Cubic critical, wide range

Percentage Accurate: 18.1% → 99.3%

Time: 13.4s

Alternatives: 6

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.1% 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: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

   1. lift-*.f64 19.0

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

   2. /-rgt-identity N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval 19.0

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

4. Applied egg-rr 19.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{1}{\frac{0.3333333333333333}{a}}}} \]
5. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r/ N/A

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

   10. /-rgt-identity N/A

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

6. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} - b\right) \cdot 0.3333333333333333\right) \cdot \frac{1}{a}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. flip-- N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

8. Applied egg-rr 19.3%

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

9. Taylor expanded in b around 0

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

    1. mul-1-neg N/A

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

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

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

    3. mul-1-neg N/A

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

    4. lower-*.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 99.3

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 2: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 95.4%

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

Alternative 3: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 95.4%

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

6. Taylor expanded in c around 0

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

   1. lower-*.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. lower-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. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 95.4

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

8. Simplified 95.4%

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

Alternative 4: 94.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. sub-neg N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r/ N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r* N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. associate-*l* N/A

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

   14. lower-fma.f64 N/A

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

5. Simplified 95.1%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 95.1

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

8. Simplified 95.1%

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

Alternative 5: 90.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 89.7

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

5. Simplified 89.7%

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

Alternative 6: 89.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 89.7

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

5. Simplified 89.7%

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

   1. associate-/l* N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 89.4

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

7. Applied egg-rr 89.4%

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

8. Final simplification 89.4%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))