Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.5% → 98.5%

Time: 7.6s

Alternatives: 11

Speedup: 1.2×

• 60.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% 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: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+106}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(4 \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+106)
   (pow a 4.0)
   (+ (pow (hypot a b) 4.0) (+ (* 4.0 (* a (* a a))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e+106) {
		tmp = pow(a, 4.0);
	} else {
		tmp = pow(hypot(a, b), 4.0) + ((4.0 * (a * (a * a))) + -1.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e+106) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(Math.hypot(a, b), 4.0) + ((4.0 * (a * (a * a))) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e+106:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(math.hypot(a, b), 4.0) + ((4.0 * (a * (a * a))) + -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e+106)
		tmp = a ^ 4.0;
	else
		tmp = Float64((hypot(a, b) ^ 4.0) + Float64(Float64(4.0 * Float64(a * Float64(a * a))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e+106)
		tmp = a ^ 4.0;
	else
		tmp = (hypot(a, b) ^ 4.0) + ((4.0 * (a * (a * a))) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e+106], N[Power[a, 4.0], $MachinePrecision], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(4.0 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000018e106

   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 4.4%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -2.00000000000000018e106 < a

   1. Initial program 85.1%

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

      1. associate--l+ 85.1%

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

      2. fma-def 85.1%

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

   3. Simplified 85.1%

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

      1. fma-def 85.1%

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

      2. add-sqr-sqrt 85.1%

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

      3. hypot-udef 85.1%

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

      4. hypot-udef 85.1%

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

      5. pow-prod-down 85.1%

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

      6. pow-prod-up 85.2%

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

      7. metadata-eval 85.2%

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

      8. expm1-log1p-u 83.4%

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

      9. expm1-udef 83.4%

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

   5. Applied egg-rr 83.4%

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

      1. expm1-def 83.4%

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

      2. expm1-log1p 85.2%

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

   7. Simplified 85.2%

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

   8. Taylor expanded in a around inf 98.4%

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

      1. unpow3 98.4%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+106}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \left(4 \cdot \left(a \cdot \left(a \cdot a\right)\right) + -1\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

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) (+ (* 4.0 (* a a)) -1.0)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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 2.6%

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

   4. Taylor expanded in b around 0 35.9%

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

      1. associate--l+ 35.9%

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

      2. associate-*r* 35.9%

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

      3. unpow2 35.9%

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

   6. Simplified 35.9%

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

   7. Taylor expanded in a around 0 95.1%

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

      1. unpow2 95.1%

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

   9. Simplified 95.1%

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

4. Final simplification 98.4%

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

Alternative 3: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + \left(4 \cdot \left(a \cdot a\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ (pow a 4.0) (+ (* 4.0 (* a a)) -1.0))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = pow(a, 4.0) + ((4.0 * (a * a)) + -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = (a ** 4.0d0) + ((4.0d0 * (a * a)) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = Math.pow(a, 4.0) + ((4.0 * (a * a)) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = math.pow(a, 4.0) + ((4.0 * (a * a)) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64((a ^ 4.0) + Float64(Float64(4.0 * Float64(a * a)) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (a ^ 4.0) + ((4.0 * (a * a)) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   1. Initial program 76.7%

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

      1. associate--l+ 76.6%

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

      2. fma-def 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in b around 0 77.5%

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

      1. associate--l+ 77.5%

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

      2. associate-*r* 77.5%

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

      3. unpow2 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in a around 0 97.1%

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

      1. unpow2 97.1%

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

   9. Simplified 97.1%

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

   if 4e8 < (*.f64 b b)

   1. Initial program 63.0%

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

      1. associate--l+ 63.0%

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

      2. fma-def 63.0%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around inf 92.9%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;{a}^{4} + \left(4 \cdot \left(a \cdot a\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 4: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + 4 \cdot \left(a \cdot a + a \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-155}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot -12\right)\right)\\ \mathbf{elif}\;a \leq 0.08:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* 4.0 (+ (* a a) (* a (* a a)))))))
   (if (<= a -2.4)
     (pow a 4.0)
     (if (<= a -3e-101)
       t_0
       (if (<= a -5.4e-155)
         (* a (* b (* b -12.0)))
         (if (<= a 0.08) t_0 (pow a 4.0)))))))

