Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 5.2s

Alternatives: 11

Speedup: 1.0×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow1 99.9%

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

   7. sqr-pow 99.9%

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

   8. unpow3 99.9%

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

   9. pow-plus 100.0%

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

   10. metadata-eval 100.0%

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

   11. unpow1/2 100.0%

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

   12. hypot-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. associate-*r* 100.0%

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

   15. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(-1.0 + 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. 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. sqr-pow 99.9%

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

   3. sqr-pow 99.9%

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

   4. fma-def 99.9%

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

   5. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4.9 \cdot 10^{-174}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 7 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4.7 \cdot 10^{+61}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0)
   -1.0
   (if (<= (* b b) 4.9e-174)
     (pow a 4.0)
     (if (<= (* b b) 7e-11)
       -1.0
       (if (<= (* b b) 4.7e+61) (pow a 4.0) (* (* b b) (+ 4.0 (* b b))))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = -1.0;
	} else if ((b * b) <= 4.9e-174) {
		tmp = pow(a, 4.0);
	} else if ((b * b) <= 7e-11) {
		tmp = -1.0;
	} else if ((b * b) <= 4.7e+61) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	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) <= 0.0d0) then
        tmp = -1.0d0
    else if ((b * b) <= 4.9d-174) then
        tmp = a ** 4.0d0
    else if ((b * b) <= 7d-11) then
        tmp = -1.0d0
    else if ((b * b) <= 4.7d+61) then
        tmp = a ** 4.0d0
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = -1.0;
	} else if ((b * b) <= 4.9e-174) {
		tmp = Math.pow(a, 4.0);
	} else if ((b * b) <= 7e-11) {
		tmp = -1.0;
	} else if ((b * b) <= 4.7e+61) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.0:
		tmp = -1.0
	elif (b * b) <= 4.9e-174:
		tmp = math.pow(a, 4.0)
	elif (b * b) <= 7e-11:
		tmp = -1.0
	elif (b * b) <= 4.7e+61:
		tmp = math.pow(a, 4.0)
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.0)
		tmp = -1.0;
	elseif (Float64(b * b) <= 4.9e-174)
		tmp = a ^ 4.0;
	elseif (Float64(b * b) <= 7e-11)
		tmp = -1.0;
	elseif (Float64(b * b) <= 4.7e+61)
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.0)
		tmp = -1.0;
	elseif ((b * b) <= 4.9e-174)
		tmp = a ^ 4.0;
	elseif ((b * b) <= 7e-11)
		tmp = -1.0;
	elseif ((b * b) <= 4.7e+61)
		tmp = a ^ 4.0;
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 4.9e-174], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 7e-11], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 4.7e+61], N[Power[a, 4.0], $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 0.0 or 4.90000000000000009e-174 < (*.f64 b b) < 7.00000000000000038e-11

   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* 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 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 59.9%

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

      1. sub-neg 59.9%

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

      2. +-commutative 59.9%

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

      3. unpow2 59.9%

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

      4. *-commutative 59.9%

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

      5. associate-*r* 59.9%

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

      6. metadata-eval 59.9%

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

      7. associate-+l+ 59.9%

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

      8. associate-*r* 59.9%

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

      9. *-commutative 59.9%

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

      10. fma-udef 59.9%

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

   6. Simplified 59.9%

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

      1. +-commutative 59.9%

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

      2. metadata-eval 59.9%

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

      3. fma-neg 59.9%

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

      4. *-commutative 59.9%

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

      5. associate-+l- 59.9%

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

      6. associate-*l* 59.9%

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

   8. Applied egg-rr 59.9%

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

   9. Taylor expanded in b around 0 59.3%

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

   if 0.0 < (*.f64 b b) < 4.90000000000000009e-174 or 7.00000000000000038e-11 < (*.f64 b b) < 4.6999999999999998e61

   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* 99.7%

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

      6. unpow1 99.7%

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

      7. sqr-pow 99.7%

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

      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 99.9%

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

      10. metadata-eval 99.9%

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

      11. unpow1/2 99.9%

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

      12. hypot-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate-*r* 99.9%

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

      15. *-commutative 99.9%

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

   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 68.4%

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

   if 4.6999999999999998e61 < (*.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 0 92.7%

