Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 98.3%

Time: 8.1s

Alternatives: 9

Speedup: 1.1×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.1 \cdot 10^{+45}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5.1e+45)
   (+ (pow (fma a a (* b b)) 2.0) (+ (* 4.0 (* (pow a 2.0) (- 1.0 a))) -1.0))
   (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5.1e+45) {
		tmp = pow(fma(a, a, (b * b)), 2.0) + ((4.0 * (pow(a, 2.0) * (1.0 - a))) + -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5.1e+45)
		tmp = Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(Float64(4.0 * Float64((a ^ 2.0) * Float64(1.0 - a))) + -1.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5.1e+45], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.0999999999999997e45

   1. Initial program 88.9%

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

      1. associate--l+ 88.9%

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

      2. fma-define 88.9%

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

      3. sqr-neg 88.9%

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

      4. fma-define 88.9%

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

      5. distribute-rgt-in 88.9%

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

      6. sqr-neg 88.9%

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

      7. distribute-rgt-in 88.9%

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

      8. fma-define 88.9%

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

      9. sqr-neg 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in b around 0 99.3%

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

   if 5.0999999999999997e45 < a

   1. Initial program 25.4%

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

      1. associate--l+ 25.4%

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

      2. fma-define 25.4%

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

      3. sqr-neg 25.4%

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

      4. fma-define 25.4%

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

      5. distribute-rgt-in 25.4%

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

      6. sqr-neg 25.4%

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

      7. distribute-rgt-in 25.4%

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

      8. fma-define 25.4%

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

      9. sqr-neg 25.4%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in a around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.1 \cdot 10^{+45}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(1 - a\right) \cdot \left(a \cdot a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* b b) (* a a)) 2.0)
          (* 4.0 (+ (* (- 1.0 a) (* a a)) (* (* b b) (+ 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(((b * b) + (a * a)), 2.0) + (4.0 * (((1.0 - a) * (a * a)) + ((b * b) * (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(((b * b) + (a * a)), 2.0) + (4.0 * (((1.0 - a) * (a * a)) + ((b * b) * (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(((b * b) + (a * a)), 2.0) + (4.0 * (((1.0 - a) * (a * a)) + ((b * b) * (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(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(1.0 - a) * Float64(a * a)) + Float64(Float64(b * b) * 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 = (((b * b) + (a * a)) ^ 2.0) + (4.0 * (((1.0 - a) * (a * a)) + ((b * b) * (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[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(1.0 - a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $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)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a)))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-define 0.0%

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

      3. sqr-neg 0.0%

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

      4. fma-define 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. fma-define 0.0%

