Cubic critical, narrow range

Percentage Accurate: 55.5% → 99.4%

Time: 10.9s

Alternatives: 10

Speedup: 2.9×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 56.4%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 56.4%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

11. Applied rewrites 99.3%

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

12. Final simplification 99.3%

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

Alternative 3: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.066)
   (/ 1.0 (* (/ a (- (sqrt (fma b b (* (* c -3.0) a))) b)) 3.0))
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.066) {
		tmp = 1.0 / ((a / (sqrt(fma(b, b, ((c * -3.0) * a))) - b)) * 3.0);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.066)
		tmp = Float64(1.0 / Float64(Float64(a / Float64(sqrt(fma(b, b, Float64(Float64(c * -3.0) * a))) - b)) * 3.0));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.066], N[(1.0 / N[(N[(a / N[(N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.066000000000000003

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 84.7%

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

   6. Applied rewrites 84.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if -0.066000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.6

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

   9. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b} \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.066)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.066) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.066)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.066], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.066000000000000003

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 85.0

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

   4. Applied rewrites 85.0%

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

   if -0.066000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.6

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

   9. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0)) -0.066)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 1.0 (fma (/ a b) 1.5 (* -2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)) <= -0.066) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / fma((a / b), 1.5, (-2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0)) <= -0.066)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / fma(Float64(a / b), 1.5, Float64(-2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.066], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] * 1.5 + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.066000000000000003

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 84.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 84.8

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

   4. Applied rewrites 84.7%

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

   if -0.066000000000000003 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 47.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 N/A

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 47.4%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.6

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

   9. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{a}{b}, 1.5, \frac{b}{c} \cdot -2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3} \leq -0.066:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{a}{b}, 1.5, -2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}} \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   (/ a (/ (fma (* -3.0 a) c 0.0) (+ (sqrt (fma (* c -3.0) a (* b b))) b)))
   3.0)))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / (fma((-3.0 * a), c, 0.0) / (sqrt(fma((c * -3.0), a, (b * b))) + b))) * 3.0);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / Float64(fma(Float64(-3.0 * a), c, 0.0) / Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b))) * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / N[(N[(N[(-3.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 54.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 56.4

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 56.4

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

8. Applied rewrites 56.4%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3 \cdot a\right), c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

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

       \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\mathsf{fma}\left(\color{blue}{-3} \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

   13. lower-*.f64 99.2

       \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\mathsf{fma}\left(\color{blue}{-3 \cdot a}, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{1}{3 \cdot \frac{a}{\frac{\color{blue}{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}} \]

11. Final simplification 99.2%

    \[\leadsto \frac{1}{\frac{a}{\frac{\mathsf{fma}\left(-3 \cdot a, c, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}} \cdot 3} \]
12. Add Preprocessing

Alternative 7: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a}{\mathsf{fma}\left(c \cdot -3, a, 0\right)} \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   (/ (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) a) (fma (* c -3.0) a 0.0))
   3.0)))

C

double code(double a, double b, double c) {
	return 1.0 / ((((sqrt(fma((c * -3.0), a, (b * b))) + b) * a) / fma((c * -3.0), a, 0.0)) * 3.0);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * a) / fma(Float64(c * -3.0), a, 0.0)) * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(c * -3.0), $MachinePrecision] * a + 0.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 54.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 56.4

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 56.4

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

8. Applied rewrites 56.4%

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

10. Applied rewrites 99.1%

    \[\leadsto \frac{1}{3 \cdot \color{blue}{\frac{a \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}{\mathsf{fma}\left(c \cdot -3, a, 0\right)}}} \]

11. Final simplification 99.1%

    \[\leadsto \frac{1}{\frac{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a}{\mathsf{fma}\left(c \cdot -3, a, 0\right)} \cdot 3} \]
12. Add Preprocessing

Alternative 8: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(c \cdot -3, a, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ 0.3333333333333333 a) (fma (* c -3.0) a 0.0))
  (+ (sqrt (fma (* c -3.0) a (* b b))) b)))

C

double code(double a, double b, double c) {
	return ((0.3333333333333333 / a) * fma((c * -3.0), a, 0.0)) / (sqrt(fma((c * -3.0), a, (b * b))) + b);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.3333333333333333 / a) * fma(Float64(c * -3.0), a, 0.0)) / Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[(c * -3.0), $MachinePrecision] * a + 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 54.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 56.4

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 56.4

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

8. Applied rewrites 56.4%

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

10. Applied rewrites 99.1%

    \[\leadsto \color{blue}{\frac{-\left(-\mathsf{fma}\left(c \cdot -3, a, 0\right)\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}} \]

11. Final simplification 99.1%

    \[\leadsto \frac{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(c \cdot -3, a, 0\right)}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b} \]
12. Add Preprocessing

Alternative 9: 81.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. unpow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-pow.f64 N/A

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

4. Applied rewrites 54.8%

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

6. Applied rewrites 54.8%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 81.4

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

9. Applied rewrites 81.4%

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

10. Final simplification 81.4%

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

Alternative 10: 64.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Final simplification 64.5%

   \[\leadsto \frac{c}{b} \cdot -0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))