Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.1% → 99.8%

Time: 7.3s

Alternatives: 12

Speedup: 1.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* b b) (- 1.0 (* a 3.0))) (* (* a a) (+ a 1.0)))))
      1e+305)
   (+
    (fma
     4.0
     (fma a (fma a a a) (* b (* b (fma a -3.0 1.0))))
     (pow (hypot a b) 4.0))
    -1.0)
   (+ (cbrt (pow (pow (hypot a b) 2.0) 6.0)) (+ (* (* b b) 4.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))))) <= 1e+305) {
		tmp = fma(4.0, fma(a, fma(a, a, a), (b * (b * fma(a, -3.0, 1.0)))), pow(hypot(a, b), 4.0)) + -1.0;
	} else {
		tmp = cbrt(pow(pow(hypot(a, b), 2.0), 6.0)) + (((b * b) * 4.0) + -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))) + Float64(Float64(a * a) * Float64(a + 1.0))))) <= 1e+305)
		tmp = Float64(fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), (hypot(a, b) ^ 4.0)) + -1.0);
	else
		tmp = Float64(cbrt(((hypot(a, b) ^ 2.0) ^ 6.0)) + Float64(Float64(Float64(b * b) * 4.0) + -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $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)))))) < 9.9999999999999994e304

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

   3. Simplified 100.0%

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

   if 9.9999999999999994e304 < (+.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 57.7%

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

      1. associate--l+ 57.7%

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

      2. fma-def 57.7%

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

   3. Simplified 60.3%

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

      1. fma-def 60.3%

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

      2. add-cbrt-cube 60.3%

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

      3. pow3 60.3%

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

      4. pow-pow 60.3%

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

      5. add-sqr-sqrt 60.3%

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

      6. hypot-udef 60.3%

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

      7. hypot-udef 60.3%

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

      8. pow2 60.3%

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

      9. metadata-eval 60.3%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), b \cdot \left(b \cdot \mathsf{fma}\left(a, -3, 1\right)\right)\right), {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\\ \mathbf{if}\;t_0 \leq 10^{+305}:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* b b) (- 1.0 (* a 3.0))) (* (* a a) (+ a 1.0)))))))
   (if (<= t_0 1e+305)
     (+ t_0 -1.0)
     (+ (cbrt (pow (pow (hypot a b) 2.0) 6.0)) (+ (* (* b b) 4.0) -1.0)))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = t_0 + -1.0;
	} else {
		tmp = cbrt(pow(pow(hypot(a, b), 2.0), 6.0)) + (((b * b) * 4.0) + -1.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.cbrt(Math.pow(Math.pow(Math.hypot(a, b), 2.0), 6.0)) + (((b * b) * 4.0) + -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))) + Float64(Float64(a * a) * Float64(a + 1.0)))))
	tmp = 0.0
	if (t_0 <= 1e+305)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(cbrt(((hypot(a, b) ^ 2.0) ^ 6.0)) + Float64(Float64(Float64(b * b) * 4.0) + -1.0));
	end
	return 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[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+305], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $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)))))) < 9.9999999999999994e304

   1. Initial program 99.6%

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

   if 9.9999999999999994e304 < (+.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 57.7%

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

      1. associate--l+ 57.7%

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

      2. fma-def 57.7%

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

   3. Simplified 60.3%

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

      1. fma-def 60.3%

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

      2. add-cbrt-cube 60.3%

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

      3. pow3 60.3%

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

      4. pow-pow 60.3%

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

      5. add-sqr-sqrt 60.3%

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

      6. hypot-udef 60.3%

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

      7. hypot-udef 60.3%

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

      8. pow2 60.3%

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

      9. metadata-eval 60.3%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 60.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right) \leq 10^{+305}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(\left(b \cdot b\right) \cdot 4 + -1\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0) + ((2.0 * ((a * b) * (a * b))) + -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(b * b) * Float64(1.0 - Float64(a * 3.0))) + Float64(Float64(a * a) * Float64(a + 1.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64((a ^ 4.0) + Float64(Float64(2.0 * Float64(Float64(a * b) * Float64(a * b))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((b * b) * (1.0 - (a * 3.0))) + ((a * a) * (a + 1.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (a ^ 4.0) + ((2.0 * ((a * b) * (a * b))) + -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[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $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[(2.0 * N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $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 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

