Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 1.5

Time: 11.6s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} t_0 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\\ t_1 := \left({a}^{4} + t_0\right) - 1\\ \mathbf{if}\;a \leq -0.15:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.00075:\\ \;\;\;\;\left({b}^{4} + t_0\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (+ (* (* a a) (+ a 1.0)) (* b b))))
        (t_1 (- (+ (pow a 4.0) t_0) 1.0)))
   (if (<= a -0.15) t_1 (if (<= a 0.00075) (- (+ (pow b 4.0) t_0) 1.0) t_1))))

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;
}

↓

double code(double a, double b) {
	double t_0 = 4.0 * (((a * a) * (a + 1.0)) + (b * b));
	double t_1 = (pow(a, 4.0) + t_0) - 1.0;
	double tmp;
	if (a <= -0.15) {
		tmp = t_1;
	} else if (a <= 0.00075) {
		tmp = (pow(b, 4.0) + t_0) - 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (((a * a) * (a + 1.0d0)) + (b * b))
    t_1 = ((a ** 4.0d0) + t_0) - 1.0d0
    if (a <= (-0.15d0)) then
        tmp = t_1
    else if (a <= 0.00075d0) then
        tmp = ((b ** 4.0d0) + t_0) - 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
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;
}

↓

public static double code(double a, double b) {
	double t_0 = 4.0 * (((a * a) * (a + 1.0)) + (b * b));
	double t_1 = (Math.pow(a, 4.0) + t_0) - 1.0;
	double tmp;
	if (a <= -0.15) {
		tmp = t_1;
	} else if (a <= 0.00075) {
		tmp = (Math.pow(b, 4.0) + t_0) - 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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

↓

def code(a, b):
	t_0 = 4.0 * (((a * a) * (a + 1.0)) + (b * b))
	t_1 = (math.pow(a, 4.0) + t_0) - 1.0
	tmp = 0
	if a <= -0.15:
		tmp = t_1
	elif a <= 0.00075:
		tmp = (math.pow(b, 4.0) + t_0) - 1.0
	else:
		tmp = t_1
	return tmp

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

↓

function code(a, b)
	t_0 = Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(b * b)))
	t_1 = Float64(Float64((a ^ 4.0) + t_0) - 1.0)
	tmp = 0.0
	if (a <= -0.15)
		tmp = t_1;
	elseif (a <= 0.00075)
		tmp = Float64(Float64((b ^ 4.0) + t_0) - 1.0);
	else
		tmp = t_1;
	end
	return tmp
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

↓

function tmp_2 = code(a, b)
	t_0 = 4.0 * (((a * a) * (a + 1.0)) + (b * b));
	t_1 = ((a ^ 4.0) + t_0) - 1.0;
	tmp = 0.0;
	if (a <= -0.15)
		tmp = t_1;
	elseif (a <= 0.00075)
		tmp = ((b ^ 4.0) + t_0) - 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
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]

↓

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[a, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -0.15], t$95$1, If[LessEqual[a, 0.00075], N[(N[(N[Power[b, 4.0], $MachinePrecision] + t$95$0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.149999999999999994 or 7.5000000000000002e-4 < a

   1. Initial program 0.5

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

   2. Taylor expanded in a around inf 6.9

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

   3. Taylor expanded in a around 0 6.8

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

   4. Applied egg-rr 6.8

      \[\leadsto \left({a}^{4} + \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + 0\right)}\right) - 1 \]

   5. Simplified 6.8

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

      [Start] 6.8	\[ \left({a}^{4} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + 0\right)\right) - 1 \]
      rational.json-simplify-4 [=>] 6.8	\[ \left({a}^{4} + \color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)}\right) - 1 \]

   if -0.149999999999999994 < a < 7.5000000000000002e-4

   1. Initial program 0.1

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

   2. Taylor expanded in a around 0 0.1

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

   3. Taylor expanded in a around 0 0.1

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

   4. Applied egg-rr 0.1

      \[\leadsto \left({b}^{4} + \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + 0\right)}\right) - 1 \]

   5. Simplified 0.1

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

      [Start] 0.1	\[ \left({b}^{4} + \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + 0\right)\right) - 1 \]
      rational.json-simplify-4 [=>] 0.1	\[ \left({b}^{4} + \color{blue}{4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)}\right) - 1 \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.15:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) - 1\\ \mathbf{elif}\;a \leq 0.00075:\\ \;\;\;\;\left({b}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) - 1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	8320

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

Alternative 2
Error	2.7
Cost	7816

\[\begin{array}{l} t_0 := \left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) - 1\\ \mathbf{if}\;a \leq -0.15:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00041:\\ \;\;\;\;{b}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.7
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00172:\\ \;\;\;\;{b}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.1
Cost	6656

\[{a}^{4} - 1 \]

Alternative 5
Error	23.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023075 
(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))