5}}\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4} \cdot 0.25 + \mathsf{fma}\left(a \cdot 1.5, c \cdot \mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right), 0\right)}{a \cdot {b}^{7}}}\right) \]
9. Step-by-step derivation

   1. distribute-lft-in 97.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right)}{a \cdot {b}^{5}}\right) + \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4} \cdot 0.25 + \mathsf{fma}\left(a \cdot 1.5, c \cdot \mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right), 0\right)}{a \cdot {b}^{7}}\right)}\right) \]

   2. associate-+l+ 97.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \color{blue}{\left(\frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \left(\frac{\mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right)}{a \cdot {b}^{5}} + \frac{{\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4} \cdot 0.25 + \mathsf{fma}\left(a \cdot 1.5, c \cdot \mathsf{fma}\left(1.5, \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right) \cdot \left(c \cdot a\right), 0\right), 0\right)}{a \cdot {b}^{7}}\right)\right)}\right) \]

10. Simplified 97.8%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)}\right) \]
11. Step-by-step derivation

    1. unpow2 97.8%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    2. *-commutative 97.8%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot \left(\color{blue}{\left(c \cdot a\right)} \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

    3. *-commutative 97.8%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot \left(\left(c \cdot a\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

12. Applied egg-rr 97.8%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(a \cdot c\right) \cdot \left(2.25 \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

13. Final simplification 97.8%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} \cdot \frac{2.25}{a} + \left(\frac{1.5 \cdot \left(\left(c \cdot a\right) \cdot \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right)\right)}{a \cdot {b}^{5}} + \frac{\mathsf{fma}\left({\left(c \cdot \left(a \cdot -1.5\right)\right)}^{4}, 0.25, \left(a \cdot 1.5\right) \cdot \left(c \cdot \left(1.5 \cdot \left(\left(c \cdot a\right) \cdot \left(2.25 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)\right)\right)\right)\right)\right)}{a \cdot {b}^{7}}\right)\right)\right) \]

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (*
     -0.16666666666666666
     (* (/ (pow (* c a) 4.0) a) (/ 6.328125 (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((c * a) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((c * a), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((c * a), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((c * a) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

2. Taylor expanded in b around inf 97.7%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

3. Taylor expanded in c around 0 97.7%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
4. Step-by-step derivation

   1. distribute-rgt-out 97.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 97.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 97.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. times-frac 97.7%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

5. Simplified 97.7%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]

6. Final simplification 97.7%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \]

Alternative 3: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

2. Taylor expanded in b around inf 96.7%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

3. Final simplification 96.7%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

2. Taylor expanded in b around inf 95.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

3. Final simplification 95.1%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]

Alternative 5: 90.0% accurate, 23.2× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

2. Taylor expanded in b around inf 90.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Step-by-step derivation

   1. *-commutative 90.8%

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

   2. associate-*l/ 90.8%

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

4. Simplified 90.8%

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

5. Final simplification 90.8%

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

Reproduce

herbie shell --seed 2023333 
(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)))