Bouland and Aaronson, Equation (25)

Percentage Accurate: 72.5% → 100.0%

Time: 9.4s

Alternatives: 7

Speedup: 6.8×

• 61.0% 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: 72.5% 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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, b \cdot \left(b \cdot \left(1 + a \cdot -3\right)\right)\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))
      INFINITY)
   (+
    (+
     (* 4.0 (fma (* a a) (+ a 1.0) (* b (* b (+ 1.0 (* a -3.0))))))
     (pow (hypot a b) 4.0))
    -1.0)
   (+ -1.0 (* (* a a) (+ 4.0 (+ (* (* b b) 2.0) (* a (+ a 4.0))))))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= ((double) INFINITY)) {
		tmp = ((4.0 * fma((a * a), (a + 1.0), (b * (b * (1.0 + (a * -3.0)))))) + pow(hypot(a, b), 4.0)) + -1.0;
	} else {
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))) <= Inf)
		tmp = Float64(Float64(Float64(4.0 * fma(Float64(a * a), Float64(a + 1.0), Float64(b * Float64(b * Float64(1.0 + Float64(a * -3.0)))))) + (hypot(a, b) ^ 4.0)) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(Float64(Float64(b * b) * 2.0) + Float64(a * Float64(a + 4.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(b * N[(b * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, b \cdot \left(b \cdot \left(1 + -3 \cdot a\right)\right)\right) + {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) + -1} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 99.8%

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

      2. unpow2 99.8%

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

      3. distribute-lft-in 86.5%

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

      4. fma-define 86.5%

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

      5. add-sqr-sqrt 86.5%

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

      6. pow2 86.5%

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

      7. fma-define 86.5%

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

      8. hypot-define 86.5%

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

      9. pow2 86.5%

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

      10. fma-define 86.5%

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

      11. add-sqr-sqrt 86.5%

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

      12. pow2 86.5%

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

      13. fma-define 86.5%

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

      14. hypot-define 86.5%

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

      15. pow2 86.5%

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

   6. Applied egg-rr 86.5%

      \[\leadsto \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, b \cdot \left(b \cdot \left(1 + -3 \cdot a\right)\right)\right) + \color{blue}{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {a}^{2} + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot {b}^{2}\right)}\right) + -1 \]
   7. Step-by-step derivation

      1. distribute-lft-out 99.8%

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

      2. rem-square-sqrt 99.8%

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

      3. unpow2 99.8%

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

      4. unpow2 99.8%

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

      5. hypot-undefine 99.8%

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

      6. unpow2 99.8%

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

      7. unpow2 99.8%

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

      8. hypot-undefine 99.8%

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

      9. unpow2 99.8%

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

      10. pow-sqr 99.9%

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

      11. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 0.0%

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

   3. Simplified 8.8%

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

   5. Taylor expanded in a around -inf 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. pow2 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(4 + \left(2 \cdot \left(b \cdot b\right) + a \cdot \left(4 + a\right)\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, b \cdot \left(b \cdot \left(1 + a \cdot -3\right)\right)\right) + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY)
     (+ t_0 -1.0)
     (+ -1.0 (* (* a a) (+ 4.0 (+ (* (* b b) 2.0) (* a (+ a 4.0)))))))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(Float64(Float64(b * b) * 2.0) + Float64(a * Float64(a + 4.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 0.0%

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

   3. Simplified 8.8%

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

   5. Taylor expanded in a around -inf 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. pow2 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(4 + \left(2 \cdot \left(b \cdot b\right) + a \cdot \left(4 + a\right)\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00058 \lor \neg \left(a \leq 2.6 \cdot 10^{+43}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.00058) (not (<= a 2.6e+43)))
   (+ -1.0 (* (* a a) (+ 4.0 (+ (* (* b b) 2.0) (* a (+ a 4.0))))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.00058) || !(a <= 2.6e+43)) {
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-0.00058d0)) .or. (.not. (a <= 2.6d+43))) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (((b * b) * 2.0d0) + (a * (a + 4.0d0)))))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.00058) || !(a <= 2.6e+43)) {
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.00058) or not (a <= 2.6e+43):
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.00058) || !(a <= 2.6e+43))
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(Float64(Float64(b * b) * 2.0) + Float64(a * Float64(a + 4.0))))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.00058) || ~((a <= 2.6e+43)))
		tmp = -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.00058], N[Not[LessEqual[a, 2.6e+43]], $MachinePrecision]], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision] + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8e-4 or 2.60000000000000021e43 < a

