Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 99.2%

Time: 6.6s

Alternatives: 9

Speedup: 1.2×

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

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate--l+ 72.5%

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

   2. fma-def 72.5%

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

3. Simplified 73.3%

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

4. Taylor expanded in a around 0 64.1%

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

   1. fma-def 64.1%

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

   2. +-commutative 64.1%

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

   3. *-commutative 64.1%

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

   4. unpow2 64.1%

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

   5. associate-*l* 73.4%

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

   6. unpow2 73.4%

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

6. Simplified 73.4%

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

7. Taylor expanded in a around 0 99.2%

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

   1. unpow2 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in a around inf 96.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-196}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left({a}^{4} + \left(\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-196)
   (+ (pow a 4.0) -1.0)
   (+
    (pow b 4.0)
    (+ (pow a 4.0) (+ (* (* b b) (+ 4.0 (* 2.0 (* a a)))) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-196) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0) + (pow(a, 4.0) + (((b * b) * (4.0 + (2.0 * (a * a)))) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-196) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + ((a ** 4.0d0) + (((b * b) * (4.0d0 + (2.0d0 * (a * a)))) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-196) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + (Math.pow(a, 4.0) + (((b * b) * (4.0 + (2.0 * (a * a)))) + -1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-196:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + (math.pow(a, 4.0) + (((b * b) * (4.0 + (2.0 * (a * a)))) + -1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-196)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + Float64((a ^ 4.0) + Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(2.0 * Float64(a * a)))) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-196)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + ((a ^ 4.0) + (((b * b) * (4.0 + (2.0 * (a * a)))) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-196], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.0000000000000005e-196

   1. Initial program 84.9%

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

      1. associate--l+ 84.9%

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

      2. fma-def 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in a around 0 77.5%

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

      1. fma-def 77.5%

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

      2. +-commutative 77.5%

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

      3. *-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. associate-*l* 85.0%

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

      6. unpow2 85.0%

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

   6. Simplified 85.0%

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

   7. Taylor expanded in a around 0 98.4%

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

      1. unpow2 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in b around 0 98.4%

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

   if 5.0000000000000005e-196 < (*.f64 b b)

   1. Initial program 66.9%

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

      1. associate--l+ 66.9%

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

      2. fma-def 66.9%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in a around 0 57.9%

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

      1. fma-def 57.9%

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

      2. +-commutative 57.9%

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

      3. *-commutative 57.9%

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

      4. unpow2 57.9%

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

      5. associate-*l* 68.2%

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

      6. unpow2 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in a around 0 99.5%

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

      1. unpow2 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in b around 0 99.5%

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

       1. associate--l+ 99.5%

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

       2. associate--l+ 99.5%

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

       3. unpow2 99.5%

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

       4. unpow2 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-196}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + \left({a}^{4} + \left(\left(b \cdot b\right) \cdot \left(4 + 2 \cdot \left(a \cdot a\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.3:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-22}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(\left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.3)
   (+ (pow a 4.0) -1.0)
   (if (<= a 6.5e-22)
     (+ (* 4.0 (* b b)) (+ (pow b 4.0) -1.0))
     (+ (pow a 4.0) (+ (* (+ a 1.0) (* (* a a) 4.0)) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.3) {
		tmp = pow(a, 4.0) + -1.0;
	} else if (a <= 6.5e-22) {
		tmp = (4.0 * (b * b)) + (pow(b, 4.0) + -1.0);
	} else {
		tmp = pow(a, 4.0) + (((a + 1.0) * ((a * a) * 4.0)) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.3d0)) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if (a <= 6.5d-22) then
        tmp = (4.0d0 * (b * b)) + ((b ** 4.0d0) + (-1.0d0))
    else
        tmp = (a ** 4.0d0) + (((a + 1.0d0) * ((a * a) * 4.0d0)) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.3) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if (a <= 6.5e-22) {
		tmp = (4.0 * (b * b)) + (Math.pow(b, 4.0) + -1.0);
	} else {
		tmp = Math.pow(a, 4.0) + (((a + 1.0) * ((a * a) * 4.0)) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.3:
		tmp = math.pow(a, 4.0) + -1.0
	elif a <= 6.5e-22:
		tmp = (4.0 * (b * b)) + (math.pow(b, 4.0) + -1.0)
	else:
		tmp = math.pow(a, 4.0) + (((a + 1.0) * ((a * a) * 4.0)) + -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.3)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif (a <= 6.5e-22)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + Float64((b ^ 4.0) + -1.0));
	else
		tmp = Float64((a ^ 4.0) + Float64(Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0)) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.3)
		tmp = (a ^ 4.0) + -1.0;
	elseif (a <= 6.5e-22)
		tmp = (4.0 * (b * b)) + ((b ^ 4.0) + -1.0);
	else
		tmp = (a ^ 4.0) + (((a + 1.0) * ((a * a) * 4.0)) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.3], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 6.5e-22], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.299999999999999989