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

      1. sub-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unpow2 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*r* 92.7%

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

      6. metadata-eval 92.7%

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

      7. associate-+l+ 92.7%

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

      8. associate-*r* 92.7%

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

      9. *-commutative 92.7%

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

      10. fma-udef 92.7%

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

   6. Simplified 92.7%

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

   7. Taylor expanded in b around inf 92.7%

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

      1. unpow2 92.7%

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

   9. Simplified 92.7%

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

       1. sqr-pow 92.7%

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

       2. metadata-eval 92.7%

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

       3. pow2 92.7%

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

       4. metadata-eval 92.7%

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

       5. pow2 92.7%

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

       6. distribute-rgt-out 92.7%

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

   11. Applied egg-rr 92.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4.9 \cdot 10^{-174}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 7 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4.7 \cdot 10^{+61}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 4: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4 \cdot 10^{-175}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+61}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.0)
   -1.0
   (if (<= (* b b) 4e-175)
     (pow a 4.0)
     (if (<= (* b b) 2e-11)
       -1.0
       (if (<= (* b b) 2e+61) (pow a 4.0) (pow b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.0) {
		tmp = -1.0;
	} else if ((b * b) <= 4e-175) {
		tmp = pow(a, 4.0);
	} else if ((b * b) <= 2e-11) {
		tmp = -1.0;
	} else if ((b * b) <= 2e+61) {
		tmp = 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) <= 0.0d0) then
        tmp = -1.0d0
    else if ((b * b) <= 4d-175) then
        tmp = a ** 4.0d0
    else if ((b * b) <= 2d-11) then
        tmp = -1.0d0
    else if ((b * b) <= 2d+61) then
        tmp = 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) <= 0.0) {
		tmp = -1.0;
	} else if ((b * b) <= 4e-175) {
		tmp = Math.pow(a, 4.0);
	} else if ((b * b) <= 2e-11) {
		tmp = -1.0;
	} else if ((b * b) <= 2e+61) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.0:
		tmp = -1.0
	elif (b * b) <= 4e-175:
		tmp = math.pow(a, 4.0)
	elif (b * b) <= 2e-11:
		tmp = -1.0
	elif (b * b) <= 2e+61:
		tmp = 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) <= 0.0)
		tmp = -1.0;
	elseif (Float64(b * b) <= 4e-175)
		tmp = a ^ 4.0;
	elseif (Float64(b * b) <= 2e-11)
		tmp = -1.0;
	elseif (Float64(b * b) <= 2e+61)
		tmp = 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) <= 0.0)
		tmp = -1.0;
	elseif ((b * b) <= 4e-175)
		tmp = a ^ 4.0;
	elseif ((b * b) <= 2e-11)
		tmp = -1.0;
	elseif ((b * b) <= 2e+61)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.0], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 4e-175], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e-11], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 2e+61], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 0.0 or 4e-175 < (*.f64 b b) < 1.99999999999999988e-11

   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* 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 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 59.9%

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

      1. sub-neg 59.9%

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

      2. +-commutative 59.9%

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

      3. unpow2 59.9%

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

      4. *-commutative 59.9%

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

      5. associate-*r* 59.9%

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

      6. metadata-eval 59.9%

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

      7. associate-+l+ 59.9%

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

      8. associate-*r* 59.9%

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

      9. *-commutative 59.9%

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

      10. fma-udef 59.9%

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

   6. Simplified 59.9%

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

      1. +-commutative 59.9%

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

      2. metadata-eval 59.9%

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

      3. fma-neg 59.9%

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

      4. *-commutative 59.9%

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

      5. associate-+l- 59.9%

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

      6. associate-*l* 59.9%

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

   8. Applied egg-rr 59.9%

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

   9. Taylor expanded in b around 0 59.3%

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

   if 0.0 < (*.f64 b b) < 4e-175 or 1.99999999999999988e-11 < (*.f64 b b) < 1.9999999999999999e61

