Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 9

Speedup: 1.0×

• 60.9% 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 99.8%

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

   5. associate-*r* 99.9%

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

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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.002:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.002)
   (+ (pow a 4.0) -1.0)
   (+ (pow b 4.0) (* (* b b) (+ 4.0 (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.002) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	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) <= 0.002d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + ((b * b) * (4.0d0 + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.002) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.002:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.002)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(4.0 + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.002)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + ((b * b) * (4.0 + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.002], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e-3

   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%

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

   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 0 100.0%

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

   if 2e-3 < (*.f64 b b)

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   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 95.1%

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

      1. *-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.002:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.002:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \mathsf{fma}\left(2, a \cdot a, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.002)
   (+ (pow a 4.0) -1.0)
   (* (* b b) (+ (* b b) (fma 2.0 (* a a) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.002) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (b * b) * ((b * b) + fma(2.0, (a * a), 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.002)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + fma(2.0, Float64(a * a), 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.002], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(2.0 * N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e-3

   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%

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

   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 0 100.0%

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

   if 2e-3 < (*.f64 b b)

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.9%

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

   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 95.1%

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

      1. *-commutative 95.1%

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

      2. unpow2 95.1%

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

      3. unpow2 95.1%

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

   6. Simplified 95.1%

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

      1. +-commutative 95.1%

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

      2. metadata-eval 95.1%

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

      3. pow-prod-up 94.9%

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

      4. pow-prod-down 94.9%

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

      5. pow2 94.9%

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

      6. distribute-lft-out 94.9%

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

      7. +-commutative 94.9%

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

      8. fma-def 94.9%

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

   8. Applied egg-rr 94.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.002:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \mathsf{fma}\left(2, a \cdot a, 4\right)\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+65) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+65) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+65:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+65)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+65)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+65], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e65

   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%

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

   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 96.4%

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

   if 4.99999999999999973e65 < (*.f64 b b)

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   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 98.4%

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

      1. *-commutative 98.4%

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

      2. unpow2 98.4%

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

      3. unpow2 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in b around inf 96.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 58.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 10^{+304}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-29)
   -1.0
   (if (<= (* b b) 1e+304) (* (* a a) (* (* b b) 2.0)) (* 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-29) {
		tmp = -1.0;
	} else if ((b * b) <= 1e+304) {
		tmp = (a * a) * ((b * b) * 2.0);
	} 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 * b) <= 5d-29) then
        tmp = -1.0d0
    else if ((b * b) <= 1d+304) then
        tmp = (a * a) * ((b * b) * 2.0d0)
    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 * b) <= 5e-29) {
		tmp = -1.0;
	} else if ((b * b) <= 1e+304) {
		tmp = (a * a) * ((b * b) * 2.0);
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-29:
		tmp = -1.0
	elif (b * b) <= 1e+304:
		tmp = (a * a) * ((b * b) * 2.0)
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-29)
		tmp = -1.0;
	elseif (Float64(b * b) <= 1e+304)
		tmp = Float64(Float64(a * a) * Float64(Float64(b * b) * 2.0));
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-29)
		tmp = -1.0;
	elseif ((b * b) <= 1e+304)
		tmp = (a * a) * ((b * b) * 2.0);
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-29], -1.0, If[LessEqual[N[(b * b), $MachinePrecision], 1e+304], N[(N[(a * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 4.99999999999999986e-29

   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%

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

      4. sqr-pow 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around -inf 46.4%

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

      1. mul-1-neg 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in b around 0 46.3%

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

   if 4.99999999999999986e-29 < (*.f64 b b) < 9.9999999999999994e303

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. unpow2 99.7%

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

      3. unpow1 99.7%

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

      4. sqr-pow 99.7%

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

      5. associate-*r* 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. unpow2 87.1%

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

      3. unpow2 87.1%

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

   6. Simplified 87.1%

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

      1. +-commutative 87.1%

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

      2. metadata-eval 87.1%

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

      3. pow-prod-up 86.9%

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

      4. pow-prod-down 86.9%

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

      5. pow2 86.9%

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

      6. distribute-lft-out 86.9%

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

      7. +-commutative 86.9%

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

      8. fma-def 86.9%

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

   8. Applied egg-rr 86.9%

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

   9. Taylor expanded in a around inf 40.2%

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

       1. unpow2 40.2%

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

       2. associate-*r* 40.2%

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

       3. *-commutative 40.2%

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

       4. associate-*l* 40.2%

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

       5. unpow2 40.2%

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

   11. Simplified 40.2%

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

   if 9.9999999999999994e303 < (*.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%

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

   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 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \cdot b \leq 10^{+304}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 7: 69.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-16) -1.0 (* (* b b) (+ 4.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-16) {
		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 * b) <= 5d-16) 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 * b) <= 5e-16) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * (4.0 + (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-16:
		tmp = -1.0
	else:
		tmp = (b * b) * (4.0 + (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-16)
		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 * b) <= 5e-16)
		tmp = -1.0;
	else
		tmp = (b * b) * (4.0 + (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-16], -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 (*.f64 b b) < 5.0000000000000004e-16

   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%

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

      4. sqr-pow 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around -inf 46.6%

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

      1. mul-1-neg 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in b around 0 46.4%

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

   if 5.0000000000000004e-16 < (*.f64 b b)

   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%

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

   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 94.4%

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

      1. *-commutative 94.4%

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

      2. unpow2 94.4%

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

      3. unpow2 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in a around 0 88.4%

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

      1. unpow2 88.4%

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

   9. Simplified 88.4%

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

       1. +-commutative 88.4%

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

       2. metadata-eval 88.4%

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

       3. pow-sqr 88.3%

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

       4. pow2 88.3%

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

       5. pow2 88.3%

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

       6. distribute-rgt-out 88.3%

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

   11. Applied egg-rr 88.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right)\\ \end{array} \]

Alternative 8: 50.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 5e-16) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-16) {
		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 * b) <= 5d-16) 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 * b) <= 5e-16) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-16:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-16)
		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 * b) <= 5e-16)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-16], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000004e-16

   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%

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

      4. sqr-pow 99.9%

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

      5. associate-*l* 99.9%

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

      6. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around -inf 46.6%

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

      1. mul-1-neg 46.6%

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

   6. Simplified 46.6%

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

   7. Taylor expanded in b around 0 46.4%

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

   if 5.0000000000000004e-16 < (*.f64 b b)

   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%

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

   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 94.4%

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

      1. *-commutative 94.4%

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

      2. unpow2 94.4%

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

      3. unpow2 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in a around 0 88.4%

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

      1. unpow2 88.4%

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

   9. Simplified 88.4%

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

   10. Taylor expanded in b around 0 51.0%

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

       1. unpow2 51.0%

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

   12. Simplified 51.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 9: 24.8% 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%

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

   4. sqr-pow 99.8%

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

   5. associate-*l* 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in b around -inf 48.6%

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

   1. mul-1-neg 48.6%

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

6. Simplified 48.6%

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

7. Taylor expanded in b around 0 23.2%

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

8. Final simplification 23.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))