   1. Initial program 40.4%

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

      1. associate--l+ 40.4%

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

      2. fma-def 40.4%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in a around 0 43.5%

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

      1. fma-def 43.5%

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

      2. +-commutative 43.5%

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

      3. *-commutative 43.5%

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

      4. unpow2 43.5%

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

      5. associate-*l* 43.5%

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

      6. unpow2 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in a around 0 97.6%

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

      1. unpow2 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in b around 0 92.3%

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

   if -0.299999999999999989 < a < 6.50000000000000043e-22

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 6.50000000000000043e-22 < a

   1. Initial program 56.6%

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

      1. associate--l+ 56.6%

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

      2. fma-def 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in b around 0 93.6%

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

      1. associate--l+ 93.6%

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

      2. associate-*r* 93.6%

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

      3. unpow2 93.6%

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

   6. Simplified 93.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.3:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-22}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(\left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right) + -1\right)\\ \end{array} \]

Alternative 5: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.3:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.3)
   (+ (pow a 4.0) -1.0)
   (if (<= a 3.5e+21) (+ (* 4.0 (* b b)) (+ (pow b 4.0) -1.0)) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.3) {
		tmp = pow(a, 4.0) + -1.0;
	} else if (a <= 3.5e+21) {
		tmp = (4.0 * (b * b)) + (pow(b, 4.0) + -1.0);
	} 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 <= (-0.3d0)) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else if (a <= 3.5d+21) then
        tmp = (4.0d0 * (b * b)) + ((b ** 4.0d0) + (-1.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.3) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else if (a <= 3.5e+21) {
		tmp = (4.0 * (b * b)) + (Math.pow(b, 4.0) + -1.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.3:
		tmp = math.pow(a, 4.0) + -1.0
	elif a <= 3.5e+21:
		tmp = (4.0 * (b * b)) + (math.pow(b, 4.0) + -1.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.3)
		tmp = Float64((a ^ 4.0) + -1.0);
	elseif (a <= 3.5e+21)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + Float64((b ^ 4.0) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.3)
		tmp = (a ^ 4.0) + -1.0;
	elseif (a <= 3.5e+21)
		tmp = (4.0 * (b * b)) + ((b ^ 4.0) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.3], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 3.5e+21], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.299999999999999989

   1. Initial program 40.4%

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

      1. associate--l+ 40.4%

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

      2. fma-def 40.4%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in a around 0 43.5%

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

      1. fma-def 43.5%

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

      2. +-commutative 43.5%

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

      3. *-commutative 43.5%

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

      4. unpow2 43.5%

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

      5. associate-*l* 43.5%

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

      6. unpow2 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in a around 0 97.6%

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

      1. unpow2 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in b around 0 92.3%