C

double code(double a, double b) {
	double t_0 = -1.0 + (4.0 * ((a * a) + (a * (a * a))));
	double tmp;
	if (a <= -2.4) {
		tmp = pow(a, 4.0);
	} else if (a <= -3e-101) {
		tmp = t_0;
	} else if (a <= -5.4e-155) {
		tmp = a * (b * (b * -12.0));
	} else if (a <= 0.08) {
		tmp = t_0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (4.0d0 * ((a * a) + (a * (a * a))))
    if (a <= (-2.4d0)) then
        tmp = a ** 4.0d0
    else if (a <= (-3d-101)) then
        tmp = t_0
    else if (a <= (-5.4d-155)) then
        tmp = a * (b * (b * (-12.0d0)))
    else if (a <= 0.08d0) then
        tmp = t_0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -1.0 + (4.0 * ((a * a) + (a * (a * a))));
	double tmp;
	if (a <= -2.4) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -3e-101) {
		tmp = t_0;
	} else if (a <= -5.4e-155) {
		tmp = a * (b * (b * -12.0));
	} else if (a <= 0.08) {
		tmp = t_0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -1.0 + (4.0 * ((a * a) + (a * (a * a))))
	tmp = 0
	if a <= -2.4:
		tmp = math.pow(a, 4.0)
	elif a <= -3e-101:
		tmp = t_0
	elif a <= -5.4e-155:
		tmp = a * (b * (b * -12.0))
	elif a <= 0.08:
		tmp = t_0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-1.0 + Float64(4.0 * Float64(Float64(a * a) + Float64(a * Float64(a * a)))))
	tmp = 0.0
	if (a <= -2.4)
		tmp = a ^ 4.0;
	elseif (a <= -3e-101)
		tmp = t_0;
	elseif (a <= -5.4e-155)
		tmp = Float64(a * Float64(b * Float64(b * -12.0)));
	elseif (a <= 0.08)
		tmp = t_0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -1.0 + (4.0 * ((a * a) + (a * (a * a))));
	tmp = 0.0;
	if (a <= -2.4)
		tmp = a ^ 4.0;
	elseif (a <= -3e-101)
		tmp = t_0;
	elseif (a <= -5.4e-155)
		tmp = a * (b * (b * -12.0));
	elseif (a <= 0.08)
		tmp = t_0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-1.0 + N[(4.0 * N[(N[(a * a), $MachinePrecision] + N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.4], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -3e-101], t$95$0, If[LessEqual[a, -5.4e-155], N[(a * N[(b * N[(b * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.08], t$95$0, N[Power[a, 4.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.39999999999999991 or 0.0800000000000000017 < a

   1. Initial program 42.7%

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

      1. associate--l+ 42.7%

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

      2. fma-def 42.7%

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

   3. Simplified 44.2%

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

   4. Taylor expanded in a around inf 90.4%

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

   if -2.39999999999999991 < a < -3.0000000000000003e-101 or -5.39999999999999962e-155 < a < 0.0800000000000000017

