Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 98.2%

Time: 8.0s

Alternatives: 14

Speedup: 1.1×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in a around inf 94.1%

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

4. Final simplification 98.5%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in a around inf 94.1%

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

4. Final simplification 98.3%

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

Alternative 3: 84.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-130}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 1000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e-130)
   (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0))))
   (if (<= (* b b) 1e-34)
     (fma 4.0 (* a a) -1.0)
     (if (<= (* b b) 1000000.0) (pow a 4.0) (* (* b b) (+ (* b b) 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e-130) {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	} else if ((b * b) <= 1e-34) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if ((b * b) <= 1000000.0) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e-130)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	elseif (Float64(b * b) <= 1e-34)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (Float64(b * b) <= 1000000.0)
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e-130], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1e-34], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], N[Power[a, 4.0], $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 b b) < 4.0000000000000003e-130

   1. Initial program 86.3%

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

      1. associate--l+ 86.3%

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

      2. fma-def 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in b around 0 86.5%

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

      1. associate--l+ 86.5%

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

      2. associate-*r* 86.5%

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

      3. unpow2 86.5%

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

   6. Simplified 86.5%

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

   if 4.0000000000000003e-130 < (*.f64 b b) < 9.99999999999999928e-35

   1. Initial program 70.8%

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

      1. associate--l+ 70.8%

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

      2. fma-def 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in b around 0 70.8%

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

      1. associate--l+ 70.8%

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

      2. associate-*r* 70.8%

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

      3. unpow2 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in a around 0 92.4%

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

      1. fma-neg 92.4%

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

      2. unpow2 92.4%

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

      3. metadata-eval 92.4%

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

   9. Simplified 92.4%

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

   if 9.99999999999999928e-35 < (*.f64 b b) < 1e6

   1. Initial program 62.3%

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

      1. associate--l+ 62.3%

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

      2. fma-def 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in a around inf 75.8%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 65.6%

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

      1. associate--l+ 65.6%

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

      2. fma-def 65.6%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in b around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. fma-def 97.7%

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

      3. unpow2 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. fma-def 97.7%

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

      7. unpow2 97.7%

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

      8. distribute-lft-in 97.7%

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

      9. metadata-eval 97.7%

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

      10. associate-*r* 97.7%

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

      11. metadata-eval 97.7%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 90.6%

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

      1. fma-def 90.6%

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

      2. unpow2 90.6%

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

   9. Simplified 90.6%

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

       1. fma-udef 90.6%

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

       2. metadata-eval 90.6%

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

       3. pow-sqr 90.5%

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

       4. pow2 90.5%

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

       5. pow2 90.5%

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

       6. distribute-rgt-out 90.5%

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

   11. Applied egg-rr 90.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{-130}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 1000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{if}\;b \cdot b \leq 10^{-286}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \cdot b \leq 10^{-180}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 1000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma 4.0 (* a a) -1.0)))
   (if (<= (* b b) 1e-286)
     t_0
     (if (<= (* b b) 1e-180)
       (pow a 4.0)
       (if (<= (* b b) 1000000.0) t_0 (* (* b b) (+ (* b b) 4.0)))))))

C

double code(double a, double b) {
	double t_0 = fma(4.0, (a * a), -1.0);
	double tmp;
	if ((b * b) <= 1e-286) {
		tmp = t_0;
	} else if ((b * b) <= 1e-180) {
		tmp = pow(a, 4.0);
	} else if ((b * b) <= 1000000.0) {
		tmp = t_0;
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(4.0, Float64(a * a), -1.0)
	tmp = 0.0
	if (Float64(b * b) <= 1e-286)
		tmp = t_0;
	elseif (Float64(b * b) <= 1e-180)
		tmp = a ^ 4.0;
	elseif (Float64(b * b) <= 1000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 1e-286], t$95$0, If[LessEqual[N[(b * b), $MachinePrecision], 1e-180], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1000000.0], t$95$0, N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 1.00000000000000005e-286 or 1e-180 < (*.f64 b b) < 1e6

   1. Initial program 82.0%

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

      1. associate--l+ 82.0%

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

      2. fma-def 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in b around 0 82.0%

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

      1. associate--l+ 82.0%

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

      2. associate-*r* 82.0%

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

      3. unpow2 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in a around 0 73.7%

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

      1. fma-neg 73.7%

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

      2. unpow2 73.7%

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

      3. metadata-eval 73.7%

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

   9. Simplified 73.7%

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

   if 1.00000000000000005e-286 < (*.f64 b b) < 1e-180

   1. Initial program 83.0%

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

      1. associate--l+ 83.0%

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

      2. fma-def 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in a around inf 78.3%

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

   if 1e6 < (*.f64 b b)

   1. Initial program 65.6%

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

      1. associate--l+ 65.6%

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

      2. fma-def 65.6%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in b around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. fma-def 97.7%

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

      3. unpow2 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. fma-def 97.7%

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

      7. unpow2 97.7%

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

      8. distribute-lft-in 97.7%

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

      9. metadata-eval 97.7%

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

      10. associate-*r* 97.7%

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

      11. metadata-eval 97.7%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 90.6%

