Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 3.7s

Alternatives: 8

Speedup: 1.0×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow1 99.9%

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

   7. sqr-pow 99.9%

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

   8. unpow3 99.9%

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

   9. pow-plus 100.0%

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

   10. metadata-eval 100.0%

       \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   11. unpow1/2 100.0%

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

   12. hypot-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. associate-*r* 100.0%

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

   15. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in a around 0 99.9%

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

   1. unpow2 99.9%

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

   2. unpow2 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-udef 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+99}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e+99)
   (+ (* b (* b (fma b b 4.0))) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e+99) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e+99)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e+99], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5.00000000000000008e99

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. unpow3 99.9%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. fma-def 98.0%

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

      3. unpow2 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in b around 0 98.0%

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

      1. unpow2 98.0%

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

      2. associate-*r* 98.0%

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

      3. metadata-eval 98.0%

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

      4. pow-plus 97.9%

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

      5. distribute-rgt-out 97.9%

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

      6. unpow3 97.9%

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

      7. distribute-rgt-in 97.9%

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

      8. +-commutative 97.9%

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

      9. fma-def 97.9%

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

   9. Simplified 97.9%

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

   if 5.00000000000000008e99 < (*.f64 a a)

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. unpow3 100.0%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+99}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 5: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e+99)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e+99) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 5d+99) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e+99) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 5e+99:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e+99)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 5e+99)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e+99], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5.00000000000000008e99

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. unpow3 99.9%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. fma-def 98.0%

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

      3. unpow2 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. metadata-eval 98.0%

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

      3. pow-pow 97.9%

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

      4. pow2 97.9%

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

      5. unpow2 97.9%

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

      6. distribute-rgt-out 97.9%

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

   8. Applied egg-rr 97.9%

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

   if 5.00000000000000008e99 < (*.f64 a a)

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. unpow3 100.0%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]

Alternative 6: 94.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 6.7 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 6.7e+99)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (+ (* (* a a) (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 6.7e+99) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((a * a) * (a * a)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 6.7d+99) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else
        tmp = ((a * a) * (a * a)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 6.7e+99) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((a * a) * (a * a)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 6.7e+99:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	else:
		tmp = ((a * a) * (a * a)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 6.7e+99)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 6.7e+99)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	else
		tmp = ((a * a) * (a * a)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 6.7e+99], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 6.70000000000000024e99

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. unpow3 99.9%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. fma-def 98.0%

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

      3. unpow2 98.0%

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

      4. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. metadata-eval 98.0%

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

      3. pow-pow 97.9%

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

      4. pow2 97.9%

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

      5. unpow2 97.9%

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

      6. distribute-rgt-out 97.9%

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

   8. Applied egg-rr 97.9%

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

   if 6.70000000000000024e99 < (*.f64 a a)

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. unpow3 100.0%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 97.4%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 97.4%

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

      2. pow-pow 97.4%

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

      3. pow2 97.4%

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

      4. pow2 97.4%

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

   6. Applied egg-rr 97.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 6.7 \cdot 10^{+99}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \end{array} \]

Alternative 7: 77.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+153) (+ (* (* a a) (* a a)) -1.0) (+ (* b (* b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+153) {
		tmp = ((a * a) * (a * a)) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.95d+153) then
        tmp = ((a * a) * (a * a)) + (-1.0d0)
    else
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+153) {
		tmp = ((a * a) * (a * a)) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.95e+153:
		tmp = ((a * a) * (a * a)) + -1.0
	else:
		tmp = (b * (b * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+153)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.95e+153)
		tmp = ((a * a) * (a * a)) + -1.0;
	else
		tmp = (b * (b * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+153], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.94999999999999992e153

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. unpow3 99.9%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 73.0%

      \[\leadsto \color{blue}{{a}^{4} - 1} \]
   5. Step-by-step derivation

      1. metadata-eval 73.0%

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

      2. pow-pow 73.0%

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

      3. pow2 73.0%

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

      4. pow2 73.0%

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

   6. Applied egg-rr 73.0%

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

   if 1.94999999999999992e153 < b

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow2 100.0%

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

      3. unpow1 100.0%

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

      4. sqr-pow 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow1 100.0%

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

      7. sqr-pow 100.0%

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

      8. unpow3 100.0%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around 0 97.6%

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

      1. unpow2 97.6%

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

      2. associate-*r* 97.6%

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

      3. *-commutative 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \]

Alternative 8: 51.2% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow1 99.9%

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

   7. sqr-pow 99.9%

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

   8. unpow3 99.9%

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

   9. pow-plus 100.0%

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

   10. metadata-eval 100.0%

       \[\leadsto {\left({\left(a \cdot a + b \cdot b\right)}^{\color{blue}{0.5}}\right)}^{\left(3 + 1\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   11. unpow1/2 100.0%

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

   12. hypot-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. associate-*r* 100.0%

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

   15. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 71.6%

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

   1. sub-neg 71.6%

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

   2. fma-def 71.6%

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

   3. unpow2 71.6%

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

   4. metadata-eval 71.6%

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

6. Simplified 71.6%

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

7. Taylor expanded in b around 0 54.7%

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

   1. unpow2 54.7%

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

   2. associate-*r* 54.7%

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

   3. *-commutative 54.7%

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

9. Simplified 54.7%

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

10. Final simplification 54.7%

    \[\leadsto b \cdot \left(b \cdot 4\right) + -1 \]

Reproduce

herbie shell --seed 2023292 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))