Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 6.2s

Alternatives: 10

Speedup: 1.0×

• 60.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(b * N[(4.0 * b), $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. Taylor expanded in a around 0 87.5%

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

   1. associate-+r+ 87.5%

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

   2. fma-def 87.5%

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

   3. unpow2 87.5%

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

   4. unpow2 87.5%

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

6. Simplified 87.5%

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

   1. fma-udef 87.5%

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

   2. +-commutative 87.5%

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

   3. *-commutative 87.5%

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

   4. unswap-sqr 100.0%

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

   5. pow2 100.0%

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

8. Applied egg-rr 100.0%

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

   1. unpow2 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-*r* 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 3: 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]

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%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\left({b}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= (* a a) 7.5e+15)
     (+ (+ (pow b 4.0) t_0) -1.0)
     (+ (+ t_0 (* (* a a) (* a a))) -1.0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 7.5e+15) {
		tmp = (pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (t_0 + ((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) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (b * b)
    if ((a * a) <= 7.5d+15) then
        tmp = ((b ** 4.0d0) + t_0) + (-1.0d0)
    else
        tmp = (t_0 + ((a * a) * (a * a))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 7.5e+15) {
		tmp = (Math.pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (t_0 + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (b * b)
	tmp = 0
	if (a * a) <= 7.5e+15:
		tmp = (math.pow(b, 4.0) + t_0) + -1.0
	else:
		tmp = (t_0 + ((a * a) * (a * a))) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(b * b))
	tmp = 0.0
	if (Float64(a * a) <= 7.5e+15)
		tmp = Float64(Float64((b ^ 4.0) + t_0) + -1.0);
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(a * a) * Float64(a * a))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (b * b);
	tmp = 0.0;
	if ((a * a) <= 7.5e+15)
		tmp = ((b ^ 4.0) + t_0) + -1.0;
	else
		tmp = (t_0 + ((a * a) * (a * a))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 7.5e+15], N[(N[(N[Power[b, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(t$95$0 + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 7.5e15

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

      1. unpow-prod-down 98.6%

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

      2. pow-prod-up 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. pow-prod-up 98.6%

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

      7. pow2 98.6%

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

      8. pow2 98.6%

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

      9. distribute-rgt-out 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. pow2 98.6%

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

      4. pow2 98.6%

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

      5. pow-sqr 98.8%

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

      6. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

   if 7.5e15 < (*.f64 a a)

   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 inf 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

      1. unpow2 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \cdot a \leq 10^{+16}:\\ \;\;\;\;\left({b}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + t_0\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= (* a a) 1e+16)
     (+ (+ (pow b 4.0) t_0) -1.0)
     (+ (+ (pow a 4.0) t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 1e+16) {
		tmp = (pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (pow(a, 4.0) + t_0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (b * b)
    if ((a * a) <= 1d+16) then
        tmp = ((b ** 4.0d0) + t_0) + (-1.0d0)
    else
        tmp = ((a ** 4.0d0) + t_0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 1e+16) {
		tmp = (Math.pow(b, 4.0) + t_0) + -1.0;
	} else {
		tmp = (Math.pow(a, 4.0) + t_0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (b * b)
	tmp = 0
	if (a * a) <= 1e+16:
		tmp = (math.pow(b, 4.0) + t_0) + -1.0
	else:
		tmp = (math.pow(a, 4.0) + t_0) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(b * b))
	tmp = 0.0
	if (Float64(a * a) <= 1e+16)
		tmp = Float64(Float64((b ^ 4.0) + t_0) + -1.0);
	else
		tmp = Float64(Float64((a ^ 4.0) + t_0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (b * b);
	tmp = 0.0;
	if ((a * a) <= 1e+16)
		tmp = ((b ^ 4.0) + t_0) + -1.0;
	else
		tmp = ((a ^ 4.0) + t_0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 1e+16], N[(N[(N[Power[b, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[Power[a, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e16

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

      1. unpow-prod-down 98.6%

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

      2. pow-prod-up 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. pow-prod-up 98.6%

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

      7. pow2 98.6%

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

      8. pow2 98.6%

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

      9. distribute-rgt-out 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. +-commutative 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. pow2 98.6%

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

      4. pow2 98.6%

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

      5. pow-sqr 98.8%

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

      6. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

   if 1e16 < (*.f64 a a)

   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 inf 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

      1. unpow-prod-down 96.4%

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

      2. pow-sqr 96.4%

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

      3. metadata-eval 96.4%

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

      4. *-un-lft-identity 96.4%

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

      5. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+16}:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 6: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 7.5e+15)
   (+ (* b (* b (fma b b 4.0))) -1.0)
   (+ (+ (* 4.0 (* b b)) (* (* a a) (* a a))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 7.5e+15) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 7.5e+15)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(a * a) * Float64(a * a))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 7.5e+15], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 7.5e15

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in b around 0 98.8%

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

      1. unpow2 98.8%

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

      2. metadata-eval 98.8%

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

      3. pow-plus 98.7%

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

      4. unpow3 98.7%

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

      5. associate-*r* 98.6%

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

      6. distribute-rgt-in 98.6%

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

      7. associate-*l* 98.7%

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

      8. +-commutative 98.7%

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

      9. fma-udef 98.7%

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

   7. Simplified 98.7%

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

   if 7.5e15 < (*.f64 a a)

   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 inf 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

      1. unpow2 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 7: 97.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 7.5e+15)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (+ (+ (* 4.0 (* b b)) (* (* a a) (* a a))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 7.5e+15) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((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) <= 7.5d+15) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * b)) + ((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) <= 7.5e+15) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 7.5e+15:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 7.5e+15)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + 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) <= 7.5e+15)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 7.5e+15], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 7.5e15

   1. Initial program 99.8%

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

   2. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

      1. unpow-prod-down 98.6%

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

      2. pow-prod-up 98.8%

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

      3. metadata-eval 98.8%

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

      4. +-commutative 98.8%

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

      5. metadata-eval 98.8%

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

      6. pow-prod-up 98.6%

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

      7. pow2 98.6%

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

      8. pow2 98.6%

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

      9. distribute-rgt-out 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 7.5e15 < (*.f64 a a)

   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 inf 96.3%

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

      1. unpow2 96.3%

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

   4. Simplified 96.3%

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

      1. unpow2 96.4%

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

   6. Applied egg-rr 96.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 8: 69.8% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $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 70.2%

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

   1. unpow2 70.2%

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

4. Simplified 70.2%

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

   1. unpow-prod-down 70.2%

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

   2. pow-prod-up 70.3%

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

   3. metadata-eval 70.3%

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

   4. +-commutative 70.3%

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

   5. metadata-eval 70.3%

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

   6. pow-prod-up 70.2%

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

   7. pow2 70.2%

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

   8. pow2 70.2%

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

   9. distribute-rgt-out 70.2%

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

6. Applied egg-rr 70.2%

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

7. Final simplification 70.2%

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

Alternative 9: 51.5% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(b * N[(4.0 * b), $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 70.2%

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

   1. unpow2 70.2%

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

4. Simplified 70.2%

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

5. Taylor expanded in b around 0 49.7%

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

   1. unpow2 49.7%

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

   2. *-commutative 49.7%

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

   3. associate-*l* 49.7%

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

7. Simplified 49.7%

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

8. Final simplification 49.7%

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

Alternative 10: 25.2% accurate, 116.0× 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 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 70.3%

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

5. Taylor expanded in b around 0 26.8%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 26.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))