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

      1. fma-def 90.6%

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

      2. unpow2 90.6%

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

   9. Simplified 90.6%

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

       1. fma-udef 90.6%

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

       2. metadata-eval 90.6%

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

       3. pow-sqr 90.5%

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

       4. pow2 90.5%

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

       5. pow2 90.5%

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

       6. distribute-rgt-out 90.5%

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

   11. Applied egg-rr 90.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-286}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;b \cdot b \leq 10^{-180}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 5: 47.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-232}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-208}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3200:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.2e-286)
   -1.0
   (if (<= b 8.5e-232)
     (pow a 4.0)
     (if (<= b 3.6e-208)
       -1.0
       (if (<= b 3200.0) (pow a 4.0) (* (* b b) (+ (* b b) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.2e-286) {
		tmp = -1.0;
	} else if (b <= 8.5e-232) {
		tmp = pow(a, 4.0);
	} else if (b <= 3.6e-208) {
		tmp = -1.0;
	} else if (b <= 3200.0) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (b * b) * ((b * 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 <= 1.2d-286) then
        tmp = -1.0d0
    else if (b <= 8.5d-232) then
        tmp = a ** 4.0d0
    else if (b <= 3.6d-208) then
        tmp = -1.0d0
    else if (b <= 3200.0d0) then
        tmp = a ** 4.0d0
    else
        tmp = (b * b) * ((b * b) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.2e-286) {
		tmp = -1.0;
	} else if (b <= 8.5e-232) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 3.6e-208) {
		tmp = -1.0;
	} else if (b <= 3200.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.2e-286:
		tmp = -1.0
	elif b <= 8.5e-232:
		tmp = math.pow(a, 4.0)
	elif b <= 3.6e-208:
		tmp = -1.0
	elif b <= 3200.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = (b * b) * ((b * b) + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.2e-286)
		tmp = -1.0;
	elseif (b <= 8.5e-232)
		tmp = a ^ 4.0;
	elseif (b <= 3.6e-208)
		tmp = -1.0;
	elseif (b <= 3200.0)
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.2e-286)
		tmp = -1.0;
	elseif (b <= 8.5e-232)
		tmp = a ^ 4.0;
	elseif (b <= 3.6e-208)
		tmp = -1.0;
	elseif (b <= 3200.0)
		tmp = a ^ 4.0;
	else
		tmp = (b * b) * ((b * b) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.2e-286], -1.0, If[LessEqual[b, 8.5e-232], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 3.6e-208], -1.0, If[LessEqual[b, 3200.0], N[Power[a, 4.0], $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.19999999999999997e-286 or 8.5e-232 < b < 3.5999999999999998e-208

   1. Initial program 78.5%

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

      1. associate--l+ 78.5%

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

      2. fma-def 78.5%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in b around 0 59.8%

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

      1. associate--l+ 59.8%

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

      2. associate-*r* 59.8%

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

      3. unpow2 59.8%

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

   6. Simplified 59.8%

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

   7. Taylor expanded in a around 0 30.1%

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

   if 1.19999999999999997e-286 < b < 8.5e-232 or 3.5999999999999998e-208 < b < 3200

   1. Initial program 81.4%

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

      1. associate--l+ 81.4%

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

      2. fma-def 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in a around inf 68.3%

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

   if 3200 < b

   1. Initial program 60.4%

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

      1. associate--l+ 60.4%

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

      2. fma-def 60.4%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. unpow2 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-*r* 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 91.6%