   3. Simplified 6.1%

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

      1. fma-def 6.1%

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

      2. add-cbrt-cube 6.1%

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

      3. pow3 6.1%

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

      4. pow-pow 6.1%

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

      5. add-sqr-sqrt 6.1%

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

      6. hypot-udef 6.1%

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

      7. hypot-udef 6.1%

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

      8. pow2 6.1%

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

      9. metadata-eval 6.1%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 6.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 100.0%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in b around 0 84.8%

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

       1. associate--l+ 84.8%

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

       2. *-commutative 84.8%

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

       3. unpow2 84.8%

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

       4. unpow2 84.8%

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

   11. Simplified 84.8%

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

   12. Taylor expanded in a around inf 84.8%

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

       1. unpow2 84.8%

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

       2. unpow2 84.8%

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

       3. unswap-sqr 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 4: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -11000000 \lor \neg \left(a \leq 210\right):\\ \;\;\;\;{a}^{4} + \left(2 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -11000000.0) (not (<= a 210.0)))
   (+ (pow a 4.0) (+ (* 2.0 (* (* a b) (* a b))) -1.0))
   (+ (+ (* (* b b) 4.0) -1.0) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -11000000.0) || !(a <= 210.0)) {
		tmp = pow(a, 4.0) + ((2.0 * ((a * b) * (a * b))) + -1.0);
	} else {
		tmp = (((b * b) * 4.0) + -1.0) + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-11000000.0d0)) .or. (.not. (a <= 210.0d0))) then
        tmp = (a ** 4.0d0) + ((2.0d0 * ((a * b) * (a * b))) + (-1.0d0))
    else
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -11000000.0) || !(a <= 210.0)) {
		tmp = Math.pow(a, 4.0) + ((2.0 * ((a * b) * (a * b))) + -1.0);
	} else {
		tmp = (((b * b) * 4.0) + -1.0) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -11000000.0) or not (a <= 210.0):
		tmp = math.pow(a, 4.0) + ((2.0 * ((a * b) * (a * b))) + -1.0)
	else:
		tmp = (((b * b) * 4.0) + -1.0) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -11000000.0) || !(a <= 210.0))
		tmp = Float64((a ^ 4.0) + Float64(Float64(2.0 * Float64(Float64(a * b) * Float64(a * b))) + -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -11000000.0) || ~((a <= 210.0)))
		tmp = (a ^ 4.0) + ((2.0 * ((a * b) * (a * b))) + -1.0);
	else
		tmp = (((b * b) * 4.0) + -1.0) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -11000000.0], N[Not[LessEqual[a, 210.0]], $MachinePrecision]], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(2.0 * N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e7 or 210 < a

   1. Initial program 51.3%

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

      1. associate--l+ 51.3%

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

      2. fma-def 51.3%

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

   3. Simplified 54.3%

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

      1. fma-def 54.3%

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

      2. add-cbrt-cube 41.4%

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

      3. pow3 41.3%

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

      4. pow-pow 41.3%

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

      5. add-sqr-sqrt 41.3%

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

      6. hypot-udef 41.3%

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

      7. hypot-udef 41.3%

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

      8. pow2 41.3%

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

      9. metadata-eval 41.3%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 41.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 85.8%

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

      1. unpow2 85.8%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 85.8%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in b around 0 80.7%

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

       1. associate--l+ 80.7%

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

       2. *-commutative 80.7%

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

       3. unpow2 80.7%

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

       4. unpow2 80.7%

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

   11. Simplified 80.7%

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

   12. Taylor expanded in a around inf 80.7%

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

       1. unpow2 80.7%

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

       2. unpow2 80.7%

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

       3. unswap-sqr 96.8%

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

   14. Simplified 96.8%

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

   if -1.1e7 < a < 210

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-def 99.9%

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

      2. add-cbrt-cube 90.5%

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

      3. pow3 90.6%

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

      4. pow-pow 90.6%

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

      5. add-sqr-sqrt 90.6%

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

      6. hypot-udef 90.6%

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

      7. hypot-udef 90.6%

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

      8. pow2 90.6%

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

      9. metadata-eval 90.6%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 90.2%