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

      9. sqr-neg 0.0%

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

   3. Simplified 12.7%

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

   5. Taylor expanded in a around inf 95.5%

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

4. Final simplification 98.8%

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

Alternative 3: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+34} \lor \neg \left(b \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a - 4\right) \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8000.0)
   (+ (* (pow a 2.0) (+ 4.0 (* a (- a 4.0)))) -1.0)
   (if (or (<= b 5.5e+34) (not (<= b 5.8e+64)))
     (pow b 4.0)
     (* (- a 4.0) (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = (pow(a, 2.0) * (4.0 + (a * (a - 4.0)))) + -1.0;
	} else if ((b <= 5.5e+34) || !(b <= 5.8e+64)) {
		tmp = pow(b, 4.0);
	} else {
		tmp = (a - 4.0) * pow(a, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8000.0d0) then
        tmp = ((a ** 2.0d0) * (4.0d0 + (a * (a - 4.0d0)))) + (-1.0d0)
    else if ((b <= 5.5d+34) .or. (.not. (b <= 5.8d+64))) then
        tmp = b ** 4.0d0
    else
        tmp = (a - 4.0d0) * (a ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = (Math.pow(a, 2.0) * (4.0 + (a * (a - 4.0)))) + -1.0;
	} else if ((b <= 5.5e+34) || !(b <= 5.8e+64)) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = (a - 4.0) * Math.pow(a, 3.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8000.0:
		tmp = (math.pow(a, 2.0) * (4.0 + (a * (a - 4.0)))) + -1.0
	elif (b <= 5.5e+34) or not (b <= 5.8e+64):
		tmp = math.pow(b, 4.0)
	else:
		tmp = (a - 4.0) * math.pow(a, 3.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8000.0)
		tmp = Float64(Float64((a ^ 2.0) * Float64(4.0 + Float64(a * Float64(a - 4.0)))) + -1.0);
	elseif ((b <= 5.5e+34) || !(b <= 5.8e+64))
		tmp = b ^ 4.0;
	else
		tmp = Float64(Float64(a - 4.0) * (a ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8000.0)
		tmp = ((a ^ 2.0) * (4.0 + (a * (a - 4.0)))) + -1.0;
	elseif ((b <= 5.5e+34) || ~((b <= 5.8e+64)))
		tmp = b ^ 4.0;
	else
		tmp = (a - 4.0) * (a ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8000.0], N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[b, 5.5e+34], N[Not[LessEqual[b, 5.8e+64]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[(N[(a - 4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8e3

   1. Initial program 78.7%

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

      1. associate--l+ 78.8%

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

      2. fma-define 78.8%

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

      3. sqr-neg 78.8%

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

      4. fma-define 78.8%

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

      5. distribute-rgt-in 78.8%

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

      6. sqr-neg 78.8%

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

      7. distribute-rgt-in 78.8%

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

      8. fma-define 78.8%

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

      9. sqr-neg 78.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in b around 0 64.4%

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

   6. Taylor expanded in a around 0 80.9%

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

   if 8e3 < b < 5.4999999999999996e34 or 5.79999999999999986e64 < b

   1. Initial program 64.1%

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

      1. associate--l+ 64.1%

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

      2. fma-define 64.1%

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

      3. sqr-neg 64.1%

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

      4. fma-define 64.1%

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

      5. distribute-rgt-in 64.1%

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

      6. sqr-neg 64.1%

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

      7. distribute-rgt-in 64.1%

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

      8. fma-define 64.1%

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

      9. sqr-neg 64.1%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in b around inf 99.3%

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

   if 5.4999999999999996e34 < b < 5.79999999999999986e64

   1. Initial program 50.0%

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

      1. associate--l+ 50.0%

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

      2. fma-define 50.0%

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

      3. sqr-neg 50.0%

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

      4. fma-define 50.0%

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

      5. distribute-rgt-in 50.0%

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

      6. sqr-neg 50.0%

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

      7. distribute-rgt-in 50.0%

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

      8. fma-define 50.0%

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

      9. sqr-neg 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in a around inf 75.6%

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

      1. associate-*r/ 75.6%

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

      2. metadata-eval 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around 0 75.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;{a}^{2} \cdot \left(4 + a \cdot \left(a - 4\right)\right) + -1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+34} \lor \neg \left(b \leq 5.8 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a - 4\right) \cdot {a}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+34} \lor \neg \left(b \leq 2.6 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a - 4\right) \cdot {a}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8000.0)
   (fma (* a 4.0) a -1.0)
   (if (or (<= b 7.6e+34) (not (<= b 2.6e+64)))
     (pow b 4.0)
     (* (- a 4.0) (pow a 3.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8000.0) {
		tmp = fma((a * 4.0), a, -1.0);
	} else if ((b <= 7.6e+34) || !(b <= 2.6e+64)) {
		tmp = pow(b, 4.0);
	} else {
		tmp = (a - 4.0) * pow(a, 3.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8000.0)
		tmp = fma(Float64(a * 4.0), a, -1.0);
	elseif ((b <= 7.6e+34) || !(b <= 2.6e+64))
		tmp = b ^ 4.0;
	else
		tmp = Float64(Float64(a - 4.0) * (a ^ 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8000.0], N[(N[(a * 4.0), $MachinePrecision] * a + -1.0), $MachinePrecision], If[Or[LessEqual[b, 7.6e+34], N[Not[LessEqual[b, 2.6e+64]], $MachinePrecision]], N[Power[b, 4.0], $MachinePrecision], N[(N[(a - 4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8e3