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

      1. fma-def 91.6%

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

      2. unpow2 91.6%

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

   9. Simplified 91.6%

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

       1. fma-udef 91.6%

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

       2. metadata-eval 91.6%

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

       3. pow-sqr 91.4%

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

       4. pow2 91.4%

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

       5. pow2 91.4%

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

       6. distribute-rgt-out 91.4%

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

   11. Applied egg-rr 91.4%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-286}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-232}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-208}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3200:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 6: 45.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-112}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3400:\\ \;\;\;\;\left(b \cdot b\right) \cdot \frac{a \cdot \left(144 - \left(a \cdot a\right) \cdot 4\right)}{-12 + a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.1e-112)
   -1.0
   (if (<= b 3400.0)
     (* (* b b) (/ (* a (- 144.0 (* (* a a) 4.0))) (+ -12.0 (* a -2.0))))
     (* (* b b) (+ (* b b) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-112) {
		tmp = -1.0;
	} else if (b <= 3400.0) {
		tmp = (b * b) * ((a * (144.0 - ((a * a) * 4.0))) / (-12.0 + (a * -2.0)));
	} else {
		tmp = (b * b) * ((b * 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 <= 3.1d-112) then
        tmp = -1.0d0
    else if (b <= 3400.0d0) then
        tmp = (b * b) * ((a * (144.0d0 - ((a * a) * 4.0d0))) / ((-12.0d0) + (a * (-2.0d0))))
    else
        tmp = (b * b) * ((b * b) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.1e-112) {
		tmp = -1.0;
	} else if (b <= 3400.0) {
		tmp = (b * b) * ((a * (144.0 - ((a * a) * 4.0))) / (-12.0 + (a * -2.0)));
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.1e-112:
		tmp = -1.0
	elif b <= 3400.0:
		tmp = (b * b) * ((a * (144.0 - ((a * a) * 4.0))) / (-12.0 + (a * -2.0)))
	else:
		tmp = (b * b) * ((b * b) + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.1e-112)
		tmp = -1.0;
	elseif (b <= 3400.0)
		tmp = Float64(Float64(b * b) * Float64(Float64(a * Float64(144.0 - Float64(Float64(a * a) * 4.0))) / Float64(-12.0 + Float64(a * -2.0))));
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.1e-112)
		tmp = -1.0;
	elseif (b <= 3400.0)
		tmp = (b * b) * ((a * (144.0 - ((a * a) * 4.0))) / (-12.0 + (a * -2.0)));
	else
		tmp = (b * b) * ((b * b) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.1e-112], -1.0, If[LessEqual[b, 3400.0], N[(N[(b * b), $MachinePrecision] * N[(N[(a * N[(144.0 - N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-12.0 + N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.0999999999999998e-112

   1. Initial program 79.9%

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

      1. associate--l+ 79.9%

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

      2. fma-def 79.9%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in b around 0 64.5%

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

      1. associate--l+ 64.5%

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

      2. associate-*r* 64.5%

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

      3. unpow2 64.5%

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

   6. Simplified 64.5%

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

   7. Taylor expanded in a around 0 30.2%

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

   if 3.0999999999999998e-112 < b < 3400

   1. Initial program 73.6%

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

      1. associate--l+ 73.6%

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

      2. fma-def 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in b around inf 44.0%

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

      1. *-commutative 44.0%

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

      2. fma-def 44.0%

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

      3. unpow2 44.0%

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

      4. +-commutative 44.0%

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

      5. +-commutative 44.0%

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

      6. fma-def 44.0%

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

      7. unpow2 44.0%

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

      8. distribute-lft-in 44.0%

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

      9. metadata-eval 44.0%

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

      10. associate-*r* 44.0%

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

      11. metadata-eval 44.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 44.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 44.3%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 44.3%

         \[\leadsto \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      2. *-commutative 44.3%

         \[\leadsto \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      3. unpow2 44.3%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + 2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right) \]

      4. associate-*r* 44.3%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(2 \cdot \left(a \cdot a\right)\right) \cdot {b}^{2}} \]

      5. *-commutative 44.3%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot a\right) \cdot 2\right)} \cdot {b}^{2} \]

      6. distribute-rgt-out 44.3%

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right)} \]

      7. unpow2 44.3%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right) \]

      8. associate-*l* 44.3%

         \[\leadsto \left(b \cdot b\right) \cdot \left(a \cdot -12 + \color{blue}{a \cdot \left(a \cdot 2\right)}\right) \]

      9. distribute-lft-out 44.3%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   9. Simplified 44.3%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 44.3%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\left(-12 + a \cdot 2\right) \cdot a\right)} \]

       2. flip-+ 44.3%

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-12 \cdot -12 - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)}{-12 - a \cdot 2}} \cdot a\right) \]

       3. associate-*l/ 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\frac{\left(-12 \cdot -12 - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2}} \]

       4. metadata-eval 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(\color{blue}{144} - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2} \]

       5. *-commutative 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{\left(2 \cdot a\right)} \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2} \]

       6. *-commutative 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \left(2 \cdot a\right) \cdot \color{blue}{\left(2 \cdot a\right)}\right) \cdot a}{-12 - a \cdot 2} \]

       7. swap-sqr 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{\left(2 \cdot 2\right) \cdot \left(a \cdot a\right)}\right) \cdot a}{-12 - a \cdot 2} \]

       8. metadata-eval 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{4} \cdot \left(a \cdot a\right)\right) \cdot a}{-12 - a \cdot 2} \]

       9. *-commutative 49.1%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 - \color{blue}{2 \cdot a}} \]

       10. cancel-sign-sub-inv 49.1%

           \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{\color{blue}{-12 + \left(-2\right) \cdot a}} \]

       11. metadata-eval 49.1%

           \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 + \color{blue}{-2} \cdot a} \]

   11. Applied egg-rr 49.1%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 + -2 \cdot a}} \]

   if 3400 < b

   1. Initial program 60.4%

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

      1. associate--l+ 60.4%

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

      2. fma-def 60.4%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. unpow2 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-*r* 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 91.6%

