Cubic critical, narrow range

Percentage Accurate: 55.5% → 99.3%

Time: 14.8s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \frac{t\_0}{\left(b + \sqrt{\mathsf{fma}\left(b, b, t\_0\right)}\right) \cdot \left(a \cdot 3\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (/ t_0 (* (+ b (sqrt (fma b b t_0))) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return t_0 / ((b + sqrt(fma(b, b, t_0))) * (a * 3.0));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	return Float64(t_0 / Float64(Float64(b + sqrt(fma(b, b, t_0))) * Float64(a * 3.0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[(b + N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.8%

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

4. Applied rewrites 53.8%

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

6. Applied rewrites 55.0%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--r+ N/A

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

   5. lower--.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.004)
   (/ (- (sqrt (+ (* c (* a -3.0)) (* b b))) b) (* a 3.0))
   (fma -0.5 (/ c b) (/ (* -0.375 (* a (* c c))) (* b (* b b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.004) {
		tmp = (sqrt(((c * (a * -3.0)) + (b * b))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, (c / b), ((-0.375 * (a * (c * c))) / (b * (b * b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.004)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * Float64(a * -3.0)) + Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(Float64(-0.375 * Float64(a * Float64(c * c))) / Float64(b * Float64(b * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(N[Sqrt[N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(-0.375 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0040000000000000001

   1. Initial program 77.3%

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

   4. Applied rewrites 77.3%

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

   6. Applied rewrites 77.4%

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

   8. Applied rewrites 77.3%

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

   if -0.0040000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.1%

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

   4. Applied rewrites 46.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 46.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.5

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

   9. Applied rewrites 88.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.004:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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)))