   1. Initial program 47.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 47.8%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in a around -inf 96.8%

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

   6. Taylor expanded in a around 0 96.7%

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

      1. pow2 96.7%

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

   8. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

   10. Applied egg-rr 96.7%

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

   if -5.8e-4 < a < 2.60000000000000021e43

   1. Initial program 98.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in b around inf 97.6%

      \[\leadsto \color{blue}{{b}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00058 \lor \neg \left(a \leq 2.6 \cdot 10^{+43}\right):\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+22}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+22)
   (+ -1.0 (* (* a a) (+ 4.0 (* a (+ a 4.0)))))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+22) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+22) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 + (a * (a + 4.0d0))))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+22) {
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+22:
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+22)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+22)
		tmp = -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+22], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e22

   1. Initial program 87.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in a around -inf 76.9%

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

   6. Taylor expanded in b around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. associate-*r/ 76.9%

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

      3. metadata-eval 76.9%

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

      4. distribute-lft-in 76.9%

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

      5. metadata-eval 76.9%

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

      6. neg-mul-1 76.9%

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

      7. distribute-neg-frac 76.9%

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

      8. metadata-eval 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in a around 0 98.2%

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

       1. unpow2 98.2%

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

   11. Applied egg-rr 98.2%

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

   if 2e22 < (*.f64 b b)

   1. Initial program 60.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 60.3%

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

      2. +-commutative 60.3%

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

      3. +-commutative 60.3%

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

      4. sub-neg 60.3%

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

      5. associate-+l+ 60.3%

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

      6. +-commutative 60.3%

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

      7. fma-define 60.3%

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

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), \mathsf{fma}\left(a, a, {a}^{3}\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + -1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 67.1%

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

      2. cube-mult 67.1%

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

      3. distribute-rgt1-in 67.1%

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

      4. +-commutative 67.1%

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

      5. associate-*r* 67.1%

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

      6. +-commutative 67.1%

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

   6. Applied egg-rr 67.1%

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

   7. Taylor expanded in b around inf 91.1%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+22}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0 + ((a * a) * (4.0 + (((b * b) * 2.0) + (a * (a + 4.0)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 73.3%

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

3. Simplified 75.6%

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

5. Taylor expanded in a around -inf 66.1%

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

6. Taylor expanded in a around 0 79.1%

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

   1. pow2 79.1%

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

8. Applied egg-rr 79.1%

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

   1. unpow2 79.1%

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

10. Applied egg-rr 79.1%

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

11. Final simplification 79.1%

    \[\leadsto -1 + \left(a \cdot a\right) \cdot \left(4 + \left(\left(b \cdot b\right) \cdot 2 + a \cdot \left(a + 4\right)\right)\right) \]
12. Add Preprocessing

Alternative 6: 70.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))));
}

Python

def code(a, b):
	return -1.0 + ((a * a) * (4.0 + (a * (a + 4.0))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 73.3%

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

3. Simplified 75.6%

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

5. Taylor expanded in a around -inf 66.1%

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

6. Taylor expanded in b around 0 57.1%

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

   1. associate-*r/ 57.1%

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

   2. associate-*r/ 57.1%

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

   3. metadata-eval 57.1%

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

   4. distribute-lft-in 57.1%

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

   5. metadata-eval 57.1%

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

   6. neg-mul-1 57.1%

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

   7. distribute-neg-frac 57.1%

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

   8. metadata-eval 57.1%

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

8. Simplified 57.1%

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

9. Taylor expanded in a around 0 67.3%

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

    1. unpow2 79.1%

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

11. Applied egg-rr 67.3%

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

12. Final simplification 67.3%

    \[\leadsto -1 + \left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) \]
13. Add Preprocessing

Alternative 7: 24.5% accurate, 130.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 73.3%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 73.3%

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

3. Simplified 75.6%

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

5. Taylor expanded in a around inf 66.4%

   \[\leadsto \color{blue}{{a}^{4}} + -1 \]

6. Taylor expanded in a around 0 24.0%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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))