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

      1. fma-def 91.6%

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

      2. unpow2 91.6%

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

   9. Simplified 91.6%

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

       1. fma-udef 91.6%

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

       2. metadata-eval 91.6%

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

       3. pow-sqr 91.4%

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

       4. pow2 91.4%

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

       5. pow2 91.4%

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

       6. distribute-rgt-out 91.4%

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

   11. Applied egg-rr 91.4%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-112}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3400:\\ \;\;\;\;\left(b \cdot b\right) \cdot \frac{a \cdot \left(144 - \left(a \cdot a\right) \cdot 4\right)}{-12 + a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 7: 41.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{-68}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-51}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.05e-68)
   -1.0
   (if (<= b 1.08e-51)
     (* (* b b) (* a (* a 2.0)))
     (if (<= b 9.8e-45) -1.0 (* a (* (* b b) (* a 2.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.05e-68) {
		tmp = -1.0;
	} else if (b <= 1.08e-51) {
		tmp = (b * b) * (a * (a * 2.0));
	} else if (b <= 9.8e-45) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * (a * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.05d-68) then
        tmp = -1.0d0
    else if (b <= 1.08d-51) then
        tmp = (b * b) * (a * (a * 2.0d0))
    else if (b <= 9.8d-45) then
        tmp = -1.0d0
    else
        tmp = a * ((b * b) * (a * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.05e-68) {
		tmp = -1.0;
	} else if (b <= 1.08e-51) {
		tmp = (b * b) * (a * (a * 2.0));
	} else if (b <= 9.8e-45) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * (a * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.05e-68:
		tmp = -1.0
	elif b <= 1.08e-51:
		tmp = (b * b) * (a * (a * 2.0))
	elif b <= 9.8e-45:
		tmp = -1.0
	else:
		tmp = a * ((b * b) * (a * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.05e-68)
		tmp = -1.0;
	elseif (b <= 1.08e-51)
		tmp = Float64(Float64(b * b) * Float64(a * Float64(a * 2.0)));
	elseif (b <= 9.8e-45)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(Float64(b * b) * Float64(a * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.05e-68)
		tmp = -1.0;
	elseif (b <= 1.08e-51)
		tmp = (b * b) * (a * (a * 2.0));
	elseif (b <= 9.8e-45)
		tmp = -1.0;
	else
		tmp = a * ((b * b) * (a * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.05e-68], -1.0, If[LessEqual[b, 1.08e-51], N[(N[(b * b), $MachinePrecision] * N[(a * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.8e-45], -1.0, N[(a * N[(N[(b * b), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.05000000000000011e-68 or 1.08000000000000004e-51 < b < 9.7999999999999996e-45

   1. Initial program 79.5%

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

      1. associate--l+ 79.5%

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

      2. fma-def 79.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 64.8%

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

      1. associate--l+ 64.8%

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

      2. associate-*r* 64.8%

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

      3. unpow2 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in a around 0 31.2%

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

   if 2.05000000000000011e-68 < b < 1.08000000000000004e-51

   1. Initial program 74.8%

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

      1. associate--l+ 74.8%

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

      2. fma-def 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in b around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. fma-def 51.3%

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

      3. unpow2 51.3%

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

      4. +-commutative 51.3%

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

      5. +-commutative 51.3%

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

      6. fma-def 51.3%

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

      7. unpow2 51.3%

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

      8. distribute-lft-in 51.3%

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

      9. metadata-eval 51.3%

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

      10. associate-*r* 51.3%

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

      11. metadata-eval 51.3%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 51.7%

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

      1. unpow2 51.7%

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

      2. associate-*r* 51.7%

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

      3. *-commutative 51.7%

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

      4. associate-*l* 51.7%

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

      5. unpow2 51.7%

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

   9. Simplified 51.7%

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

   if 9.7999999999999996e-45 < b

   1. Initial program 61.3%

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

      1. associate--l+ 61.3%

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

      2. fma-def 61.3%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in b around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. fma-def 95.9%

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

      3. unpow2 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. fma-def 95.9%

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

      7. unpow2 95.9%

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

      8. distribute-lft-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. associate-*r* 95.9%

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

      11. metadata-eval 95.9%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 60.0%

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

      1. unpow2 60.0%

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

      2. associate-*r* 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-*l* 60.0%

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

      5. associate-*l* 65.7%

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

      6. unpow2 65.7%

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

   9. Simplified 65.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{-68}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.08 \cdot 10^{-51}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\\ \end{array} \]

Alternative 8: 45.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-68}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e-68)
   -1.0
   (if (<= b 6.8e-9)
     (* (* b b) (* a (* a 2.0)))
     (if (<= b 0.48) -1.0 (* (* b b) (+ (* b b) 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-68) {
		tmp = -1.0;
	} else if (b <= 6.8e-9) {
		tmp = (b * b) * (a * (a * 2.0));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * ((b * 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 <= 2.1d-68) then
        tmp = -1.0d0
    else if (b <= 6.8d-9) then
        tmp = (b * b) * (a * (a * 2.0d0))
    else if (b <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * ((b * b) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e-68) {
		tmp = -1.0;
	} else if (b <= 6.8e-9) {
		tmp = (b * b) * (a * (a * 2.0));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e-68:
		tmp = -1.0
	elif b <= 6.8e-9:
		tmp = (b * b) * (a * (a * 2.0))
	elif b <= 0.48:
		tmp = -1.0
	else:
		tmp = (b * b) * ((b * b) + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e-68)
		tmp = -1.0;
	elseif (b <= 6.8e-9)
		tmp = Float64(Float64(b * b) * Float64(a * Float64(a * 2.0)));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e-68)
		tmp = -1.0;
	elseif (b <= 6.8e-9)
		tmp = (b * b) * (a * (a * 2.0));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = (b * b) * ((b * b) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e-68], -1.0, If[LessEqual[b, 6.8e-9], N[(N[(b * b), $MachinePrecision] * N[(a * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.48], -1.0, N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.10000000000000008e-68 or 6.7999999999999997e-9 < b < 0.47999999999999998

