Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 6.1s

Alternatives: 9

Speedup: 1.0×

• 61.4% 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

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

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 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. 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(-1.0 + 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]), $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%

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

Alternative 3: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-32}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4 + b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-32)
   (+ -1.0 (pow a 4.0))
   (fma (* b b) (+ 4.0 (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-32) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = fma((b * b), (4.0 + (b * b)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-32)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = fma(Float64(b * b), Float64(4.0 + Float64(b * b)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-32], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.00000000000000022e-32

   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 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 inf 100.0%

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

   5. Taylor expanded in b around 0 100.0%

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

   if 4.00000000000000022e-32 < (*.f64 b b)

   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. Step-by-step derivation

      1. hypot-udef 100.0%

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

      2. sqrt-pow2 99.9%

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

      3. metadata-eval 99.9%

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

      4. unpow2 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 87.5%

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

      7. add-sqr-sqrt 87.5%

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

      8. hypot-udef 87.5%

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

      9. hypot-udef 87.5%

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

      10. pow2 87.5%

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

      11. pow2 87.5%

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

      12. pow-prod-down 87.5%

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

      13. add-sqr-sqrt 87.5%

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

      14. hypot-udef 87.5%

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

      15. hypot-udef 87.5%

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

      16. pow2 87.5%

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

      17. pow2 87.5%

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

      18. pow-prod-down 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 85.4%

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

      1. unpow2 85.4%

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

      2. unpow2 85.4%

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

   8. Simplified 85.4%

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

   9. Taylor expanded in a around 0 92.5%

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

       1. +-commutative 92.5%

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

       2. metadata-eval 92.5%

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

       3. pow-sqr 92.4%

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

       4. distribute-rgt-out 92.4%

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

       5. fma-neg 92.4%

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

       6. unpow2 92.4%

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

       7. unpow2 92.4%

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

       8. metadata-eval 92.4%

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

   11. Simplified 92.4%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-32}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4 + b \cdot b, -1\right)\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5000.0) (+ -1.0 (pow a 4.0)) (+ (pow b 4.0) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0) + (b * (b * 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (b ** 4.0d0) + (b * (b * 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0) + (b * (b * 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0) + (b * (b * 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64((b ^ 4.0) + Float64(b * Float64(b * 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (b ^ 4.0) + (b * (b * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e3

   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 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 inf 99.1%

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

   5. Taylor expanded in b around 0 98.6%

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

   if 5e3 < (*.f64 b b)

   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 93.0%

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

   5. Taylor expanded in b around inf 93.0%

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

      1. *-commutative 93.0%

         \[\leadsto \color{blue}{{b}^{2} \cdot 4} + {b}^{4} \]

      2. unpow2 93.0%

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

      3. associate-*r* 93.0%

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

      4. fma-def 93.0%

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

   7. Simplified 93.0%

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

      1. fma-udef 93.0%

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

      2. +-commutative 93.0%

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

   9. Applied egg-rr 93.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 5: 94.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5000.0) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e3

   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 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 inf 99.1%

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

   5. Taylor expanded in b around 0 98.6%

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

   if 5e3 < (*.f64 b b)

   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. Step-by-step derivation

      1. hypot-udef 100.0%

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

      2. sqrt-pow2 99.9%

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

      3. metadata-eval 99.9%

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

      4. unpow2 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 87.0%

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

      7. add-sqr-sqrt 87.0%

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

      8. hypot-udef 87.0%

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

      9. hypot-udef 87.0%

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

      10. pow2 87.0%

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

      11. pow2 87.0%

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

      12. pow-prod-down 87.0%

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

      13. add-sqr-sqrt 87.0%

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

      14. hypot-udef 87.0%

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

      15. hypot-udef 87.0%

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

      16. pow2 87.0%

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

      17. pow2 87.0%

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

      18. pow-prod-down 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 84.8%

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

      1. unpow2 84.8%

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

      2. unpow2 84.8%

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

   8. Simplified 84.8%

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

   9. Taylor expanded in b around inf 93.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 1e-5) -1.0 (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-5) then
        tmp = -1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-5:
		tmp = -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-5)
		tmp = -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-5], -1.0, N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000008e-5

   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 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 52.7%

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

   5. Taylor expanded in b around 0 52.0%

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

   if 1.00000000000000008e-5 < (*.f64 b b)

