Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.4% → 99.2%

Time: 6.7s

Alternatives: 7

Speedup: 1.2×

• 61.2% 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: 73.4% 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} \cdot \left(a + 1\right)\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+103)
   (pow a 4.0)
   (+ (pow (fma a a (* b b)) 2.0) (+ (* 4.0 (* (pow a 2.0) (+ a 1.0))) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+103) {
		tmp = pow(a, 4.0);
	} else {
		tmp = pow(fma(a, a, (b * b)), 2.0) + ((4.0 * (pow(a, 2.0) * (a + 1.0))) + -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+103)
		tmp = a ^ 4.0;
	else
		tmp = Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(Float64(4.0 * Float64((a ^ 2.0) * Float64(a + 1.0))) + -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+103], N[Power[a, 4.0], $MachinePrecision], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5e103

   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 -5e103 < a

   1. Initial program 84.6%

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

      1. associate--l+ 84.6%

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

      2. fma-def 84.6%

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

      3. distribute-rgt-in 84.6%

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

      4. sqr-neg 84.6%

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

      5. distribute-rgt-in 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in b around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} \cdot \left(a + 1\right)\right) + -1\right)\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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 3.8%

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

   4. Taylor expanded in a around inf 95.2%

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

4. Final simplification 98.4%

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

Alternative 3: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 6.5 \cdot 10^{-26}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.4) (not (<= a 6.5e-26))) (pow a 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.4) || !(a <= 6.5e-26)) {
		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 <= (-2.4d0)) .or. (.not. (a <= 6.5d-26))) 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 <= -2.4) || !(a <= 6.5e-26)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.4) or not (a <= 6.5e-26):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.4) || !(a <= 6.5e-26))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.4) || ~((a <= 6.5e-26)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.4], N[Not[LessEqual[a, 6.5e-26]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -2.39999999999999991 or 6.5e-26 < a

   1. Initial program 44.1%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in a around inf 86.5%

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

   if -2.39999999999999991 < a < 6.5e-26

   1. Initial program 99.8%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 56.4%

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

   5. Taylor expanded in a around 0 55.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \lor \neg \left(a \leq 6.5 \cdot 10^{-26}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.85e+41) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.85e+41) {
		tmp = pow(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 <= 1.85d+41) then
        tmp = (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 <= 1.85e+41) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.85e+41:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.85e+41)
		tmp = Float64((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 <= 1.85e+41)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.85e+41], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.84999999999999991e41

   1. Initial program 70.4%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in b around 0 63.6%

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

      1. cube-mult 63.6%

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

      2. unpow2 63.6%

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

      3. distribute-rgt1-in 63.5%

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

      4. +-commutative 63.5%

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

      5. unpow2 63.5%

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

      6. associate-*r* 63.5%

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

      7. +-commutative 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in a around inf 81.4%

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

   if 1.84999999999999991e41 < b

   1. Initial program 65.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in b around inf 94.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+41}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+41}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e+41) (* (+ 1.0 (* a 2.0)) (+ (* a 2.0) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+41) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.65d+41) then
        tmp = (1.0d0 + (a * 2.0d0)) * ((a * 2.0d0) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+41) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e+41:
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e+41)
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(Float64(a * 2.0) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e+41)
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e+41], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65e41

   1. Initial program 70.4%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in b around 0 63.6%

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

   5. Taylor expanded in a around 0 63.6%

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

      1. add-sqr-sqrt 63.6%

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

      2. difference-of-sqr-1 63.6%

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

      3. *-commutative 63.6%

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

      4. sqrt-prod 63.6%

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

      5. unpow2 63.6%

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

      6. sqrt-prod 29.3%

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

      7. add-sqr-sqrt 46.8%

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

      8. metadata-eval 46.8%

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

      9. *-commutative 46.8%

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

      10. sqrt-prod 46.8%

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

      11. unpow2 46.8%

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

      12. sqrt-prod 29.3%

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

      13. add-sqr-sqrt 63.6%

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

      14. metadata-eval 63.6%

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

   7. Applied egg-rr 63.6%

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

   if 1.65e41 < b

   1. Initial program 65.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in b around inf 94.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+41}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 6: 51.4% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

3. Simplified 70.7%

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

4. Taylor expanded in b around 0 55.6%

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

5. Taylor expanded in a around 0 55.7%

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

   1. add-sqr-sqrt 55.7%

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

   2. difference-of-sqr-1 55.7%

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

   3. *-commutative 55.7%

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

   4. sqrt-prod 55.7%

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

   5. unpow2 55.7%

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

   6. sqrt-prod 26.3%

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

   7. add-sqr-sqrt 40.2%

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

   8. metadata-eval 40.2%

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

   9. *-commutative 40.2%

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

   10. sqrt-prod 40.2%

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

   11. unpow2 40.2%

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

   12. sqrt-prod 26.3%

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

   13. add-sqr-sqrt 55.7%

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

   14. metadata-eval 55.7%

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

7. Applied egg-rr 55.7%

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

8. Final simplification 55.7%

   \[\leadsto \left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right) \]

Alternative 7: 24.9% 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 69.4%

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

3. Simplified 70.7%

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

4. Taylor expanded in b around 0 55.6%

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

5. Taylor expanded in a around 0 25.3%

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

6. Final simplification 25.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023298 
(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))