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 50.3%

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

      1. associate--l+ 50.2%

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

      2. associate-*r* 50.2%

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

      3. unpow2 50.2%

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

   6. Simplified 50.2%

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

   7. Taylor expanded in a around 0 48.8%

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

      1. sub-neg 48.8%

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

      2. distribute-lft-out 48.8%

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

      3. metadata-eval 48.8%

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

      4. unpow2 48.8%

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

   9. Simplified 48.8%

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

       1. unpow3 97.3%

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

   11. Applied egg-rr 48.8%

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

   if -3.0000000000000003e-101 < a < -5.39999999999999962e-155

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 78.4%

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

      1. *-commutative 78.4%

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

      2. fma-def 78.4%

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

      3. unpow2 78.4%

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

      4. +-commutative 78.4%

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

      5. +-commutative 78.4%

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

      6. fma-def 78.4%

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

      7. unpow2 78.4%

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

      8. distribute-lft-in 78.4%

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

      9. metadata-eval 78.4%

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

      10. associate-*r* 78.4%

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

      11. metadata-eval 78.4%

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

   6. Simplified 78.4%

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

   7. Taylor expanded in a around inf 67.9%

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

      1. associate-*r* 67.9%

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

      2. *-commutative 67.9%

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

      3. associate-*r* 67.9%

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

      4. distribute-rgt-out 67.9%

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

      5. *-commutative 67.9%

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

      6. +-commutative 67.9%

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

      7. unpow2 67.9%

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

      8. unpow2 67.9%

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

      9. associate-*r* 67.9%

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

      10. *-commutative 67.9%

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

      11. distribute-rgt-out 67.9%

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

      12. *-commutative 67.9%

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

   9. Simplified 67.9%

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

   10. Taylor expanded in a around 0 67.9%

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

       1. unpow2 67.9%

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

   12. Simplified 67.9%

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

   13. Taylor expanded in a around 0 67.9%

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

       1. *-commutative 67.9%

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

       2. unpow2 67.9%

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

       3. associate-*l* 78.4%

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

       4. associate-*r* 78.4%

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

   15. Simplified 78.4%

       \[\leadsto \color{blue}{a \cdot \left(b \cdot \left(b \cdot -12\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-101}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a + a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-155}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot -12\right)\right)\\ \mathbf{elif}\;a \leq 0.08:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a + a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 5: 82.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0) (+ (* 4.0 (* a a)) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 400000000.0d0) then
        tmp = (4.0d0 * (a * a)) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = (4.0 * (a * a)) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64(Float64(4.0 * Float64(a * a)) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = (4.0 * (a * a)) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   1. Initial program 76.7%

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

      1. associate--l+ 76.6%

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

      2. fma-def 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in b around 0 77.5%

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

      1. associate--l+ 77.5%

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

      2. associate-*r* 77.5%

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

      3. unpow2 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in a around 0 58.4%

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

      1. sub-neg 58.4%

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

      2. distribute-lft-out 58.4%

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

      3. metadata-eval 58.4%

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

      4. unpow2 58.4%

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

   9. Simplified 58.4%

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

   10. Taylor expanded in a around 0 73.9%

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

       1. unpow2 73.9%

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

   12. Simplified 73.9%

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

   if 4e8 < (*.f64 b b)

   1. Initial program 63.0%

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

      1. associate--l+ 63.0%

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

      2. fma-def 63.0%

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

   3. Simplified 64.6%

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

   4. Taylor expanded in b around inf 92.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 62.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-34) (+ (* 4.0 (* a a)) -1.0) (* 2.0 (* b (* b (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-34) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = 2.0 * (b * (b * (a * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-34) then
        tmp = (4.0d0 * (a * a)) + (-1.0d0)
    else
        tmp = 2.0d0 * (b * (b * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-34) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = 2.0 * (b * (b * (a * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-34:
		tmp = (4.0 * (a * a)) + -1.0
	else:
		tmp = 2.0 * (b * (b * (a * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-34)
		tmp = Float64(Float64(4.0 * Float64(a * a)) + -1.0);
	else
		tmp = Float64(2.0 * Float64(b * Float64(b * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-34)
		tmp = (4.0 * (a * a)) + -1.0;
	else
		tmp = 2.0 * (b * (b * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-34], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(2.0 * N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.6%

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

      1. associate--l+ 77.6%

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

      2. fma-def 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around 0 78.5%

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

      1. associate--l+ 78.5%

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

      2. associate-*r* 78.5%

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

      3. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in a around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. distribute-lft-out 59.0%

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

      3. metadata-eval 59.0%

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

      4. unpow2 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in a around 0 74.0%

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

       1. unpow2 74.0%

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

   12. Simplified 74.0%

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

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

   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 63.9%

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

   4. Taylor expanded in b around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. fma-def 97.4%

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

      3. unpow2 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. fma-def 97.4%

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

      7. unpow2 97.4%

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

      8. distribute-lft-in 97.4%

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

      9. metadata-eval 97.4%

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

      10. associate-*r* 97.4%

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

      11. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in a around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. unpow2 57.4%

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

      3. associate-*l* 49.3%

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

      4. unpow2 49.3%

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

   9. Simplified 49.3%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(b \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e-34) (+ (* 4.0 (* a a)) -1.0) (* (* b b) (* (* a a) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-34) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = (b * b) * ((a * a) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d-34) then
        tmp = (4.0d0 * (a * a)) + (-1.0d0)
    else
        tmp = (b * b) * ((a * a) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e-34) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = (b * b) * ((a * a) * 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e-34:
		tmp = (4.0 * (a * a)) + -1.0
	else:
		tmp = (b * b) * ((a * a) * 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e-34)
		tmp = Float64(Float64(4.0 * Float64(a * a)) + -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(a * a) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e-34)
		tmp = (4.0 * (a * a)) + -1.0;
	else
		tmp = (b * b) * ((a * a) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e-34], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.6%

