Bouland and Aaronson, Equation (25)

Percentage Accurate: 72.9% → 99.1%

Time: 6.4s

Alternatives: 8

Speedup: 0.5×

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

Specification

Math

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

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: 72.9% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 270:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, -3, 1\right), \mathsf{fma}\left(a, a, {a}^{3}\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left({a}^{4} + {a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (pow a 4.0)
   (if (<= a 270.0)
     (+
      (pow (hypot a b) 4.0)
      (fma 4.0 (fma b (* b (fma a -3.0 1.0)) (fma a a (pow a 3.0))) -1.0))
     (+
      (* 4.0 (pow a 3.0))
      (+ (pow a 4.0) (* (pow a 2.0) (+ 4.0 (* 2.0 (pow b 2.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = pow(a, 4.0);
	} else if (a <= 270.0) {
		tmp = pow(hypot(a, b), 4.0) + fma(4.0, fma(b, (b * fma(a, -3.0, 1.0)), fma(a, a, pow(a, 3.0))), -1.0);
	} else {
		tmp = (4.0 * pow(a, 3.0)) + (pow(a, 4.0) + (pow(a, 2.0) * (4.0 + (2.0 * pow(b, 2.0)))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = a ^ 4.0;
	elseif (a <= 270.0)
		tmp = Float64((hypot(a, b) ^ 4.0) + fma(4.0, fma(b, Float64(b * fma(a, -3.0, 1.0)), fma(a, a, (a ^ 3.0))), -1.0));
	else
		tmp = Float64(Float64(4.0 * (a ^ 3.0)) + Float64((a ^ 4.0) + Float64((a ^ 2.0) * Float64(4.0 + Float64(2.0 * (b ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 270.0], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(a * a + N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(2.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e103

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -1e103 < a < 270

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

   if 270 < a

   1. Initial program 58.4%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in a around inf 98.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 270:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(a, -3, 1\right), \mathsf{fma}\left(a, a, {a}^{3}\right)\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left({a}^{4} + {a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3750:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left({a}^{4} + {a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (pow a 4.0)
   (if (<= a 3750.0)
     (+
      -1.0
      (+
       (pow (+ (* a a) (* b b)) 2.0)
       (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0)))))))
     (+
      (* 4.0 (pow a 3.0))
      (+ (pow a 4.0) (* (pow a 2.0) (+ 4.0 (* 2.0 (pow b 2.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = pow(a, 4.0);
	} else if (a <= 3750.0) {
		tmp = -1.0 + (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0))))));
	} else {
		tmp = (4.0 * pow(a, 3.0)) + (pow(a, 4.0) + (pow(a, 2.0) * (4.0 + (2.0 * pow(b, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1d+103)) then
        tmp = a ** 4.0d0
    else if (a <= 3750.0d0) then
        tmp = (-1.0d0) + ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (a + 1.0d0)) + ((b * b) * (1.0d0 - (a * 3.0d0))))))
    else
        tmp = (4.0d0 * (a ** 3.0d0)) + ((a ** 4.0d0) + ((a ** 2.0d0) * (4.0d0 + (2.0d0 * (b ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 3750.0) {
		tmp = -1.0 + (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0))))));
	} else {
		tmp = (4.0 * Math.pow(a, 3.0)) + (Math.pow(a, 4.0) + (Math.pow(a, 2.0) * (4.0 + (2.0 * Math.pow(b, 2.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+103:
		tmp = math.pow(a, 4.0)
	elif a <= 3750.0:
		tmp = -1.0 + (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0))))))
	else:
		tmp = (4.0 * math.pow(a, 3.0)) + (math.pow(a, 4.0) + (math.pow(a, 2.0) * (4.0 + (2.0 * math.pow(b, 2.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = a ^ 4.0;
	elseif (a <= 3750.0)
		tmp = Float64(-1.0 + Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0)))))));
	else
		tmp = Float64(Float64(4.0 * (a ^ 3.0)) + Float64((a ^ 4.0) + Float64((a ^ 2.0) * Float64(4.0 + Float64(2.0 * (b ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+103)
		tmp = a ^ 4.0;
	elseif (a <= 3750.0)
		tmp = -1.0 + ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0))))));
	else
		tmp = (4.0 * (a ^ 3.0)) + ((a ^ 4.0) + ((a ^ 2.0) * (4.0 + (2.0 * (b ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3750.0], N[(-1.0 + N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(2.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e103

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -1e103 < a < 3750

   1. Initial program 99.9%

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

   if 3750 < a

   1. Initial program 58.4%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in a around inf 98.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3750:\\ \;\;\;\;-1 + \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {a}^{3} + \left({a}^{4} + {a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right)\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in a around inf 90.2%

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

4. Final simplification 97.3%

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

Alternative 4: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{if}\;b \leq 1.15 \cdot 10^{-212}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-130}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.00033:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 185000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (pow a 3.0) (+ a 4.0))))
   (if (<= b 1.15e-212)
     -1.0
     (if (<= b 1.9e-147)
       t_0
       (if (<= b 6.3e-130)
         -1.0
         (if (<= b 3.1e-34)
           t_0
           (if (<= b 0.00033)
             -1.0
             (if (<= b 185000000.0) t_0 (pow b 4.0)))))))))