   1. Initial program 79.5%

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

      1. associate--l+ 79.5%

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

      2. fma-def 79.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 64.8%

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

      1. associate--l+ 64.8%

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

      2. associate-*r* 64.8%

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

      3. unpow2 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in a around 0 31.2%

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

   if 2.10000000000000008e-68 < b < 6.7999999999999997e-9

   1. Initial program 81.7%

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

      1. associate--l+ 81.7%

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

      2. fma-def 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in b around inf 47.3%

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

      1. *-commutative 47.3%

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

      2. fma-def 47.3%

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

      3. unpow2 47.3%

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

      4. +-commutative 47.3%

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

      5. +-commutative 47.3%

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

      6. fma-def 47.3%

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

      7. unpow2 47.3%

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

      8. distribute-lft-in 47.3%

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

      9. metadata-eval 47.3%

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

      10. associate-*r* 47.3%

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

      11. metadata-eval 47.3%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 47.5%

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

      1. unpow2 47.5%

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

      2. associate-*r* 47.5%

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

      3. *-commutative 47.5%

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

      4. associate-*l* 47.5%

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

      5. unpow2 47.5%

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

   9. Simplified 47.5%

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

   if 0.47999999999999998 < b

   1. Initial program 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. fma-def 98.6%

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

      3. unpow2 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

      6. fma-def 98.6%

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

      7. unpow2 98.6%

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

      8. distribute-lft-in 98.6%

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

      9. metadata-eval 98.6%

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

      10. associate-*r* 98.6%

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

      11. metadata-eval 98.6%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 90.3%

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

      1. fma-def 90.3%

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

      2. unpow2 90.3%

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

   9. Simplified 90.3%

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

       1. fma-udef 90.3%

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

       2. metadata-eval 90.3%

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

       3. pow-sqr 90.1%

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

       4. pow2 90.1%

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

       5. pow2 90.1%

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

       6. distribute-rgt-out 90.1%

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

   11. Applied egg-rr 90.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-68}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 9: 45.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.1e-69)
   -1.0
   (if (<= b 8e-10)
     (* (* b b) (* a (+ -12.0 (* a 2.0))))
     (if (<= b 0.48) -1.0 (* (* b b) (+ (* b b) 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.1e-69) {
		tmp = -1.0;
	} else if (b <= 8e-10) {
		tmp = (b * b) * (a * (-12.0 + (a * 2.0)));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * ((b * 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 <= 4.1d-69) then
        tmp = -1.0d0
    else if (b <= 8d-10) then
        tmp = (b * b) * (a * ((-12.0d0) + (a * 2.0d0)))
    else if (b <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = (b * b) * ((b * b) + 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.1e-69) {
		tmp = -1.0;
	} else if (b <= 8e-10) {
		tmp = (b * b) * (a * (-12.0 + (a * 2.0)));
	} else if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * ((b * b) + 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.1e-69:
		tmp = -1.0
	elif b <= 8e-10:
		tmp = (b * b) * (a * (-12.0 + (a * 2.0)))
	elif b <= 0.48:
		tmp = -1.0
	else:
		tmp = (b * b) * ((b * b) + 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.1e-69)
		tmp = -1.0;
	elseif (b <= 8e-10)
		tmp = Float64(Float64(b * b) * Float64(a * Float64(-12.0 + Float64(a * 2.0))));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.1e-69)
		tmp = -1.0;
	elseif (b <= 8e-10)
		tmp = (b * b) * (a * (-12.0 + (a * 2.0)));
	elseif (b <= 0.48)
		tmp = -1.0;
	else
		tmp = (b * b) * ((b * b) + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.1e-69], -1.0, If[LessEqual[b, 8e-10], N[(N[(b * b), $MachinePrecision] * N[(a * N[(-12.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.48], -1.0, N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.0999999999999999e-69 or 8.00000000000000029e-10 < b < 0.47999999999999998

   1. Initial program 79.5%

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

      1. associate--l+ 79.5%

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

      2. fma-def 79.5%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 64.8%

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

      1. associate--l+ 64.8%

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

      2. associate-*r* 64.8%

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

      3. unpow2 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in a around 0 31.2%

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

   if 4.0999999999999999e-69 < b < 8.00000000000000029e-10

   1. Initial program 81.7%

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

      1. associate--l+ 81.7%

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

      2. fma-def 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in b around inf 47.3%