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

      1. associate--l+ 77.6%

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

      2. fma-def 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around 0 78.5%

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

      1. associate--l+ 78.5%

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

      2. associate-*r* 78.5%

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

      3. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in a around 0 59.0%

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

      1. sub-neg 59.0%

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

      2. distribute-lft-out 59.0%

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

      3. metadata-eval 59.0%

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

      4. unpow2 59.0%

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

   9. Simplified 59.0%

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

   10. Taylor expanded in a around 0 74.0%

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

       1. unpow2 74.0%

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

   12. Simplified 74.0%

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

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

   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 63.9%

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

   4. Taylor expanded in b around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. fma-def 97.4%

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

      3. unpow2 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. fma-def 97.4%

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

      7. unpow2 97.4%

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

      8. distribute-lft-in 97.4%

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

      9. metadata-eval 97.4%

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

      10. associate-*r* 97.4%

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

      11. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in a around inf 57.4%

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

      1. associate-*r* 57.4%

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

      2. unpow2 57.4%

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

      3. unpow2 57.4%

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

   9. Simplified 57.4%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{-34}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 8: 39.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 7.5e-28) -1.0 (* -12.0 (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.5e-28) {
		tmp = -1.0;
	} else {
		tmp = -12.0 * (a * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 7.5d-28) then
        tmp = -1.0d0
    else
        tmp = (-12.0d0) * (a * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.5e-28) {
		tmp = -1.0;
	} else {
		tmp = -12.0 * (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 7.5e-28:
		tmp = -1.0
	else:
		tmp = -12.0 * (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 7.5e-28)
		tmp = -1.0;
	else
		tmp = Float64(-12.0 * Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 7.5e-28)
		tmp = -1.0;
	else
		tmp = -12.0 * (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 7.5e-28], -1.0, N[(-12.0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 7.5000000000000003e-28

   1. Initial program 77.6%

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

      1. associate--l+ 77.6%

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

      2. fma-def 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around 0 78.5%

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

      1. associate--l+ 78.5%

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

      2. associate-*r* 78.5%

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

      3. unpow2 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in a around 0 43.2%

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

   if 7.5000000000000003e-28 < (*.f64 b b)

   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 63.9%

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

   4. Taylor expanded in b around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. fma-def 97.4%

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

      3. unpow2 97.4%

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

      4. +-commutative 97.4%

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

      5. +-commutative 97.4%

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

      6. fma-def 97.4%

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

      7. unpow2 97.4%

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

      8. distribute-lft-in 97.4%

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

      9. metadata-eval 97.4%

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

      10. associate-*r* 97.4%

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

      11. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in a around inf 26.8%

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

      1. associate-*r* 26.8%

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

      2. *-commutative 26.8%

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

      3. associate-*r* 26.8%

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

      4. distribute-rgt-out 56.4%

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

      5. *-commutative 56.4%

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

      6. +-commutative 56.4%

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

      7. unpow2 56.4%

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

      8. unpow2 56.4%

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

      9. associate-*r* 56.4%

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

      10. *-commutative 56.4%

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

      11. distribute-rgt-out 56.4%

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

      12. *-commutative 56.4%

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

   9. Simplified 56.4%

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

   10. Taylor expanded in a around 0 28.1%

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

       1. unpow2 28.1%

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

   12. Simplified 28.1%

       \[\leadsto \color{blue}{-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 9: 33.0% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot -12\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.7e-14) -1.0 (* a (* b (* b -12.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.7e-14) {
		tmp = -1.0;
	} else {
		tmp = a * (b * (b * -12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.7d-14) then
        tmp = -1.0d0
    else
        tmp = a * (b * (b * (-12.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.7e-14) {
		tmp = -1.0;
	} else {
		tmp = a * (b * (b * -12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.7e-14:
		tmp = -1.0
	else:
		tmp = a * (b * (b * -12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.7e-14)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(b * Float64(b * -12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.7e-14)
		tmp = -1.0;
	else
		tmp = a * (b * (b * -12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.7e-14], -1.0, N[(a * N[(b * N[(b * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999999e-14