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

   if -0.299999999999999989 < a < 3.5e21

   1. Initial program 99.1%

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

      1. associate--l+ 99.1%

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

      2. fma-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 98.9%

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

      1. associate--l+ 98.9%

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

      2. fma-def 98.9%

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

      3. unpow2 98.9%

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

      4. sub-neg 98.9%

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

      5. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

      1. fma-udef 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 3.5e21 < a

   1. Initial program 53.3%

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

      1. associate--l+ 53.3%

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

      2. fma-def 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in a around inf 94.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.3:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + \left({b}^{4} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -19000000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-249}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-246}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -19000000000000.0)
   (pow a 4.0)
   (if (<= a -9.2e-249)
     (pow b 4.0)
     (if (<= a 1.2e-246)
       -1.0
       (if (<= a 2.2e-113) (pow b 4.0) (if (<= a 0.42) -1.0 (pow a 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -19000000000000.0) {
		tmp = pow(a, 4.0);
	} else if (a <= -9.2e-249) {
		tmp = pow(b, 4.0);
	} else if (a <= 1.2e-246) {
		tmp = -1.0;
	} else if (a <= 2.2e-113) {
		tmp = pow(b, 4.0);
	} else if (a <= 0.42) {
		tmp = -1.0;
	} 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 <= (-19000000000000.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= (-9.2d-249)) then
        tmp = b ** 4.0d0
    else if (a <= 1.2d-246) then
        tmp = -1.0d0
    else if (a <= 2.2d-113) then
        tmp = b ** 4.0d0
    else if (a <= 0.42d0) then
        tmp = -1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -19000000000000.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -9.2e-249) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 1.2e-246) {
		tmp = -1.0;
	} else if (a <= 2.2e-113) {
		tmp = Math.pow(b, 4.0);
	} else if (a <= 0.42) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -19000000000000.0:
		tmp = math.pow(a, 4.0)
	elif a <= -9.2e-249:
		tmp = math.pow(b, 4.0)
	elif a <= 1.2e-246:
		tmp = -1.0
	elif a <= 2.2e-113:
		tmp = math.pow(b, 4.0)
	elif a <= 0.42:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -19000000000000.0)
		tmp = a ^ 4.0;
	elseif (a <= -9.2e-249)
		tmp = b ^ 4.0;
	elseif (a <= 1.2e-246)
		tmp = -1.0;
	elseif (a <= 2.2e-113)
		tmp = b ^ 4.0;
	elseif (a <= 0.42)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -19000000000000.0)
		tmp = a ^ 4.0;
	elseif (a <= -9.2e-249)
		tmp = b ^ 4.0;
	elseif (a <= 1.2e-246)
		tmp = -1.0;
	elseif (a <= 2.2e-113)
		tmp = b ^ 4.0;
	elseif (a <= 0.42)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -19000000000000.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -9.2e-249], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 1.2e-246], -1.0, If[LessEqual[a, 2.2e-113], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[a, 0.42], -1.0, N[Power[a, 4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9e13 or 0.419999999999999984 < a

   1. Initial program 45.6%

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

      1. associate--l+ 45.6%

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

      2. fma-def 45.6%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in a around inf 93.8%

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

   if -1.9e13 < a < -9.1999999999999993e-249 or 1.1999999999999999e-246 < a < 2.20000000000000004e-113

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around inf 65.6%

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

   if -9.1999999999999993e-249 < a < 1.1999999999999999e-246 or 2.20000000000000004e-113 < a < 0.419999999999999984

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 69.9%

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

      1. associate--l+ 69.9%

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

      2. associate-*r* 69.9%

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

      3. unpow2 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in a around 0 68.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -19000000000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-249}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-246}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-113}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 93.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+65) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+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) <= 1d+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) <= 1e+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) <= 1e+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) <= 1e+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) <= 1e+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], 1e+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) < 9.9999999999999999e64

   1. Initial program 83.4%

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

      1. associate--l+ 83.4%

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

      2. fma-def 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in a around 0 79.1%

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

      1. fma-def 79.1%

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

      2. +-commutative 79.1%

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

      3. *-commutative 79.1%

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

      4. unpow2 79.1%

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

      5. associate-*l* 83.6%

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

      6. unpow2 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in a around 0 98.4%

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

      1. unpow2 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in b around 0 94.2%

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

   if 9.9999999999999999e64 < (*.f64 b b)

   1. Initial program 60.5%

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

      1. associate--l+ 60.5%

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

      2. fma-def 60.5%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in b around inf 93.9%

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

4. Final simplification 94.1%

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

Alternative 8: 69.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.5) (pow a 4.0) (if (<= a 0.42) -1.0 (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.5) {
		tmp = pow(a, 4.0);
	} else if (a <= 0.42) {
		tmp = -1.0;
	} 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 <= (-2.5d0)) then
        tmp = a ** 4.0d0
    else if (a <= 0.42d0) then
        tmp = -1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 0.42) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.5:
		tmp = math.pow(a, 4.0)
	elif a <= 0.42:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.5)
		tmp = a ^ 4.0;
	elseif (a <= 0.42)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5)
		tmp = a ^ 4.0;
	elseif (a <= 0.42)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.5], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 0.42], -1.0, N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5 or 0.419999999999999984 < a

   1. Initial program 45.6%

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

      1. associate--l+ 45.6%

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

      2. fma-def 45.6%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in a around inf 93.8%

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

   if -2.5 < a < 0.419999999999999984

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 47.4%

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

      1. associate--l+ 47.4%

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

      2. associate-*r* 47.4%

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

      3. unpow2 47.4%

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

   6. Simplified 47.4%

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

   7. Taylor expanded in a around 0 45.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 9: 25.1% accurate, 130.0× 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 72.5%

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

   1. associate--l+ 72.5%

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

   2. fma-def 72.5%

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

3. Simplified 73.3%

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

4. Taylor expanded in b around 0 54.8%

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

   1. associate--l+ 54.8%

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

   2. associate-*r* 54.8%

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

   3. unpow2 54.8%

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

6. Simplified 54.8%

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

7. Taylor expanded in a around 0 23.1%

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

8. Final simplification 23.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023176 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))