Cubic critical, narrow range

Percentage Accurate: 55.7% → 99.3%

Time: 14.9s

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: 55.7% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 55.3%

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

6. Applied rewrites 57.2%

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

7. Taylor expanded in a around 0

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

   1. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 6.2:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (if (<= b 6.2)
     (/ (- (* b b) t_0) (* (* a -3.0) (+ b (sqrt t_0))))
     (fma a (/ (* -0.375 (* c c)) (* b (* b b))) (/ (* c -0.5) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (b <= 6.2) {
		tmp = ((b * b) - t_0) / ((a * -3.0) * (b + sqrt(t_0)));
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * (b * b))), ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 6.2)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))));
	else
		tmp = fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * Float64(b * b))), Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.2], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.20000000000000018

   1. Initial program 80.1%

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

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\frac{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}{a}}{-3}} \]
   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. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 81.6%

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

   if 6.20000000000000018 < b

   1. Initial program 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. 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. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r/ N/A

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

      18. lower-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

Reproduce

herbie shell --seed 2024228 
(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)))