   1. Initial program 73.2%

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

      1. associate--l+ 73.2%

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

      2. fma-def 73.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in b around 0 57.8%

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

      1. associate--l+ 57.8%

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

      2. associate-*r* 57.8%

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

      3. unpow2 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in a around 0 27.6%

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

   if 2.6999999999999999e-14 < b

   1. Initial program 57.0%

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

      1. associate--l+ 57.0%

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

      2. fma-def 57.0%

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

   3. Simplified 57.0%

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

   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 inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. associate-*r* 27.0%

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

      4. distribute-rgt-out 65.8%

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

      5. *-commutative 65.8%

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

      6. +-commutative 65.8%

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

      7. unpow2 65.8%

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

      8. unpow2 65.8%

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

      9. associate-*r* 65.8%

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

      10. *-commutative 65.8%

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

      11. distribute-rgt-out 65.8%

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

      12. *-commutative 65.8%

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

   9. Simplified 65.8%

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

   10. Taylor expanded in a around 0 29.2%

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

       1. unpow2 29.2%

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

   12. Simplified 29.2%

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

   13. Taylor expanded in a around 0 29.2%

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

       1. *-commutative 29.2%

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

       2. unpow2 29.2%

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

       3. associate-*l* 29.2%

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

       4. associate-*r* 29.2%

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

   15. Simplified 29.2%

       \[\leadsto \color{blue}{a \cdot \left(b \cdot \left(b \cdot -12\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot -12\right)\right)\\ \end{array} \]

Alternative 10: 53.2% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{+203}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.06e+203) (+ (* 4.0 (* a a)) -1.0) (* -12.0 (* a (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.06e+203) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = -12.0 * (a * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.06d+203) then
        tmp = (4.0d0 * (a * a)) + (-1.0d0)
    else
        tmp = (-12.0d0) * (a * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.06e+203) {
		tmp = (4.0 * (a * a)) + -1.0;
	} else {
		tmp = -12.0 * (a * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.06e+203:
		tmp = (4.0 * (a * a)) + -1.0
	else:
		tmp = -12.0 * (a * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.06e+203)
		tmp = Float64(Float64(4.0 * Float64(a * a)) + -1.0);
	else
		tmp = Float64(-12.0 * Float64(a * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.06e+203)
		tmp = (4.0 * (a * a)) + -1.0;
	else
		tmp = -12.0 * (a * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.06e+203], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-12.0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05999999999999994e203

   1. Initial program 70.1%

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

      1. associate--l+ 70.1%

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

      2. fma-def 70.1%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in b around 0 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 41.7%

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

      1. sub-neg 41.7%

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

      2. distribute-lft-out 41.7%

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

      3. metadata-eval 41.7%

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

      4. unpow2 41.7%

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

   9. Simplified 41.7%

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

   10. Taylor expanded in a around 0 53.4%

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

       1. unpow2 53.4%

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

   12. Simplified 53.4%

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

   if 1.05999999999999994e203 < b

   1. Initial program 70.0%

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

      1. associate--l+ 70.0%

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

      2. fma-def 70.0%

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

   3. Simplified 70.0%

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

   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 inf 30.0%

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

      1. associate-*r* 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-*r* 30.0%

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

      4. distribute-rgt-out 65.0%

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

      5. *-commutative 65.0%

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

      6. +-commutative 65.0%

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

      7. unpow2 65.0%

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

      8. unpow2 65.0%

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

      9. associate-*r* 65.0%

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

      10. *-commutative 65.0%

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

      11. distribute-rgt-out 65.0%

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

      12. *-commutative 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in a around 0 40.0%

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

       1. unpow2 40.0%

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

   12. Simplified 40.0%

       \[\leadsto \color{blue}{-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.06 \cdot 10^{+203}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) + -1\\ \mathbf{else}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

Alternative 11: 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 70.1%

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

   1. associate--l+ 70.1%

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

   2. fma-def 70.1%

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

3. Simplified 70.9%

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

4. Taylor expanded in b around 0 53.1%

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

   1. associate--l+ 53.1%

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

   2. associate-*r* 53.1%

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

   3. unpow2 53.1%

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

6. Simplified 53.1%

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

7. Taylor expanded in a around 0 22.5%

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

8. Final simplification 22.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023256 
(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))