   1. Initial program 78.7%

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

      1. associate--l+ 78.8%

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

      2. fma-define 78.8%

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

      3. sqr-neg 78.8%

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

      4. fma-define 78.8%

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

      5. distribute-rgt-in 78.8%

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

      6. sqr-neg 78.8%

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

      7. distribute-rgt-in 78.8%

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

      8. fma-define 78.8%

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

      9. sqr-neg 78.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in b around 0 64.4%

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

   6. Taylor expanded in a around 0 58.7%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 58.7%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 58.6%

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

      3. *-commutative 58.6%

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

      4. sqrt-prod 58.6%

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

      5. sqrt-pow1 42.3%

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

      6. metadata-eval 42.3%

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

      7. pow1 42.3%

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

      8. metadata-eval 42.3%

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

      9. *-commutative 42.3%

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

      10. sqrt-prod 42.3%

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

      11. sqrt-pow1 58.6%

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

      12. metadata-eval 58.6%

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

      13. pow1 58.6%

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

      14. metadata-eval 58.6%

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

   8. Applied egg-rr 58.6%

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

      1. difference-of-sqr-1 58.7%

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

      2. swap-sqr 58.7%

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

      3. unpow2 58.7%

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

      4. metadata-eval 58.7%

         \[\leadsto {a}^{2} \cdot \color{blue}{4} - 1 \]

      5. *-commutative 58.7%

         \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]

      6. unpow2 58.7%

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

      7. associate-*r* 58.7%

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

      8. fma-neg 58.7%

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

      9. metadata-eval 58.7%

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

   10. Applied egg-rr 58.7%

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

   if 8e3 < b < 7.6000000000000003e34 or 2.59999999999999997e64 < b

   1. Initial program 64.1%

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

      1. associate--l+ 64.1%

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

      2. fma-define 64.1%

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

      3. sqr-neg 64.1%

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

      4. fma-define 64.1%

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

      5. distribute-rgt-in 64.1%

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

      6. sqr-neg 64.1%

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

      7. distribute-rgt-in 64.1%

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

      8. fma-define 64.1%

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

      9. sqr-neg 64.1%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in b around inf 99.3%

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

   if 7.6000000000000003e34 < b < 2.59999999999999997e64

   1. Initial program 50.0%

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

      1. associate--l+ 50.0%

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

      2. fma-define 50.0%

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

      3. sqr-neg 50.0%

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

      4. fma-define 50.0%

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

      5. distribute-rgt-in 50.0%

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

      6. sqr-neg 50.0%

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

      7. distribute-rgt-in 50.0%

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

      8. fma-define 50.0%

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

      9. sqr-neg 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in a around inf 75.6%

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

      1. associate-*r/ 75.6%

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

      2. metadata-eval 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in a around 0 75.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+34} \lor \neg \left(b \leq 2.6 \cdot 10^{+64}\right):\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(a - 4\right) \cdot {a}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-8} \lor \neg \left(a \leq 5\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.8e-8) (not (<= a 5.0)))
   (pow a 4.0)
   (* (+ 1.0 (* a 2.0)) (+ (* a 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9.8e-8) || !(a <= 5.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -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 <= (-9.8d-8)) .or. (.not. (a <= 5.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (1.0d0 + (a * 2.0d0)) * ((a * 2.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -9.8e-8) || !(a <= 5.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -9.8e-8) or not (a <= 5.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9.8e-8) || !(a <= 5.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(Float64(a * 2.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -9.8e-8) || ~((a <= 5.0)))
		tmp = a ^ 4.0;
	else
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9.8e-8], N[Not[LessEqual[a, 5.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.8000000000000004e-8 or 5 < a