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

      1. *-commutative 47.3%

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

      2. fma-def 47.3%

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

      3. unpow2 47.3%

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

      4. +-commutative 47.3%

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

      5. +-commutative 47.3%

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

      6. fma-def 47.3%

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

      7. unpow2 47.3%

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

      8. distribute-lft-in 47.3%

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

      9. metadata-eval 47.3%

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

      10. associate-*r* 47.3%

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

      11. metadata-eval 47.3%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 47.5%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 47.5%

         \[\leadsto \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      2. *-commutative 47.5%

         \[\leadsto \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      3. unpow2 47.5%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + 2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right) \]

      4. associate-*r* 47.5%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(2 \cdot \left(a \cdot a\right)\right) \cdot {b}^{2}} \]

      5. *-commutative 47.5%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot a\right) \cdot 2\right)} \cdot {b}^{2} \]

      6. distribute-rgt-out 47.5%

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right)} \]

      7. unpow2 47.5%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right) \]

      8. associate-*l* 47.5%

         \[\leadsto \left(b \cdot b\right) \cdot \left(a \cdot -12 + \color{blue}{a \cdot \left(a \cdot 2\right)}\right) \]

      9. distribute-lft-out 47.5%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   9. Simplified 47.5%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   if 0.47999999999999998 < b

   1. Initial program 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. fma-def 98.6%

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

      3. unpow2 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

      6. fma-def 98.6%

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

      7. unpow2 98.6%

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

      8. distribute-lft-in 98.6%

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

      9. metadata-eval 98.6%

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

      10. associate-*r* 98.6%

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

      11. metadata-eval 98.6%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 90.3%

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

      1. fma-def 90.3%

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

      2. unpow2 90.3%

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

   9. Simplified 90.3%

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

       1. fma-udef 90.3%

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

       2. metadata-eval 90.3%

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

       3. pow-sqr 90.1%

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

       4. pow2 90.1%

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

       5. pow2 90.1%

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

       6. distribute-rgt-out 90.1%

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

   11. Applied egg-rr 90.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)\\ \mathbf{elif}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 4\right)\\ \end{array} \]

Alternative 10: 41.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.8e-45)
   -1.0
   (if (<= b 6.8e+153) (* 2.0 (* a (* b (* a b)))) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-45) {
		tmp = -1.0;
	} else if (b <= 6.8e+153) {
		tmp = 2.0 * (a * (b * (a * b)));
	} else {
		tmp = b * (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 <= 9.8d-45) then
        tmp = -1.0d0
    else if (b <= 6.8d+153) then
        tmp = 2.0d0 * (a * (b * (a * b)))
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-45) {
		tmp = -1.0;
	} else if (b <= 6.8e+153) {
		tmp = 2.0 * (a * (b * (a * b)));
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.8e-45:
		tmp = -1.0
	elif b <= 6.8e+153:
		tmp = 2.0 * (a * (b * (a * b)))
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.8e-45)
		tmp = -1.0;
	elseif (b <= 6.8e+153)
		tmp = Float64(2.0 * Float64(a * Float64(b * Float64(a * b))));
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.8e-45)
		tmp = -1.0;
	elseif (b <= 6.8e+153)
		tmp = 2.0 * (a * (b * (a * b)));
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.8e-45], -1.0, If[LessEqual[b, 6.8e+153], N[(2.0 * N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.7999999999999996e-45

   1. Initial program 79.3%

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

      1. associate--l+ 79.3%

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

      2. fma-def 79.3%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in b around 0 65.2%

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

      1. associate--l+ 65.2%

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

      2. associate-*r* 65.2%

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

      3. unpow2 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in a around 0 31.5%

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

   if 9.7999999999999996e-45 < b < 6.7999999999999995e153

   1. Initial program 77.2%

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

      1. associate--l+ 77.2%

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

      2. fma-def 77.2%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around inf 92.9%

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

      1. *-commutative 92.9%

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

      2. fma-def 92.9%

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

      3. unpow2 92.9%

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

      4. +-commutative 92.9%

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

      5. +-commutative 92.9%

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

      6. fma-def 92.9%

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

      7. unpow2 92.9%

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

      8. distribute-lft-in 92.9%

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

      9. metadata-eval 92.9%

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

      10. associate-*r* 92.9%

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

      11. metadata-eval 92.9%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 37.1%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 37.1%

         \[\leadsto \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      2. *-commutative 37.1%

         \[\leadsto \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      3. unpow2 37.1%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + 2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right) \]

      4. associate-*r* 37.1%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(2 \cdot \left(a \cdot a\right)\right) \cdot {b}^{2}} \]

      5. *-commutative 37.1%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot a\right) \cdot 2\right)} \cdot {b}^{2} \]

      6. distribute-rgt-out 39.6%

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right)} \]

      7. unpow2 39.6%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right) \]