   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* 99.7%

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

      6. unpow1 99.7%

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

      7. sqr-pow 99.7%

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

      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 99.9%

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

      10. metadata-eval 99.9%

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

      11. unpow1/2 99.9%

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

      12. hypot-def 99.9%

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

      13. metadata-eval 99.9%

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

      14. associate-*r* 99.9%

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

      15. *-commutative 99.9%

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

   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 68.4%

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

   if 1.9999999999999999e61 < (*.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 b around inf 92.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 4 \cdot 10^{-175}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+61}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 480000000:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 480000000.0) (+ (* b (* b 4.0)) (+ -1.0 (pow b 4.0))) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 480000000.0) {
		tmp = (b * (b * 4.0)) + (-1.0 + pow(b, 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 480000000.0d0) then
        tmp = (b * (b * 4.0d0)) + ((-1.0d0) + (b ** 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 480000000.0) {
		tmp = (b * (b * 4.0)) + (-1.0 + Math.pow(b, 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 480000000.0:
		tmp = (b * (b * 4.0)) + (-1.0 + math.pow(b, 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 480000000.0)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + Float64(-1.0 + (b ^ 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 480000000.0)
		tmp = (b * (b * 4.0)) + (-1.0 + (b ^ 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 480000000.0], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.8e8

   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 79.4%

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

      1. sub-neg 79.4%

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

      2. +-commutative 79.4%

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

      3. unpow2 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*r* 79.4%

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

      6. metadata-eval 79.4%

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

      7. associate-+l+ 79.4%

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

      8. associate-*r* 79.4%

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

      9. *-commutative 79.4%

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

      10. fma-udef 79.4%

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

   6. Simplified 79.4%

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

      1. +-commutative 79.4%

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

      2. metadata-eval 79.4%

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

      3. fma-neg 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-+l- 79.4%

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

      6. associate-*l* 79.4%

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

   8. Applied egg-rr 79.4%

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

   if 4.8e8 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      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 91.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 480000000:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 550000000:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 550000000.0) (+ -1.0 (* (* b b) (fma b b 4.0))) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 550000000.0) {
		tmp = -1.0 + ((b * b) * fma(b, b, 4.0));
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 550000000.0)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * fma(b, b, 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 550000000.0], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.5e8

   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 79.4%

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

      1. sub-neg 79.4%

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

      2. +-commutative 79.4%

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

      3. unpow2 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*r* 79.4%

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

      6. metadata-eval 79.4%

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

      7. associate-+l+ 79.4%

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

      8. associate-*r* 79.4%

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

      9. *-commutative 79.4%

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

      10. fma-udef 79.4%

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

   6. Simplified 79.4%

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

      1. +-commutative 79.4%

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

      2. metadata-eval 79.4%

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

      3. fma-neg 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-+l- 79.4%

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

      6. associate-*l* 79.4%

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

   8. Applied egg-rr 79.4%

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

   9. Taylor expanded in b around 0 79.4%

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

       1. sub-neg 79.4%

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

       2. fma-def 79.4%

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

       3. unpow2 79.4%

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

       4. metadata-eval 79.4%

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

       5. +-commutative 79.4%

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

       6. fma-udef 79.4%

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

       7. metadata-eval 79.4%

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

       8. pow-plus 79.4%

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

       9. unpow3 79.4%

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

       10. associate-*r* 79.4%

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

       11. distribute-rgt-in 79.4%

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

       12. +-commutative 79.4%

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

       13. fma-def 79.4%

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

   11. Simplified 79.4%

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

   if 5.5e8 < a

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      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 91.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 550000000:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+61}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+61) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+61) {
		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) <= 2d+61) 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) <= 2e+61) {
		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) <= 2e+61:
		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) <= 2e+61)
		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) <= 2e+61)
		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], 2e+61], 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) < 1.9999999999999999e61

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. unpow3 99.9%

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

      9. pow-plus 100.0%

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

      10. metadata-eval 100.0%

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

      11. unpow1/2 100.0%

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

      12. hypot-def 100.0%

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

      13. metadata-eval 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 96.1%

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

   if 1.9999999999999999e61 < (*.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 b around inf 92.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+61}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9: 46.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00155) -1.0 (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.00155d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * (4.0d0 + (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00155:
		tmp = -1.0
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * Float64(4.0 + Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00155], -1.0, N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00154999999999999995