   1. Initial program 53.5%

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

      1. associate--l+ 53.5%

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

      2. fma-define 53.5%

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

      3. sqr-neg 53.5%

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

      4. fma-define 53.5%

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

      5. distribute-rgt-in 53.5%

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

      6. sqr-neg 53.5%

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

      7. distribute-rgt-in 53.5%

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

      8. fma-define 53.5%

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

      9. sqr-neg 53.5%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in a around inf 88.1%

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

   if -9.8000000000000004e-8 < a < 5

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-define 99.9%

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

      3. sqr-neg 99.9%

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

      4. fma-define 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. sqr-neg 99.9%

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

      7. distribute-rgt-in 99.9%

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

      8. fma-define 99.9%

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

      9. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 52.6%

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

   6. Taylor expanded in a around 0 51.9%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 51.9%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 51.9%

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

      3. *-commutative 51.9%

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

      4. sqrt-prod 51.9%

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

      5. sqrt-pow1 51.8%

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

      6. metadata-eval 51.8%

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

      7. pow1 51.8%

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

      8. metadata-eval 51.8%

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

      9. *-commutative 51.8%

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

      10. sqrt-prod 51.8%

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

      11. sqrt-pow1 51.9%

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

      12. metadata-eval 51.9%

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

      13. pow1 51.9%

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

      14. metadata-eval 51.9%

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

   8. Applied egg-rr 51.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-8} \lor \neg \left(a \leq 5\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7200.0) (fma (* a 4.0) a -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = fma((a * 4.0), a, -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7200.0)
		tmp = fma(Float64(a * 4.0), a, -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7200.0], N[(N[(a * 4.0), $MachinePrecision] * a + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7200

   1. Initial program 78.7%

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

      1. associate--l+ 78.8%

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

      2. fma-define 78.8%

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

      3. sqr-neg 78.8%

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

      4. fma-define 78.8%

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

      5. distribute-rgt-in 78.8%

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

      6. sqr-neg 78.8%

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

      7. distribute-rgt-in 78.8%

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

      8. fma-define 78.8%

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

      9. sqr-neg 78.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in b around 0 64.4%

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

   6. Taylor expanded in a around 0 58.7%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 58.7%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 58.6%

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

      3. *-commutative 58.6%

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

      4. sqrt-prod 58.6%

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

      5. sqrt-pow1 42.3%

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

      6. metadata-eval 42.3%

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

      7. pow1 42.3%

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

      8. metadata-eval 42.3%

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

      9. *-commutative 42.3%

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

      10. sqrt-prod 42.3%

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

      11. sqrt-pow1 58.6%

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

      12. metadata-eval 58.6%

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

      13. pow1 58.6%

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

      14. metadata-eval 58.6%

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

   8. Applied egg-rr 58.6%

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

      1. difference-of-sqr-1 58.7%

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

      2. swap-sqr 58.7%

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

      3. unpow2 58.7%

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

      4. metadata-eval 58.7%

         \[\leadsto {a}^{2} \cdot \color{blue}{4} - 1 \]

      5. *-commutative 58.7%

         \[\leadsto \color{blue}{4 \cdot {a}^{2}} - 1 \]

      6. unpow2 58.7%

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

      7. associate-*r* 58.7%

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

      8. fma-neg 58.7%

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

      9. metadata-eval 58.7%

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

   10. Applied egg-rr 58.7%

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

   if 7200 < b

   1. Initial program 63.1%

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

      1. associate--l+ 63.1%

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

      2. fma-define 63.1%

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

      3. sqr-neg 63.1%

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

      4. fma-define 63.1%

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

      5. distribute-rgt-in 63.1%

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

      6. sqr-neg 63.1%

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

      7. distribute-rgt-in 63.1%

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

      8. fma-define 63.1%

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

      9. sqr-neg 63.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in b around inf 94.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 4, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7800:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7800.0) (* (+ 1.0 (* a 2.0)) (+ (* a 2.0) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7800.0) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -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 <= 7800.0d0) then
        tmp = (1.0d0 + (a * 2.0d0)) * ((a * 2.0d0) + (-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 <= 7800.0) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7800.0:
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7800.0)
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(Float64(a * 2.0) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7800.0)
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7800.0], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7800

