Cubic critical, medium range

Percentage Accurate: 31.1% → 99.4%

Time: 14.2s

Alternatives: 6

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{c}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (* a (* (+ b (sqrt (fma c (* a -3.0) (* b b)))) (/ -1.0 a)))))

C

double code(double a, double b, double c) {
	return c / (a * ((b + sqrt(fma(c, (a * -3.0), (b * b)))) * (-1.0 / a)));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(a * Float64(Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))) * Float64(-1.0 / a))))
end

Wolfram

code[a_, b_, c_] := N[(c / N[(a * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

4. Applied egg-rr 30.3%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

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

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

6. Applied egg-rr 30.1%

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

7. Taylor expanded in b around 0

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

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

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

   2. /-lowering-/.f64 99.2

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

9. Simplified 99.2%

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

    1. associate-/l/ N/A

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

    2. times-frac N/A

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

    3. metadata-eval N/A

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

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

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

    5. associate-/l/ N/A

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

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

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

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

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \frac{c}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \frac{-1}{a}\right)} \]
13. Add Preprocessing

Alternative 2: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 96.5%

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

6. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

   4. unpow2 N/A

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

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

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 91.3

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

8. Simplified 91.3%

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

Alternative 3: 91.0% 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 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 91.2%

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

6. Final simplification 91.2%

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

Alternative 4: 91.0% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 96.5%

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

7. Applied egg-rr 96.5%

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

8. Taylor expanded in b around inf

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

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

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

   2. *-commutative N/A

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

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

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 91.2

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

10. Simplified 91.2%

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

11. Taylor expanded in c around 0

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. unpow2 N/A

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

    9. *-lowering-*.f64 91.2

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

13. Simplified 91.2%

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

Alternative 5: 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 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 82.0

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

5. Simplified 82.0%

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

Alternative 6: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 30.3%

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

4. Applied egg-rr 30.3%

   \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
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. associate-/r* N/A

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

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

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

6. Applied egg-rr 30.3%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. sub-div N/A

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

   4. sub-div N/A

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

   5. sub-neg N/A

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

   6. div-inv N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

8. Applied egg-rr 31.6%

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

9. Taylor expanded in c around 0

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

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

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

    3. mul0-rgt 3.2

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

11. Simplified 3.2%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))