      8. associate-*l* 39.6%

         \[\leadsto \left(b \cdot b\right) \cdot \left(a \cdot -12 + \color{blue}{a \cdot \left(a \cdot 2\right)}\right) \]

      9. distribute-lft-out 39.6%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   9. Simplified 39.6%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative 39.6%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\left(-12 + a \cdot 2\right) \cdot a\right)} \]

       2. flip-+ 39.6%

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{\frac{-12 \cdot -12 - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)}{-12 - a \cdot 2}} \cdot a\right) \]

       3. associate-*l/ 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\frac{\left(-12 \cdot -12 - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2}} \]

       4. metadata-eval 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(\color{blue}{144} - \left(a \cdot 2\right) \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2} \]

       5. *-commutative 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{\left(2 \cdot a\right)} \cdot \left(a \cdot 2\right)\right) \cdot a}{-12 - a \cdot 2} \]

       6. *-commutative 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \left(2 \cdot a\right) \cdot \color{blue}{\left(2 \cdot a\right)}\right) \cdot a}{-12 - a \cdot 2} \]

       7. swap-sqr 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{\left(2 \cdot 2\right) \cdot \left(a \cdot a\right)}\right) \cdot a}{-12 - a \cdot 2} \]

       8. metadata-eval 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - \color{blue}{4} \cdot \left(a \cdot a\right)\right) \cdot a}{-12 - a \cdot 2} \]

       9. *-commutative 41.9%

          \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 - \color{blue}{2 \cdot a}} \]

       10. cancel-sign-sub-inv 41.9%

           \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{\color{blue}{-12 + \left(-2\right) \cdot a}} \]

       11. metadata-eval 41.9%

           \[\leadsto \left(b \cdot b\right) \cdot \frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 + \color{blue}{-2} \cdot a} \]

   11. Applied egg-rr 41.9%

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\frac{\left(144 - 4 \cdot \left(a \cdot a\right)\right) \cdot a}{-12 + -2 \cdot a}} \]

   12. Taylor expanded in a around inf 40.0%

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

       1. unpow2 40.0%

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

       2. unpow2 40.0%

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

       3. associate-*l* 40.0%

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

       4. *-commutative 40.0%

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

       5. associate-*l* 40.0%

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

   14. Simplified 40.0%

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

   if 6.7999999999999995e153 < b

   1. Initial program 40.0%

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

      1. associate--l+ 40.0%

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

      2. fma-def 40.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. unpow2 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-*r* 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(a \cdot \left(b \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 11: 39.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.48)
   -1.0
   (if (<= b 6.8e+153) (* a (* (* b b) -12.0)) (* b (* b 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else if (b <= 6.8e+153) {
		tmp = a * ((b * b) * -12.0);
	} else {
		tmp = b * (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 <= 0.48d0) then
        tmp = -1.0d0
    else if (b <= 6.8d+153) then
        tmp = a * ((b * b) * (-12.0d0))
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else if (b <= 6.8e+153) {
		tmp = a * ((b * b) * -12.0);
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	elif b <= 6.8e+153:
		tmp = a * ((b * b) * -12.0)
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		tmp = -1.0;
	elseif (b <= 6.8e+153)
		tmp = Float64(a * Float64(Float64(b * b) * -12.0));
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.48)
		tmp = -1.0;
	elseif (b <= 6.8e+153)
		tmp = a * ((b * b) * -12.0);
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, If[LessEqual[b, 6.8e+153], N[(a * N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 0.47999999999999998

   1. Initial program 79.7%

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

      1. associate--l+ 79.7%

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

      2. fma-def 79.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 65.8%

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

      1. associate--l+ 65.8%

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

      2. associate-*r* 65.8%

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

      3. unpow2 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in a around 0 31.0%

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

   if 0.47999999999999998 < b < 6.7999999999999995e153

   1. Initial program 75.3%

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

      1. associate--l+ 75.3%

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

      2. fma-def 75.3%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in b around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. fma-def 97.5%

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

      3. unpow2 97.5%

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

      4. +-commutative 97.5%

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

      5. +-commutative 97.5%

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

      6. fma-def 97.5%

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

      7. unpow2 97.5%

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

      8. distribute-lft-in 97.5%

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

      9. metadata-eval 97.5%

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

      10. associate-*r* 97.5%

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

      11. metadata-eval 97.5%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 37.2%

      \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right) + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 37.2%

         \[\leadsto \color{blue}{\left(-12 \cdot a\right) \cdot {b}^{2}} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      2. *-commutative 37.2%

         \[\leadsto \color{blue}{\left(a \cdot -12\right)} \cdot {b}^{2} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right) \]

      3. unpow2 37.2%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + 2 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right) \]

      4. associate-*r* 37.2%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(2 \cdot \left(a \cdot a\right)\right) \cdot {b}^{2}} \]

      5. *-commutative 37.2%

         \[\leadsto \left(a \cdot -12\right) \cdot {b}^{2} + \color{blue}{\left(\left(a \cdot a\right) \cdot 2\right)} \cdot {b}^{2} \]