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

      1. unpow2 90.2%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 90.2%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 99.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -11000000 \lor \neg \left(a \leq 210\right):\\ \;\;\;\;{a}^{4} + \left(2 \cdot \left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + {b}^{4}\\ \end{array} \]

Alternative 5: 93.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + \left(\left(a \cdot a\right) \cdot 4 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+65) (+ (pow a 4.0) (+ (* (* a a) 4.0) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = pow(a, 4.0) + (((a * a) * 4.0) + -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+65) then
        tmp = (a ** 4.0d0) + (((a * a) * 4.0d0) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = Math.pow(a, 4.0) + (((a * a) * 4.0) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+65:
		tmp = math.pow(a, 4.0) + (((a * a) * 4.0) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+65)
		tmp = Float64((a ^ 4.0) + Float64(Float64(Float64(a * a) * 4.0) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+65)
		tmp = (a ^ 4.0) + (((a * a) * 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+65], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e65

   1. Initial program 82.1%

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

      1. associate--l+ 82.1%

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

      2. fma-def 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in b around 0 80.1%

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

      1. associate--l+ 80.1%

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

      2. associate-*r* 80.1%

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

      3. unpow2 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in a around 0 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%

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

   if 4.99999999999999973e65 < (*.f64 b b)

   1. Initial program 64.3%

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

      1. associate--l+ 64.3%

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

      2. fma-def 64.3%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in b around inf 96.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + \left(\left(a \cdot a\right) \cdot 4 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 93.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + \left(\left(a \cdot a\right) \cdot 4 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+65)
   (+ (pow a 4.0) (+ (* (* a a) 4.0) -1.0))
   (+ (+ (* (* b b) 4.0) -1.0) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = pow(a, 4.0) + (((a * a) * 4.0) + -1.0);
	} else {
		tmp = (((b * b) * 4.0) + -1.0) + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+65) then
        tmp = (a ** 4.0d0) + (((a * a) * 4.0d0) + (-1.0d0))
    else
        tmp = (((b * b) * 4.0d0) + (-1.0d0)) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+65) {
		tmp = Math.pow(a, 4.0) + (((a * a) * 4.0) + -1.0);
	} else {
		tmp = (((b * b) * 4.0) + -1.0) + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+65:
		tmp = math.pow(a, 4.0) + (((a * a) * 4.0) + -1.0)
	else:
		tmp = (((b * b) * 4.0) + -1.0) + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+65)
		tmp = Float64((a ^ 4.0) + Float64(Float64(Float64(a * a) * 4.0) + -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + -1.0) + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+65)
		tmp = (a ^ 4.0) + (((a * a) * 4.0) + -1.0);
	else
		tmp = (((b * b) * 4.0) + -1.0) + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+65], N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e65

   1. Initial program 82.1%

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

      1. associate--l+ 82.1%

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

      2. fma-def 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in b around 0 80.1%

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

      1. associate--l+ 80.1%

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

      2. associate-*r* 80.1%

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

      3. unpow2 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in a around 0 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%

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

   if 4.99999999999999973e65 < (*.f64 b b)

   1. Initial program 64.3%

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

      1. associate--l+ 64.3%

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

      2. fma-def 64.3%

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

   3. Simplified 67.7%

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

      1. fma-def 67.7%

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

      2. add-cbrt-cube 56.4%

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

      3. pow3 56.4%

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

      4. pow-pow 56.4%

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

      5. add-sqr-sqrt 56.4%

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

      6. hypot-udef 56.4%

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

      7. hypot-udef 56.4%

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

      8. pow2 56.4%

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

      9. metadata-eval 56.4%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 88.6%

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

      1. unpow2 88.6%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 88.6%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 96.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4} + \left(\left(a \cdot a\right) \cdot 4 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + -1\right) + {b}^{4}\\ \end{array} \]

