Cubic critical, narrow range

Percentage Accurate: 55.7% → 99.3%

Time: 18.9s

Alternatives: 9

Speedup: 1.1×

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: 55.7% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 56.6%

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

6. Applied egg-rr 57.5%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, -3 \cdot c, 0\right)}{a} \cdot \frac{\frac{0.3333333333333333}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{1}} \]

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 56.6%

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

6. Applied egg-rr 57.5%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, -3 \cdot c, 0\right)}{a} \cdot \frac{\frac{0.3333333333333333}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{1}} \]

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 56.6%

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

6. Applied egg-rr 57.5%

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in a around 0

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

    1. *-commutative N/A

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

    2. rem-square-sqrt N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. rem-square-sqrt 99.2

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 55.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 55.9

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

4. Applied egg-rr 55.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-rgt-identity N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 56.0

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

   8. lift-fma.f64 N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 56.0

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

6. Applied egg-rr 56.0%

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

7. Final simplification 56.0%

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

Alternative 5: 55.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 55.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 55.9

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

6. Applied egg-rr 55.9%

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

Alternative 6: 55.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 55.9%

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

5. Final simplification 55.9%

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

Alternative 7: 15.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. unsub-neg N/A

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

   10. lower--.f64 55.9

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

4. Applied egg-rr 55.9%

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

5. Taylor expanded in c around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 15.1

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

7. Simplified 15.1%

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

8. Final simplification 15.1%

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

Alternative 8: 11.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return -1.0 / (b * (b * (b * 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 = (-1.0d0) / (b * (b * (b * b)))
end function

Java

public static double code(double a, double b, double c) {
	return -1.0 / (b * (b * (b * b)));
}

Python

def code(a, b, c):
	return -1.0 / (b * (b * (b * b)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -1.0 / (b * (b * (b * b)));
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 56.6%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{{b}^{4}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{{b}^{4}}} \]

   2. metadata-eval N/A

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

   3. pow-plus N/A

      \[\leadsto \frac{-1}{\color{blue}{{b}^{3} \cdot b}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{{b}^{3} \cdot b}} \]

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 10.9

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto \frac{-1}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} \]
9. Add Preprocessing

Alternative 9: 10.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in b around inf

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

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b}}{3 \cdot a} \]

   2. lower-*.f64 1.6

      \[\leadsto \frac{\color{blue}{b \cdot b}}{3 \cdot a} \]

5. Simplified 1.6%

   \[\leadsto \frac{\color{blue}{b \cdot b}}{3 \cdot a} \]

6. Taylor expanded in a around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{b \cdot b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

   2. lower-neg.f64 10.9

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

8. Simplified 10.9%

   \[\leadsto \frac{b \cdot b}{\color{blue}{-a}} \]
9. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \frac{b \cdot b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{b \cdot \frac{b}{\mathsf{neg}\left(a\right)}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)} \cdot b} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{\mathsf{neg}\left(a\right)} \cdot b} \]

   5. lower-/.f64 10.9

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

10. Applied egg-rr 10.9%

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

11. Final simplification 10.9%

    \[\leadsto b \cdot \frac{b}{-a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))