   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 66.8%

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

      1. sub-neg 66.8%

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

      2. +-commutative 66.8%

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

      3. unpow2 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-*r* 66.8%

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

      6. metadata-eval 66.8%

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

      7. associate-+l+ 66.8%

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

      8. associate-*r* 66.8%

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

      9. *-commutative 66.8%

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

      10. fma-udef 66.8%

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

   6. Simplified 66.8%

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

      1. +-commutative 66.8%

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

      2. metadata-eval 66.8%

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

      3. fma-neg 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-+l- 66.8%

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

      6. associate-*l* 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in b around 0 35.4%

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

   if 0.00154999999999999995 < 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 81.3%

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

      1. sub-neg 81.3%

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

      2. +-commutative 81.3%

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

      3. unpow2 81.3%

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

      4. *-commutative 81.3%

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

      5. associate-*r* 81.3%

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

      6. metadata-eval 81.3%

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

      7. associate-+l+ 81.3%

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

      8. associate-*r* 81.3%

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

      9. *-commutative 81.3%

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

      10. fma-udef 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in b around inf 78.1%

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

      1. unpow2 78.1%

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

   9. Simplified 78.1%

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

       1. sqr-pow 78.0%

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

       2. metadata-eval 78.0%

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

       3. pow2 78.0%

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

       4. metadata-eval 78.0%

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

       5. pow2 78.0%

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

       6. distribute-rgt-out 78.0%

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

   11. Applied egg-rr 78.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 10: 37.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.00155) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 0.00155d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.00155) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00155:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.00155)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00155], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00154999999999999995

   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 66.8%

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

      1. sub-neg 66.8%

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

      2. +-commutative 66.8%

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

      3. unpow2 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-*r* 66.8%

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

      6. metadata-eval 66.8%

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

      7. associate-+l+ 66.8%

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

      8. associate-*r* 66.8%

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

      9. *-commutative 66.8%

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

      10. fma-udef 66.8%

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

   6. Simplified 66.8%

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

      1. +-commutative 66.8%

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

      2. metadata-eval 66.8%

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

      3. fma-neg 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-+l- 66.8%

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

      6. associate-*l* 66.8%

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

   8. Applied egg-rr 66.8%

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

   9. Taylor expanded in b around 0 35.4%

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

   if 0.00154999999999999995 < 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 81.3%

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

      1. sub-neg 81.3%

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

      2. +-commutative 81.3%

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

      3. unpow2 81.3%

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

      4. *-commutative 81.3%

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

      5. associate-*r* 81.3%

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

      6. metadata-eval 81.3%

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

      7. associate-+l+ 81.3%

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

      8. associate-*r* 81.3%

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

      9. *-commutative 81.3%

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

      10. fma-udef 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in b around inf 78.1%

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

      1. unpow2 78.1%

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

   9. Simplified 78.1%

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

       1. sqr-pow 78.0%

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

       2. metadata-eval 78.0%

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

       3. pow2 78.0%

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

       4. metadata-eval 78.0%

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

       5. pow2 78.0%

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

       6. distribute-rgt-out 78.0%

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

   11. Applied egg-rr 78.0%

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

   12. Taylor expanded in b around 0 49.2%

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

       1. unpow2 49.2%

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

   14. Simplified 49.2%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00155:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 11: 24.5% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.9%

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

   2. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow1 99.9%

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

   7. sqr-pow 99.9%

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

   8. unpow3 99.9%

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

   9. pow-plus 100.0%

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

   10. metadata-eval 100.0%

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

   11. unpow1/2 100.0%

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

   12. hypot-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. associate-*r* 100.0%

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

   15. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 70.5%

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

   1. sub-neg 70.5%

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

   2. +-commutative 70.5%

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

   3. unpow2 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-*r* 70.5%

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

   6. metadata-eval 70.5%

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

   7. associate-+l+ 70.5%

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

   8. associate-*r* 70.5%

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

   9. *-commutative 70.5%

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

   10. fma-udef 70.5%

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

6. Simplified 70.5%

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

   1. +-commutative 70.5%

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

   2. metadata-eval 70.5%

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

   3. fma-neg 70.5%

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

   4. *-commutative 70.5%

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

   5. associate-+l- 70.5%

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

   6. associate-*l* 70.5%

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

8. Applied egg-rr 70.5%

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

9. Taylor expanded in b around 0 26.5%

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

10. Final simplification 26.5%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))