   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. Step-by-step derivation

      1. hypot-udef 100.0%

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

      2. sqrt-pow2 99.9%

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

      3. metadata-eval 99.9%

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

      4. unpow2 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-lft-in 87.2%

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

      7. add-sqr-sqrt 87.2%

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

      8. hypot-udef 87.2%

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

      9. hypot-udef 87.2%

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

      10. pow2 87.2%

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

      11. pow2 87.2%

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

      12. pow-prod-down 87.2%

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

      13. add-sqr-sqrt 87.2%

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

      14. hypot-udef 87.2%

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

      15. hypot-udef 87.2%

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

      16. pow2 87.2%

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

      17. pow2 87.2%

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

      18. pow-prod-down 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in a around 0 85.0%

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

      1. unpow2 85.0%

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

      2. unpow2 85.0%

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

   8. Simplified 85.0%

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

   9. Taylor expanded in b around inf 91.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 59.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 10^{+301}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-5)
   -1.0
   (if (<= (* b b) 1e+301) (* 2.0 (* (* a a) (* b b))) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else if ((b * b) <= 1e+301) {
		tmp = 2.0 * ((a * a) * (b * b));
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-5) then
        tmp = -1.0d0
    else if ((b * b) <= 1d+301) then
        tmp = 2.0d0 * ((a * a) * (b * b))
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else if ((b * b) <= 1e+301) {
		tmp = 2.0 * ((a * a) * (b * b));
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-5:
		tmp = -1.0
	elif (b * b) <= 1e+301:
		tmp = 2.0 * ((a * a) * (b * b))
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-5)
		tmp = -1.0;
	elseif (Float64(b * b) <= 1e+301)
		tmp = Float64(2.0 * Float64(Float64(a * a) * Float64(b * b)));
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-5)
		tmp = -1.0;
	elseif ((b * b) <= 1e+301)
		tmp = 2.0 * ((a * a) * (b * b));
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-5], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 1e+301], N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.00000000000000008e-5

   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 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 52.7%

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

   5. Taylor expanded in b around 0 52.0%

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

   if 1.00000000000000008e-5 < (*.f64 b b) < 1.00000000000000005e301

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. unpow2 99.7%

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

      3. unpow1 99.7%

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

      4. sqr-pow 99.7%

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

      5. associate-*r* 99.8%

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

      6. unpow1 99.8%

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

      7. sqr-pow 99.8%

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

      8. unpow3 99.8%

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

      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. Step-by-step derivation

      1. hypot-udef 100.0%

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

      2. sqrt-pow2 99.7%

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

      3. metadata-eval 99.7%

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

      4. unpow2 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. hypot-udef 99.7%

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

      9. hypot-udef 99.7%

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

      10. pow2 99.7%

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

      11. pow2 99.7%

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

      12. pow-prod-down 99.7%

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

      13. add-sqr-sqrt 99.7%

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

      14. hypot-udef 99.7%

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

      15. hypot-udef 99.7%

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

      16. pow2 99.7%

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

      17. pow2 99.7%

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

      18. pow-prod-down 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in a around 0 94.8%

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

      1. unpow2 94.8%

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

      2. unpow2 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in a around inf 35.7%

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

       1. unpow2 35.7%

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

       2. *-commutative 35.7%

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

       3. unpow2 35.7%

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

   11. Simplified 35.7%

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

   if 1.00000000000000005e301 < (*.f64 b 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 inf 98.8%

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

   5. Taylor expanded in b around inf 98.8%

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

      1. *-commutative 98.8%

         \[\leadsto \color{blue}{{b}^{2} \cdot 4} \]

      2. unpow2 98.8%

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

      3. associate-*r* 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 10^{+301}:\\ \;\;\;\;2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 8: 51.7% accurate, 12.8× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 1e-5) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-5:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-5)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-5], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.00000000000000008e-5

   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 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 52.7%

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

   5. Taylor expanded in b around 0 52.0%

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

   if 1.00000000000000008e-5 < (*.f64 b b)

   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 inf 73.3%

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

   5. Taylor expanded in b around inf 58.4%

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

      1. *-commutative 58.4%

         \[\leadsto \color{blue}{{b}^{2} \cdot 4} \]

      2. unpow2 58.4%

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

      3. associate-*r* 58.4%

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

   7. Simplified 58.4%

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

4. Final simplification 55.1%

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

Alternative 9: 24.6% 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 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 72.2%

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

5. Taylor expanded in b around 0 26.7%

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

6. Final simplification 26.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023287 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))