C

double code(double a, double b) {
	double t_0 = pow(a, 3.0) * (a + 4.0);
	double tmp;
	if (b <= 1.15e-212) {
		tmp = -1.0;
	} else if (b <= 1.9e-147) {
		tmp = t_0;
	} else if (b <= 6.3e-130) {
		tmp = -1.0;
	} else if (b <= 3.1e-34) {
		tmp = t_0;
	} else if (b <= 0.00033) {
		tmp = -1.0;
	} else if (b <= 185000000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (a ** 3.0d0) * (a + 4.0d0)
    if (b <= 1.15d-212) then
        tmp = -1.0d0
    else if (b <= 1.9d-147) then
        tmp = t_0
    else if (b <= 6.3d-130) then
        tmp = -1.0d0
    else if (b <= 3.1d-34) then
        tmp = t_0
    else if (b <= 0.00033d0) then
        tmp = -1.0d0
    else if (b <= 185000000.0d0) then
        tmp = t_0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(a, 3.0) * (a + 4.0);
	double tmp;
	if (b <= 1.15e-212) {
		tmp = -1.0;
	} else if (b <= 1.9e-147) {
		tmp = t_0;
	} else if (b <= 6.3e-130) {
		tmp = -1.0;
	} else if (b <= 3.1e-34) {
		tmp = t_0;
	} else if (b <= 0.00033) {
		tmp = -1.0;
	} else if (b <= 185000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(a, 3.0) * (a + 4.0)
	tmp = 0
	if b <= 1.15e-212:
		tmp = -1.0
	elif b <= 1.9e-147:
		tmp = t_0
	elif b <= 6.3e-130:
		tmp = -1.0
	elif b <= 3.1e-34:
		tmp = t_0
	elif b <= 0.00033:
		tmp = -1.0
	elif b <= 185000000.0:
		tmp = t_0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64((a ^ 3.0) * Float64(a + 4.0))
	tmp = 0.0
	if (b <= 1.15e-212)
		tmp = -1.0;
	elseif (b <= 1.9e-147)
		tmp = t_0;
	elseif (b <= 6.3e-130)
		tmp = -1.0;
	elseif (b <= 3.1e-34)
		tmp = t_0;
	elseif (b <= 0.00033)
		tmp = -1.0;
	elseif (b <= 185000000.0)
		tmp = t_0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a ^ 3.0) * (a + 4.0);
	tmp = 0.0;
	if (b <= 1.15e-212)
		tmp = -1.0;
	elseif (b <= 1.9e-147)
		tmp = t_0;
	elseif (b <= 6.3e-130)
		tmp = -1.0;
	elseif (b <= 3.1e-34)
		tmp = t_0;
	elseif (b <= 0.00033)
		tmp = -1.0;
	elseif (b <= 185000000.0)
		tmp = t_0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.15e-212], -1.0, If[LessEqual[b, 1.9e-147], t$95$0, If[LessEqual[b, 6.3e-130], -1.0, If[LessEqual[b, 3.1e-34], t$95$0, If[LessEqual[b, 0.00033], -1.0, If[LessEqual[b, 185000000.0], t$95$0, N[Power[b, 4.0], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.15e-212 or 1.90000000000000014e-147 < b < 6.2999999999999997e-130 or 3.0999999999999998e-34 < b < 3.3e-4

   1. Initial program 78.5%

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

      1. associate--l+ 78.5%

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

      2. fma-def 78.5%

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

      3. distribute-rgt-in 78.5%

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

      4. sqr-neg 78.5%

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

      5. distribute-rgt-in 78.5%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in b around 0 58.6%

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

   5. Taylor expanded in a around 0 51.5%

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

   6. Taylor expanded in a around 0 30.1%

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

   if 1.15e-212 < b < 1.90000000000000014e-147 or 6.2999999999999997e-130 < b < 3.0999999999999998e-34 or 3.3e-4 < b < 1.85e8

   1. Initial program 74.8%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in a around inf 37.3%

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

   5. Taylor expanded in b around inf 37.3%

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

      1. associate-*r* 37.3%

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

      2. *-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in a around inf 37.2%

      \[\leadsto \color{blue}{4 \cdot {a}^{3} + {a}^{4}} \]
   9. Step-by-step derivation

      1. metadata-eval 37.2%

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

      2. pow-sqr 37.1%

         \[\leadsto 4 \cdot {a}^{3} + \color{blue}{{a}^{2} \cdot {a}^{2}} \]

      3. unpow2 37.1%

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

      4. associate-*l* 37.1%

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

      5. unpow2 37.1%

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

      6. cube-mult 37.2%

         \[\leadsto 4 \cdot {a}^{3} + a \cdot \color{blue}{{a}^{3}} \]

