Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.3% → 99.8%

Time: 9.4s

Alternatives: 12

Speedup: 5.9×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(2 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right), 4, b \cdot b\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma
  (* (fma (* 2.0 a) a (fma (fma -3.0 a 1.0) 4.0 (* b b))) b)
  b
  (fma (* a a) (fma a a (fma a 4.0 4.0)) -1.0)))

C

double code(double a, double b) {
	return fma((fma((2.0 * a), a, fma(fma(-3.0, a, 1.0), 4.0, (b * b))) * b), b, fma((a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0));
}

Julia

function code(a, b)
	return fma(Float64(fma(Float64(2.0 * a), a, fma(fma(-3.0, a, 1.0), 4.0, Float64(b * b))) * b), b, fma(Float64(a * a), fma(a, a, fma(a, 4.0, 4.0)), -1.0))
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(2.0 * a), $MachinePrecision] * a + N[(N[(-3.0 * a + 1.0), $MachinePrecision] * 4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(a * 4.0 + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)\right) + {a}^{4}\right)\right) - 1} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right), 4, b \cdot b\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right)} \]

6. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right), 4, b \cdot b\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right) \]
7. Add Preprocessing

Alternative 2: 96.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, 4 + a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-7)
   (fma (* (fma a (+ 4.0 a) 4.0) a) a -1.0)
   (fma (* (fma (* 2.0 a) a (fma (fma (/ b a) b -12.0) a 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-7) {
		tmp = fma((fma(a, (4.0 + a), 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma((2.0 * a), a, fma(fma((b / a), b, -12.0), a, 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-7)
		tmp = fma(Float64(fma(a, Float64(4.0 + a), 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(Float64(2.0 * a), a, fma(fma(Float64(b / a), b, -12.0), a, 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-7], N[(N[(N[(a * N[(4.0 + a), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(2.0 * a), $MachinePrecision] * a + N[(N[(N[(b / a), $MachinePrecision] * b + -12.0), $MachinePrecision] * a + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999977e-7

   1. Initial program 83.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right) - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({a}^{\color{blue}{\left(2 \cdot 2\right)}} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{a}^{2} \cdot {a}^{2}} + 4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + 4 \cdot \color{blue}{\left(\left(1 + a\right) \cdot {a}^{2}\right)}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \left({a}^{2} \cdot {a}^{2} + \color{blue}{\left(4 \cdot \left(1 + a\right)\right) \cdot {a}^{2}}\right) + \left(\mathsf{neg}\left(1\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 + a\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left({a}^{2} + 4 \cdot \left(1 + a\right)\right) + \color{blue}{-1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, {a}^{2} + 4 \cdot \left(1 + a\right), -1\right)} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, {a}^{2} + 4 \cdot \left(1 + a\right), -1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a}, {a}^{2} + 4 \cdot \left(1 + a\right), -1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot a} + 4 \cdot \left(1 + a\right), -1\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(a, a, 4 \cdot \left(1 + a\right)\right)}, -1\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, 4 \cdot \color{blue}{\left(a + 1\right)}\right), -1\right) \]

      15. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \color{blue}{a \cdot 4 + 1 \cdot 4}\right), -1\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, a \cdot 4 + \color{blue}{4}\right), -1\right) \]

      17. lower-fma.f64 99.6

          \[\leadsto \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \color{blue}{\mathsf{fma}\left(a, 4, 4\right)}\right), -1\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a + 4, 4\right) \cdot a, \color{blue}{a}, -1\right) \]

   if 4.99999999999999977e-7 < (*.f64 b b)

   1. Initial program 65.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + \left({b}^{2} \cdot \left(2 \cdot {a}^{2} + \left(4 \cdot \left(1 - 3 \cdot a\right) + {b}^{2}\right)\right) + {a}^{4}\right)\right) - 1} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right), 4, b \cdot b\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, a \cdot \left(\left(4 \cdot \frac{1}{a} + \frac{{b}^{2}}{a}\right) - 12\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 4, 4\right)\right), -1\right)\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, 4, -1\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 93.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, \mathsf{fma}\left(a \cdot a, 4, -1\right)\right) \]

   12. Taylor expanded in a around 0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, -1\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 93.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot 2, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, 4 + a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2 \cdot a, a, \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{a}, b, -12\right), a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))