Alternative 7: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+39}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+30}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.1e+39)
   (pow a 4.0)
   (if (<= a 7.6e+30) (+ (* (* b b) (fma b b 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.1e+39) {
		tmp = pow(a, 4.0);
	} else if (a <= 7.6e+30) {
		tmp = ((b * b) * fma(b, b, 4.0)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.1e+39)
		tmp = a ^ 4.0;
	elseif (a <= 7.6e+30)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.1e+39], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 7.6e+30], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.10000000000000004e39 or 7.6000000000000003e30 < a

   1. Initial program 46.9%

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

      1. associate--l+ 46.9%

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

      2. fma-def 46.9%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in a around inf 94.5%

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

   if -4.10000000000000004e39 < a < 7.6000000000000003e30

   1. Initial program 97.7%

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

      1. associate--l+ 97.7%

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

      2. fma-def 97.7%

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

   3. Simplified 97.7%

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

      1. fma-def 97.7%

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

      2. add-cbrt-cube 88.8%

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

      3. pow3 88.8%

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

      4. pow-pow 88.8%

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

      5. add-sqr-sqrt 88.8%

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

      6. hypot-udef 88.8%

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

      7. hypot-udef 88.8%

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

      8. pow2 88.8%

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

      9. metadata-eval 88.8%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 88.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 89.6%

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

      1. unpow2 89.6%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   8. Simplified 89.6%

      \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}} + \left(4 \cdot \color{blue}{\left(b \cdot b\right)} - 1\right) \]

   9. Taylor expanded in a around 0 95.4%

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

       1. associate-+r- 95.4%

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

       2. metadata-eval 95.4%

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

       3. pow-sqr 95.3%

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

       4. pow2 95.3%

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

       5. pow2 95.3%

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

       6. distribute-rgt-out 95.3%

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

       7. fma-def 95.3%

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

   11. Applied egg-rr 95.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+39}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{+30}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-16)
   (+ (* (* a a) 4.0) -1.0)
   (if (<= (* b b) 5e+65) (pow a 4.0) (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-16) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if ((b * b) <= 5e+65) {
		tmp = pow(a, 4.0);
	} 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) <= 5d-16) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else if ((b * b) <= 5d+65) then
        tmp = a ** 4.0d0
    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) <= 5e-16) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else if ((b * b) <= 5e+65) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-16:
		tmp = ((a * a) * 4.0) + -1.0
	elif (b * b) <= 5e+65:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-16)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	elseif (Float64(b * b) <= 5e+65)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-16)
		tmp = ((a * a) * 4.0) + -1.0;
	elseif ((b * b) <= 5e+65)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-16], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+65], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 5.0000000000000004e-16

   1. Initial program 82.4%

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

      1. associate--l+ 82.4%

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

      2. fma-def 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in b around 0 82.5%

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

      1. associate--l+ 82.5%

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

      2. associate-*r* 82.5%

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

      3. unpow2 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in a around 0 64.4%

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

      1. sub-neg 64.4%

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

      2. distribute-lft-out 64.4%

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

      3. unpow2 64.4%

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

      4. metadata-eval 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in a around 0 76.1%

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

       1. unpow2 76.1%

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

   12. Simplified 76.1%

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

   if 5.0000000000000004e-16 < (*.f64 b b) < 4.99999999999999973e65

   1. Initial program 79.4%

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

      1. associate--l+ 79.4%

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

      2. fma-def 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in a around inf 66.8%

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

   if 4.99999999999999973e65 < (*.f64 b b)

   1. Initial program 64.3%

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

      1. associate--l+ 64.3%

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

      2. fma-def 64.3%

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

   3. Simplified 67.7%

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

   4. Taylor expanded in b around inf 96.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9: 68.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 3.6 \cdot 10^{-11}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.4) (not (<= a 3.6e-11)))
   (pow a 4.0)
   (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 3.6e-11)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2.4d0)) .or. (.not. (a <= 3.6d-11))) then
        tmp = a ** 4.0d0
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 3.6e-11)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.4) or not (a <= 3.6e-11):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4) || !(a <= 3.6e-11))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.4) || ~((a <= 3.6e-11)))
		tmp = a ^ 4.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.4], N[Not[LessEqual[a, 3.6e-11]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991 or 3.59999999999999985e-11 < a

