Cubic critical, medium range

Percentage Accurate: 31.1% → 99.4%

Time: 13.9s

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: 31.1% 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: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

4. Applied egg-rr 30.1%

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

   1. associate-/l/ N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

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

6. Applied egg-rr 31.2%

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

   3. --lowering--.f64 N/A

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

   4. --lowering--.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 99.4

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

8. Applied egg-rr 99.4%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

4. Applied egg-rr 30.1%

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

   1. associate-/l/ N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

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

6. Applied egg-rr 31.2%

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

7. Taylor expanded in b around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 99.1

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \frac{3 \cdot \left(c \cdot a\right)}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \]
11. Add Preprocessing

Alternative 3: 91.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

3. Taylor expanded in b around inf

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

5. Simplified 96.5%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 N/A

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

   19. associate-*r/ N/A

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

   20. /-lowering-/.f64 N/A

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

   21. *-lowering-*.f64 92.2

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

8. Simplified 92.2%

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

9. Final simplification 92.2%

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

Alternative 4: 91.1% 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 30.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. /-lowering-/.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 92.2%

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

6. Final simplification 92.2%

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

Alternative 5: 91.0% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

3. Taylor expanded in b around inf

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

5. Simplified 96.5%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. associate-*r* N/A

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

   11. associate-*r/ N/A

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

   12. distribute-rgt-in N/A

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

   13. distribute-lft-in N/A

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

8. Simplified 92.1%

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

Alternative 6: 90.8% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

3. Taylor expanded in c around 0

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

5. Simplified 94.8%

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

6. Taylor expanded in b around inf

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

   1. /-lowering-/.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. accelerator-lowering-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. associate-/l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 91.9

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

8. Simplified 91.9%

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

Alternative 7: 81.6% 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 30.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-lowering-*.f64 N/A

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

   2. /-lowering-/.f64 82.5

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

5. Simplified 82.5%

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

Reproduce

herbie shell --seed 2024204 
(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)))