Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 98.6%

Time: 8.9s

Alternatives: 6

Speedup: 1.2×

• 60.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 74.1% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+102}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot {a}^{3} + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+102)
   (+ (pow a 4.0) -1.0)
   (+ -1.0 (+ (* 4.0 (pow a 3.0)) (pow (hypot a b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+102) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + ((4.0 * pow(a, 3.0)) + pow(hypot(a, b), 4.0));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+102) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + ((4.0 * Math.pow(a, 3.0)) + Math.pow(Math.hypot(a, b), 4.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+102:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + ((4.0 * math.pow(a, 3.0)) + math.pow(math.hypot(a, b), 4.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+102)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(4.0 * (a ^ 3.0)) + (hypot(a, b) ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+102)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + ((4.0 * (a ^ 3.0)) + (hypot(a, b) ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+102], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e102

   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. sub-neg 0.0%

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

   3. Simplified 16.2%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -5e102 < a

   1. Initial program 86.7%

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

      1. sub-neg 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. fma-def 99.2%

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

      2. expm1-log1p-u 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)\right)}\right) + -1 \]

      3. expm1-udef 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)} - 1\right)}\right) + -1 \]

      4. fma-def 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}}^{2}\right)} - 1\right)\right) + -1 \]

      5. add-sqr-sqrt 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}}^{2}\right)} - 1\right)\right) + -1 \]

      6. pow2 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(a, a, b \cdot b\right)}\right)}^{2}\right)}}^{2}\right)} - 1\right)\right) + -1 \]

      7. fma-def 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \left(e^{\mathsf{log1p}\left({\left({\left(\sqrt{\color{blue}{a \cdot a + b \cdot b}}\right)}^{2}\right)}^{2}\right)} - 1\right)\right) + -1 \]

      8. hypot-def 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \left(e^{\mathsf{log1p}\left({\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right)} - 1\right)\right) + -1 \]

   7. Applied egg-rr 97.8%

      \[\leadsto \left(4 \cdot {a}^{3} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} - 1\right)}\right) + -1 \]
   8. Step-by-step derivation

      1. expm1-def 97.8%

         \[\leadsto \left(4 \cdot {a}^{3} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)\right)}\right) + -1 \]

      2. expm1-log1p 99.2%

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

      3. unpow2 99.2%

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

      4. pow-sqr 99.3%

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

      5. metadata-eval 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+102}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(4 \cdot {a}^{3} + {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

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) (+ -1.0 t_0) (+ (pow a 4.0) -1.0))))

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 = -1.0 + t_0;
	} else {
		tmp = pow(a, 4.0) + -1.0;
	}
	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 = -1.0 + t_0;
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	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 = -1.0 + t_0
	else:
		tmp = math.pow(a, 4.0) + -1.0
	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(-1.0 + t_0);
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	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 = -1.0 + t_0;
	else
		tmp = (a ^ 4.0) + -1.0;
	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[(-1.0 + t$95$0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $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 99.9%

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

   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. sub-neg 0.0%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in a around inf 90.0%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

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

Alternative 3: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 17.0) (+ (pow a 4.0) -1.0) (+ -1.0 (* b (* b (fma b b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 17.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (b * (b * fma(b, b, 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 17.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * fma(b, b, 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 17.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 17

   1. Initial program 78.7%

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

      1. sub-neg 78.7%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in a around inf 80.7%

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

   if 17 < b

   1. Initial program 61.1%

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

      1. associate--l+ 61.1%

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

      2. +-commutative 61.1%

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

      3. +-commutative 61.1%

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

      4. sub-neg 61.1%

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

      5. associate-+l+ 61.1%

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

      6. +-commutative 61.1%

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

      7. fma-def 62.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in a around 0 88.7%

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

      1. flip-+ 15.4%

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

      2. frac-2neg 15.4%

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

      3. pow2 15.4%

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

      4. *-commutative 15.4%

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

      5. pow2 15.4%

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

      6. *-commutative 15.4%

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

      7. swap-sqr 15.4%

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

      8. pow2 15.4%

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

      9. pow-prod-down 15.4%

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

      10. pow-prod-up 15.4%

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

      11. metadata-eval 15.4%

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

      12. metadata-eval 15.4%

          \[\leadsto \frac{-\left({b}^{4} \cdot \color{blue}{16} - {b}^{4} \cdot {b}^{4}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      13. pow-prod-up 15.3%

          \[\leadsto \frac{-\left({b}^{4} \cdot 16 - \color{blue}{{b}^{\left(4 + 4\right)}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      14. metadata-eval 15.3%

          \[\leadsto \frac{-\left({b}^{4} \cdot 16 - {b}^{\color{blue}{8}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   7. Applied egg-rr 15.3%

      \[\leadsto \color{blue}{\frac{-\left({b}^{4} \cdot 16 - {b}^{8}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]
   8. Step-by-step derivation