      7. distribute-rgt-out 62.2%

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

   10. Simplified 62.2%

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

   if 1.85e8 < b

   1. Initial program 58.8%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around inf 88.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-212}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-147}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-130}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{elif}\;b \leq 0.00033:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 185000000:\\ \;\;\;\;{a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 47.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 0.00033:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1350000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e-134)
   -1.0
   (if (<= b 5.5e-35)
     (pow a 4.0)
     (if (<= b 0.00033)
       -1.0
       (if (<= b 1350000000.0) (pow a 4.0) (pow b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e-134) {
		tmp = -1.0;
	} else if (b <= 5.5e-35) {
		tmp = pow(a, 4.0);
	} else if (b <= 0.00033) {
		tmp = -1.0;
	} else if (b <= 1350000000.0) {
		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 <= 6.5d-134) then
        tmp = -1.0d0
    else if (b <= 5.5d-35) then
        tmp = a ** 4.0d0
    else if (b <= 0.00033d0) then
        tmp = -1.0d0
    else if (b <= 1350000000.0d0) 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 <= 6.5e-134) {
		tmp = -1.0;
	} else if (b <= 5.5e-35) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 0.00033) {
		tmp = -1.0;
	} else if (b <= 1350000000.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5e-134:
		tmp = -1.0
	elif b <= 5.5e-35:
		tmp = math.pow(a, 4.0)
	elif b <= 0.00033:
		tmp = -1.0
	elif b <= 1350000000.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5e-134)
		tmp = -1.0;
	elseif (b <= 5.5e-35)
		tmp = a ^ 4.0;
	elseif (b <= 0.00033)
		tmp = -1.0;
	elseif (b <= 1350000000.0)
		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 <= 6.5e-134)
		tmp = -1.0;
	elseif (b <= 5.5e-35)
		tmp = a ^ 4.0;
	elseif (b <= 0.00033)
		tmp = -1.0;
	elseif (b <= 1350000000.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5e-134], -1.0, If[LessEqual[b, 5.5e-35], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 0.00033], -1.0, If[LessEqual[b, 1350000000.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.4999999999999998e-134 or 5.4999999999999997e-35 < b < 3.3e-4

   1. Initial program 78.2%

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

      1. associate--l+ 78.1%

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

      2. fma-def 78.1%

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

      3. distribute-rgt-in 78.1%

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

      4. sqr-neg 78.1%

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

      5. distribute-rgt-in 78.1%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in b around 0 59.5%

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

   5. Taylor expanded in a around 0 53.5%

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

   6. Taylor expanded in a around 0 30.5%

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

   if 6.4999999999999998e-134 < b < 5.4999999999999997e-35 or 3.3e-4 < b < 1.35e9

   1. Initial program 75.8%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in a around inf 60.0%

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

   if 1.35e9 < b

   1. Initial program 58.8%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around inf 88.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 0.00033:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 1350000000:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 66.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 66000000:\\ \;\;\;\;-1 + 4 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 66000000.0) (+ -1.0 (* 4.0 (pow a 2.0))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 66000000.0) {
		tmp = -1.0 + (4.0 * pow(a, 2.0));
	} 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 <= 66000000.0d0) then
        tmp = (-1.0d0) + (4.0d0 * (a ** 2.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 66000000.0) {
		tmp = -1.0 + (4.0 * Math.pow(a, 2.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 66000000.0:
		tmp = -1.0 + (4.0 * math.pow(a, 2.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 66000000.0)
		tmp = Float64(-1.0 + Float64(4.0 * (a ^ 2.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 66000000.0)
		tmp = -1.0 + (4.0 * (a ^ 2.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 66000000.0], N[(-1.0 + N[(4.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6e7

   1. Initial program 77.9%

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

      1. associate--l+ 77.9%

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

      2. fma-def 77.9%

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

      3. distribute-rgt-in 77.9%

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

      4. sqr-neg 77.9%

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

      5. distribute-rgt-in 77.9%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in b around 0 60.6%

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

   5. Taylor expanded in a around 0 55.5%

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

   if 6.6e7 < b

   1. Initial program 58.8%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in b around inf 88.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 66000000:\\ \;\;\;\;-1 + 4 \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 69.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.3) (not (<= a 0.41))) (pow a 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.3) || !(a <= 0.41)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -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 <= (-2.3d0)) .or. (.not. (a <= 0.41d0))) then
        tmp = a ** 4.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.3) || !(a <= 0.41)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.3) or not (a <= 0.41):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.3) || !(a <= 0.41))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.3) || ~((a <= 0.41)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.3], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -2.2999999999999998 or 0.409999999999999976 < a

   1. Initial program 47.1%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in a around inf 87.0%

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

   if -2.2999999999999998 < a < 0.409999999999999976

   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. distribute-rgt-in 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-in 99.9%

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

   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 47.3%

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

   5. Taylor expanded in a around 0 47.3%

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

   6. Taylor expanded in a around 0 47.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 24.5% 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 73.7%

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

   1. associate--l+ 73.7%

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

   2. fma-def 73.7%

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

   3. distribute-rgt-in 73.7%

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

   4. sqr-neg 73.7%

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

   5. distribute-rgt-in 73.7%

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

3. Simplified 74.9%

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

4. Taylor expanded in b around 0 52.6%

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

5. Taylor expanded in a around 0 50.8%

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

6. Taylor expanded in a around 0 24.2%

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

7. Final simplification 24.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023300 
(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))