Cubic critical, wide range

Percentage Accurate: 17.7% → 99.4%

Time: 13.6s

Alternatives: 5

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: 17.7% 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: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

4. Applied egg-rr 15.1%

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

6. Applied egg-rr 15.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. lift-*.f64 N/A

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in c around 0

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

    1. mul-1-neg N/A

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

    2. *-commutative N/A

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

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

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

    4. mul-1-neg N/A

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

    5. lower-*.f64 N/A

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

    6. mul-1-neg N/A

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

    7. lower-neg.f64 99.5

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

11. Simplified 99.5%

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

Alternative 2: 95.3% 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 15.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.6%

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

Alternative 3: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

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 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. cube-mult N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 95.1

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

8. Simplified 95.1%

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

9. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 95.2

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

11. Simplified 95.2%

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

Alternative 4: 95.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. distribute-lft-in N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*l/ N/A

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

   11. associate-*r* N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

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. Final simplification 95.1%

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

Alternative 5: 90.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

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

   2. lower-/.f64 92.1

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

5. Simplified 92.1%

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

Reproduce

herbie shell --seed 2024207 
(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)))