      1. neg-sub0 15.3%

         \[\leadsto \frac{\color{blue}{0 - \left({b}^{4} \cdot 16 - {b}^{8}\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      2. associate--r- 15.3%

         \[\leadsto \frac{\color{blue}{\left(0 - {b}^{4} \cdot 16\right) + {b}^{8}}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      3. neg-sub0 15.3%

         \[\leadsto \frac{\color{blue}{\left(-{b}^{4} \cdot 16\right)} + {b}^{8}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      4. +-commutative 15.3%

         \[\leadsto \frac{\color{blue}{{b}^{8} + \left(-{b}^{4} \cdot 16\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      5. sub-neg 15.3%

         \[\leadsto \frac{\color{blue}{{b}^{8} - {b}^{4} \cdot 16}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      6. neg-sub0 15.3%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{0 - \left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]

      7. associate--r- 15.3%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(0 - 4 \cdot {b}^{2}\right) + {b}^{4}}} - 1 \]

      8. neg-sub0 15.3%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(-4 \cdot {b}^{2}\right)} + {b}^{4}} - 1 \]

      9. +-commutative 15.3%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} + \left(-4 \cdot {b}^{2}\right)}} - 1 \]

      10. sub-neg 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} - 4 \cdot {b}^{2}}} - 1 \]

      11. *-commutative 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{b}^{2} \cdot 4}} - 1 \]

      12. unpow2 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot b\right)} \cdot 4} - 1 \]

      13. metadata-eval 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \left(b \cdot b\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} - 1 \]

      14. swap-sqr 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot 2\right) \cdot \left(b \cdot 2\right)}} - 1 \]

      15. unpow2 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{\left(b \cdot 2\right)}^{2}}} - 1 \]

   9. Simplified 15.3%

      \[\leadsto \color{blue}{\frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}}} - 1 \]
   10. Step-by-step derivation

       1. metadata-eval 15.3%

          \[\leadsto \frac{{b}^{8} - {b}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       2. pow-sqr 15.3%

          \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot {b}^{2}\right)} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       3. metadata-eval 15.3%

          \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left(4 \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       4. swap-sqr 15.3%

          \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot 4\right) \cdot \left({b}^{2} \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       5. metadata-eval 15.3%

          \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right) \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       6. unpow-prod-down 15.3%

          \[\leadsto \frac{{b}^{8} - \color{blue}{{\left(b \cdot 2\right)}^{2}} \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       7. metadata-eval 15.3%

          \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       8. unpow-prod-down 15.3%

          \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \color{blue}{{\left(b \cdot 2\right)}^{2}}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       9. metadata-eval 15.3%

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

       10. pow-sqr 15.4%

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

       11. flip-+ 88.7%

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

       12. metadata-eval 88.7%

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

       13. pow-sqr 88.6%

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

       14. unpow-prod-down 88.6%

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

       15. metadata-eval 88.6%

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

       16. distribute-lft-in 88.6%

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

       17. *-commutative 88.6%

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

       18. unpow2 88.6%

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

       19. associate-*r* 88.6%

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

   11. Applied egg-rr 88.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+144}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(4 \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.36e+144) (+ (pow a 4.0) -1.0) (+ -1.0 (* b (* 4.0 b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.36e+144) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (b * (4.0 * 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 <= 1.36d+144) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + (b * (4.0d0 * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.36e+144) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (b * (4.0 * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.36e+144:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + (b * (4.0 * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.36e+144)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(b * Float64(4.0 * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.36e+144)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + (b * (4.0 * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.36e+144], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(b * N[(4.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35999999999999993e144

   1. Initial program 76.2%

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

      1. sub-neg 76.2%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in a around inf 75.3%

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

   if 1.35999999999999993e144 < b

   1. Initial program 62.2%

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

      1. associate--l+ 62.2%

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

      2. +-commutative 62.2%

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

      3. +-commutative 62.2%

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

      4. sub-neg 62.2%

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

      5. associate-+l+ 62.2%

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

      6. +-commutative 62.2%

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

      7. fma-def 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. flip-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. pow2 0.0%

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

      4. *-commutative 0.0%

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

      5. pow2 0.0%

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

      6. *-commutative 0.0%

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

      7. swap-sqr 0.0%

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

      8. pow2 0.0%

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

      9. pow-prod-down 0.0%

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

      10. pow-prod-up 0.0%

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

      11. metadata-eval 0.0%

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

      12. metadata-eval 0.0%

          \[\leadsto \frac{-\left({b}^{4} \cdot \color{blue}{16} - {b}^{4} \cdot {b}^{4}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      13. pow-prod-up 0.0%

          \[\leadsto \frac{-\left({b}^{4} \cdot 16 - \color{blue}{{b}^{\left(4 + 4\right)}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      14. metadata-eval 0.0%

          \[\leadsto \frac{-\left({b}^{4} \cdot 16 - {b}^{\color{blue}{8}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{-\left({b}^{4} \cdot 16 - {b}^{8}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]
   8. Step-by-step derivation