   1. Initial program 53.4%

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

      1. associate--l+ 53.4%

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

      2. fma-def 53.4%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in a around inf 82.7%

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

   if -2.39999999999999991 < a < 3.59999999999999985e-11

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 51.2%

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

      1. associate--l+ 51.2%

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

      2. associate-*r* 51.2%

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

      3. unpow2 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in a around 0 51.2%

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

      1. sub-neg 51.2%

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

      2. distribute-lft-out 51.2%

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

      3. unpow2 51.2%

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

      4. metadata-eval 51.2%

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

   9. Simplified 51.2%

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

       1. cube-mult 51.2%

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

       2. distribute-rgt1-in 51.2%

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

   11. Applied egg-rr 51.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 3.6 \cdot 10^{-11}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 10: 55.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4)
   (+ (* (* a a) 4.0) -1.0)
   (+ (* 4.0 (* (* a a) (+ a 1.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.4d0)) then
        tmp = ((a * a) * 4.0d0) + (-1.0d0)
    else
        tmp = (4.0d0 * ((a * a) * (a + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.4) {
		tmp = ((a * a) * 4.0) + -1.0;
	} else {
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.4:
		tmp = ((a * a) * 4.0) + -1.0
	else:
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4)
		tmp = Float64(Float64(Float64(a * a) * 4.0) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(Float64(a * a) * Float64(a + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.4)
		tmp = ((a * a) * 4.0) + -1.0;
	else
		tmp = (4.0 * ((a * a) * (a + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991

   1. Initial program 43.4%

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

      1. associate--l+ 43.4%

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

      2. fma-def 43.4%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in b around 0 23.6%

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

      1. associate--l+ 23.6%

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

      2. associate-*r* 23.6%

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

      3. unpow2 23.6%

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

   6. Simplified 23.6%

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

   7. Taylor expanded in a around 0 0.2%

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

      1. sub-neg 0.2%

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

      2. distribute-lft-out 0.2%

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

      3. unpow2 0.2%

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

      4. metadata-eval 0.2%

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

   9. Simplified 0.2%

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

   10. Taylor expanded in a around 0 45.4%

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

       1. unpow2 45.4%

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

   12. Simplified 45.4%

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

   if -2.39999999999999991 < a

   1. Initial program 83.9%

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

      1. associate--l+ 83.9%

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

      2. fma-def 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in b around 0 65.8%

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

      1. associate--l+ 65.8%

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

      2. associate-*r* 65.8%

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

      3. unpow2 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in a around 0 54.9%

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

      1. sub-neg 54.9%

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

      2. distribute-lft-out 54.9%

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

      3. unpow2 54.9%

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

      4. metadata-eval 54.9%

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

   9. Simplified 54.9%

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

       1. cube-mult 54.9%

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

       2. distribute-rgt1-in 54.9%

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

   11. Applied egg-rr 54.9%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right) + -1\\ \end{array} \]

Alternative 11: 51.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. associate--l+ 74.1%

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

   2. fma-def 74.1%

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

3. Simplified 75.6%

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

4. Taylor expanded in b around 0 55.6%

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

   1. associate--l+ 55.6%

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

   2. associate-*r* 55.6%

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

   3. unpow2 55.6%

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

6. Simplified 55.6%

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

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. unpow2 41.7%

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

   4. metadata-eval 41.7%

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

9. Simplified 41.7%

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

10. Taylor expanded in a around 0 49.8%

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

    1. unpow2 49.8%

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

12. Simplified 49.8%

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

13. Final simplification 49.8%

    \[\leadsto \left(a \cdot a\right) \cdot 4 + -1 \]

Alternative 12: 24.7% accurate, 130.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 74.1%

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

   1. associate--l+ 74.1%

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

   2. fma-def 74.1%

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

3. Simplified 75.6%

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

4. Taylor expanded in b around 0 55.6%

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

   1. associate--l+ 55.6%

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

   2. associate-*r* 55.6%

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

   3. unpow2 55.6%

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

6. Simplified 55.6%

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

7. Taylor expanded in a around 0 23.0%

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

8. Final simplification 23.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023178 
(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))