Cubic critical, medium range

Percentage Accurate: 31.1% → 99.2%

Time: 14.2s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

4. Applied egg-rr 30.3%

   \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \frac{1}{a}}{-3}} \]
5. Step-by-step derivation

   1. un-div-inv N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

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

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

6. Applied egg-rr 31.3%

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

7. Taylor expanded in b around 0

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

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

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

   2. *-lowering-*.f64 99.2

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

9. Simplified 99.2%

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

Alternative 2: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

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

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

   13. associate-*r/ N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 91.3

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

5. Simplified 91.3%

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

Alternative 3: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in b around inf

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

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

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

5. Simplified 91.2%

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

Alternative 4: 91.0% 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 30.3%

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

3. Taylor expanded in b around inf

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

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

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

5. Simplified 91.2%

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

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. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. 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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 91.2

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

8. Simplified 91.2%

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

Alternative 5: 81.6% 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 30.3%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.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. *-lowering-*.f64 82.0

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

5. Simplified 82.0%

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

Alternative 6: 81.4% 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 30.3%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.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. *-lowering-*.f64 82.0

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

5. Simplified 82.0%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

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

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

   4. /-lowering-/.f64 81.8

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

7. Applied egg-rr 81.8%

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

8. Final simplification 81.8%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))