      6. distribute-rgt-out 39.9%

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right)} \]

      7. unpow2 39.9%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(a \cdot -12 + \left(a \cdot a\right) \cdot 2\right) \]

      8. associate-*l* 39.9%

         \[\leadsto \left(b \cdot b\right) \cdot \left(a \cdot -12 + \color{blue}{a \cdot \left(a \cdot 2\right)}\right) \]

      9. distribute-lft-out 39.9%

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   9. Simplified 39.9%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(a \cdot \left(-12 + a \cdot 2\right)\right)} \]

   10. Taylor expanded in a around 0 24.2%

       \[\leadsto \color{blue}{-12 \cdot \left(a \cdot {b}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 24.2%

          \[\leadsto -12 \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right) \]

       2. *-commutative 24.2%

          \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot -12} \]

       3. associate-*l* 24.2%

          \[\leadsto \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot -12\right)} \]

   12. Simplified 24.2%

       \[\leadsto \color{blue}{a \cdot \left(\left(b \cdot b\right) \cdot -12\right)} \]

   if 6.7999999999999995e153 < b

   1. Initial program 40.0%

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

      1. associate--l+ 40.0%

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

      2. fma-def 40.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-def 100.0%

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

      7. unpow2 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-*r* 100.0%

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

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot -12\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 12: 41.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.8e-45) -1.0 (* a (* (* b b) (* a 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-45) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * (a * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9.8d-45) then
        tmp = -1.0d0
    else
        tmp = a * ((b * b) * (a * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-45) {
		tmp = -1.0;
	} else {
		tmp = a * ((b * b) * (a * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.8e-45:
		tmp = -1.0
	else:
		tmp = a * ((b * b) * (a * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.8e-45)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(Float64(b * b) * Float64(a * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.8e-45)
		tmp = -1.0;
	else
		tmp = a * ((b * b) * (a * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.8e-45], -1.0, N[(a * N[(N[(b * b), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.7999999999999996e-45

   1. Initial program 79.3%

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

      1. associate--l+ 79.3%

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

      2. fma-def 79.3%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in b around 0 65.2%

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

      1. associate--l+ 65.2%

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

      2. associate-*r* 65.2%

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

      3. unpow2 65.2%

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

   6. Simplified 65.2%

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

   7. Taylor expanded in a around 0 31.5%

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

   if 9.7999999999999996e-45 < b

   1. Initial program 61.3%

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

      1. associate--l+ 61.3%

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

      2. fma-def 61.3%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in b around inf 95.9%

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

      1. *-commutative 95.9%

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

      2. fma-def 95.9%

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

      3. unpow2 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. fma-def 95.9%

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

      7. unpow2 95.9%

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

      8. distribute-lft-in 95.9%

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

      9. metadata-eval 95.9%

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

      10. associate-*r* 95.9%

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

      11. metadata-eval 95.9%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around inf 60.0%

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

      1. unpow2 60.0%

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

      2. associate-*r* 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-*l* 60.0%

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

      5. associate-*l* 65.7%

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

      6. unpow2 65.7%

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

   9. Simplified 65.7%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot 2\right)\right)\\ \end{array} \]

Alternative 13: 38.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.48) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (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 <= 0.48d0) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.48) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.48:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.48)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.48], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.47999999999999998

   1. Initial program 79.7%

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

      1. associate--l+ 79.7%

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

      2. fma-def 79.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in b around 0 65.8%

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

      1. associate--l+ 65.8%

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

      2. associate-*r* 65.8%

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

      3. unpow2 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in a around 0 31.0%

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

   if 0.47999999999999998 < b

   1. Initial program 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in b around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. fma-def 98.6%

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

      3. unpow2 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

      6. fma-def 98.6%

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

      7. unpow2 98.6%

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

      8. distribute-lft-in 98.6%

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

      9. metadata-eval 98.6%

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

      10. associate-*r* 98.6%

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

      11. metadata-eval 98.6%

          \[\leadsto \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + \color{blue}{-12} \cdot a\right), {b}^{4}\right) \]

   6. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(2, a \cdot a, 4 + -12 \cdot a\right), {b}^{4}\right)} \]

   7. Taylor expanded in a around 0 90.3%

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

      1. fma-def 90.3%

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

      2. unpow2 90.3%

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

   9. Simplified 90.3%

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

   10. Taylor expanded in b around 0 48.2%

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

       1. unpow2 48.2%

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

       2. *-commutative 48.2%

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

       3. associate-*l* 48.2%

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

   12. Simplified 48.2%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.48:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 14: 25.1% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.4%

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

   1. associate--l+ 74.4%

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

   2. fma-def 74.4%

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

3. Simplified 76.3%

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

4. Taylor expanded in b around 0 55.0%

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

   1. associate--l+ 55.0%

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

   2. associate-*r* 55.0%

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

   3. unpow2 55.0%

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

6. Simplified 55.0%

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

7. Taylor expanded in a around 0 23.1%

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

8. Final simplification 23.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023222 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))