   1. Initial program 78.7%

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

      1. associate--l+ 78.8%

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

      2. fma-define 78.8%

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

      3. sqr-neg 78.8%

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

      4. fma-define 78.8%

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

      5. distribute-rgt-in 78.8%

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

      6. sqr-neg 78.8%

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

      7. distribute-rgt-in 78.8%

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

      8. fma-define 78.8%

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

      9. sqr-neg 78.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in b around 0 64.4%

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

   6. Taylor expanded in a around 0 58.7%

      \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 58.7%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 58.6%

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

      3. *-commutative 58.6%

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

      4. sqrt-prod 58.6%

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

      5. sqrt-pow1 42.3%

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

      6. metadata-eval 42.3%

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

      7. pow1 42.3%

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

      8. metadata-eval 42.3%

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

      9. *-commutative 42.3%

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

      10. sqrt-prod 42.3%

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

      11. sqrt-pow1 58.6%

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

      12. metadata-eval 58.6%

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

      13. pow1 58.6%

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

      14. metadata-eval 58.6%

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

   8. Applied egg-rr 58.6%

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

   if 7800 < b

   1. Initial program 63.1%

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

      1. associate--l+ 63.1%

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

      2. fma-define 63.1%

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

      3. sqr-neg 63.1%

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

      4. fma-define 63.1%

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

      5. distribute-rgt-in 63.1%

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

      6. sqr-neg 63.1%

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

      7. distribute-rgt-in 63.1%

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

      8. fma-define 63.1%

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

      9. sqr-neg 63.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in b around inf 94.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7800:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. associate--l+ 75.3%

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

   2. fma-define 75.3%

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

   3. sqr-neg 75.3%

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

   4. fma-define 75.3%

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

   5. distribute-rgt-in 75.3%

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

   6. sqr-neg 75.3%

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

   7. distribute-rgt-in 75.3%

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

   8. fma-define 75.3%

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

   9. sqr-neg 75.3%

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

3. Simplified 78.4%

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

5. Taylor expanded in b around 0 56.1%

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

6. Taylor expanded in a around 0 51.5%

   \[\leadsto \color{blue}{4 \cdot {a}^{2} - 1} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 51.5%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

   2. difference-of-sqr-1 51.5%

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

   3. *-commutative 51.5%

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

   4. sqrt-prod 51.5%

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

   5. sqrt-pow1 34.8%

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

   6. metadata-eval 34.8%

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

   7. pow1 34.8%

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

   8. metadata-eval 34.8%

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

   9. *-commutative 34.8%

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

   10. sqrt-prod 34.8%

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

   11. sqrt-pow1 51.5%

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

   12. metadata-eval 51.5%

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

   13. pow1 51.5%

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

   14. metadata-eval 51.5%

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

8. Applied egg-rr 51.5%

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

9. Final simplification 51.5%

   \[\leadsto \left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right) \]
10. Add Preprocessing

Alternative 9: 24.5% accurate, 128.0× 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 75.3%

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

   1. associate--l+ 75.3%

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

   2. fma-define 75.3%

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

   3. sqr-neg 75.3%

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

   4. fma-define 75.3%

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

   5. distribute-rgt-in 75.3%

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

   6. sqr-neg 75.3%

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

   7. distribute-rgt-in 75.3%

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

   8. fma-define 75.3%

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

   9. sqr-neg 75.3%

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

3. Simplified 78.4%

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

5. Taylor expanded in b around 0 56.1%

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

6. Taylor expanded in a around 0 24.5%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024083 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))