      1. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{0 - \left({b}^{4} \cdot 16 - {b}^{8}\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      2. associate--r- 0.0%

         \[\leadsto \frac{\color{blue}{\left(0 - {b}^{4} \cdot 16\right) + {b}^{8}}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      3. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{\left(-{b}^{4} \cdot 16\right)} + {b}^{8}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      4. +-commutative 0.0%

         \[\leadsto \frac{\color{blue}{{b}^{8} + \left(-{b}^{4} \cdot 16\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      5. sub-neg 0.0%

         \[\leadsto \frac{\color{blue}{{b}^{8} - {b}^{4} \cdot 16}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

      6. neg-sub0 0.0%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{0 - \left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]

      7. associate--r- 0.0%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(0 - 4 \cdot {b}^{2}\right) + {b}^{4}}} - 1 \]

      8. neg-sub0 0.0%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(-4 \cdot {b}^{2}\right)} + {b}^{4}} - 1 \]

      9. +-commutative 0.0%

         \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} + \left(-4 \cdot {b}^{2}\right)}} - 1 \]

      10. sub-neg 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} - 4 \cdot {b}^{2}}} - 1 \]

      11. *-commutative 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{b}^{2} \cdot 4}} - 1 \]

      12. unpow2 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot b\right)} \cdot 4} - 1 \]

      13. metadata-eval 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \left(b \cdot b\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} - 1 \]

      14. swap-sqr 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot 2\right) \cdot \left(b \cdot 2\right)}} - 1 \]

      15. unpow2 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{\left(b \cdot 2\right)}^{2}}} - 1 \]

   9. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}}} - 1 \]
   10. Step-by-step derivation

       1. metadata-eval 0.0%

          \[\leadsto \frac{{b}^{8} - {b}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       2. pow-sqr 0.0%

          \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot {b}^{2}\right)} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       3. metadata-eval 0.0%

          \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left(4 \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       4. swap-sqr 0.0%

          \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot 4\right) \cdot \left({b}^{2} \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       5. metadata-eval 0.0%

          \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right) \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       6. unpow-prod-down 0.0%

          \[\leadsto \frac{{b}^{8} - \color{blue}{{\left(b \cdot 2\right)}^{2}} \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       7. metadata-eval 0.0%

          \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       8. unpow-prod-down 0.0%

          \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \color{blue}{{\left(b \cdot 2\right)}^{2}}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

       9. metadata-eval 0.0%

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

       10. pow-sqr 0.0%

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

       11. flip-+ 100.0%

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

       12. metadata-eval 100.0%

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

       13. pow-sqr 100.0%

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

       14. unpow-prod-down 100.0%

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

       15. metadata-eval 100.0%

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

       16. distribute-lft-in 100.0%

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

       17. *-commutative 100.0%

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

       18. unpow2 100.0%

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

       19. associate-*r* 100.0%

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

   11. Applied egg-rr 100.0%

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

   12. Taylor expanded in b around 0 97.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+144}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(4 \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3100:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3100.0) (+ (pow a 4.0) -1.0) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3100.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + 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 <= 3100.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3100.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3100.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3100.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3100.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3100.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3100

   1. Initial program 78.7%

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

      1. sub-neg 78.7%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in a around inf 80.7%

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

   if 3100 < b

   1. Initial program 61.1%

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

      1. sub-neg 61.1%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in b around inf 87.9%

      \[\leadsto \color{blue}{{b}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3100:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate--l+ 74.1%

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

   2. +-commutative 74.1%

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

   3. +-commutative 74.1%

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

   4. sub-neg 74.1%

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

   5. associate-+l+ 74.1%

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

   6. +-commutative 74.1%

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

   7. fma-def 74.5%

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

3. Simplified 76.9%

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

5. Taylor expanded in a around 0 72.3%

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

   1. flip-+ 22.6%

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

   2. frac-2neg 22.6%

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

   3. pow2 22.6%

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

   4. *-commutative 22.6%

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

   5. pow2 22.6%

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

   6. *-commutative 22.6%

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

   7. swap-sqr 22.6%

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

   8. pow2 22.6%

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

   9. pow-prod-down 22.6%

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

   10. pow-prod-up 22.6%

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

   11. metadata-eval 22.6%

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

   12. metadata-eval 22.6%

       \[\leadsto \frac{-\left({b}^{4} \cdot \color{blue}{16} - {b}^{4} \cdot {b}^{4}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   13. pow-prod-up 22.6%

       \[\leadsto \frac{-\left({b}^{4} \cdot 16 - \color{blue}{{b}^{\left(4 + 4\right)}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   14. metadata-eval 22.6%

       \[\leadsto \frac{-\left({b}^{4} \cdot 16 - {b}^{\color{blue}{8}}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

7. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{\frac{-\left({b}^{4} \cdot 16 - {b}^{8}\right)}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]
8. Step-by-step derivation

   1. neg-sub0 22.6%

      \[\leadsto \frac{\color{blue}{0 - \left({b}^{4} \cdot 16 - {b}^{8}\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   2. associate--r- 22.6%

      \[\leadsto \frac{\color{blue}{\left(0 - {b}^{4} \cdot 16\right) + {b}^{8}}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   3. neg-sub0 22.6%

      \[\leadsto \frac{\color{blue}{\left(-{b}^{4} \cdot 16\right)} + {b}^{8}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   4. +-commutative 22.6%

      \[\leadsto \frac{\color{blue}{{b}^{8} + \left(-{b}^{4} \cdot 16\right)}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   5. sub-neg 22.6%

      \[\leadsto \frac{\color{blue}{{b}^{8} - {b}^{4} \cdot 16}}{-\left(4 \cdot {b}^{2} - {b}^{4}\right)} - 1 \]

   6. neg-sub0 22.6%

      \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{0 - \left(4 \cdot {b}^{2} - {b}^{4}\right)}} - 1 \]

   7. associate--r- 22.6%

      \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(0 - 4 \cdot {b}^{2}\right) + {b}^{4}}} - 1 \]

   8. neg-sub0 22.6%

      \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{\left(-4 \cdot {b}^{2}\right)} + {b}^{4}} - 1 \]

   9. +-commutative 22.6%

      \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} + \left(-4 \cdot {b}^{2}\right)}} - 1 \]

   10. sub-neg 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{\color{blue}{{b}^{4} - 4 \cdot {b}^{2}}} - 1 \]

   11. *-commutative 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{b}^{2} \cdot 4}} - 1 \]

   12. unpow2 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot b\right)} \cdot 4} - 1 \]

   13. metadata-eval 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \left(b \cdot b\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} - 1 \]

   14. swap-sqr 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{\left(b \cdot 2\right) \cdot \left(b \cdot 2\right)}} - 1 \]

   15. unpow2 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - \color{blue}{{\left(b \cdot 2\right)}^{2}}} - 1 \]

9. Simplified 22.6%

   \[\leadsto \color{blue}{\frac{{b}^{8} - {b}^{4} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}}} - 1 \]
10. Step-by-step derivation

    1. metadata-eval 22.6%

       \[\leadsto \frac{{b}^{8} - {b}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    2. pow-sqr 22.6%

       \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot {b}^{2}\right)} \cdot 16}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    3. metadata-eval 22.6%

       \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot {b}^{2}\right) \cdot \color{blue}{\left(4 \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    4. swap-sqr 22.6%

       \[\leadsto \frac{{b}^{8} - \color{blue}{\left({b}^{2} \cdot 4\right) \cdot \left({b}^{2} \cdot 4\right)}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    5. metadata-eval 22.6%

       \[\leadsto \frac{{b}^{8} - \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right) \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    6. unpow-prod-down 22.6%

       \[\leadsto \frac{{b}^{8} - \color{blue}{{\left(b \cdot 2\right)}^{2}} \cdot \left({b}^{2} \cdot 4\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    7. metadata-eval 22.6%

       \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \left({b}^{2} \cdot \color{blue}{{2}^{2}}\right)}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    8. unpow-prod-down 22.6%

       \[\leadsto \frac{{b}^{8} - {\left(b \cdot 2\right)}^{2} \cdot \color{blue}{{\left(b \cdot 2\right)}^{2}}}{{b}^{4} - {\left(b \cdot 2\right)}^{2}} - 1 \]

    9. metadata-eval 22.6%

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

    10. pow-sqr 22.6%

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

    11. flip-+ 72.3%

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

    12. metadata-eval 72.3%

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

    13. pow-sqr 72.3%

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

    14. unpow-prod-down 72.3%

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

    15. metadata-eval 72.3%

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

    16. distribute-lft-in 72.3%

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

    17. *-commutative 72.3%

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

    18. unpow2 72.3%

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

    19. associate-*r* 72.3%

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

11. Applied egg-rr 72.3%

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

12. Taylor expanded in b around 0 55.2%

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

13. Final simplification 55.2%

    \[\leadsto -1 + b \cdot \left(4 \cdot b\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024016 
(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))