Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.2

Time: 19.8s

Precision: binary64

Cost: 7684

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left({a}^{4} + \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-23)
   (- (+ (pow a 4.0) (* (- 1.0 a) (* 4.0 (* a a)))) 1.0)
   (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 12.0 (* b b))) 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) * (3.0 + a))))) - 1.0;
}

↓

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-23) {
		tmp = (pow(a, 4.0) + ((1.0 - a) * (4.0 * (a * a)))) - 1.0;
	} else {
		tmp = (pow(((a * a) + (b * b)), 2.0) + (12.0 * (b * b))) - 1.0;
	}
	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) * (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) :: tmp
    if ((b * b) <= 2d-23) then
        tmp = ((a ** 4.0d0) + ((1.0d0 - a) * (4.0d0 * (a * a)))) - 1.0d0
    else
        tmp = ((((a * a) + (b * b)) ** 2.0d0) + (12.0d0 * (b * b))) - 1.0d0
    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) * (3.0 + a))))) - 1.0;
}

↓

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-23) {
		tmp = (Math.pow(a, 4.0) + ((1.0 - a) * (4.0 * (a * a)))) - 1.0;
	} else {
		tmp = (Math.pow(((a * a) + (b * b)), 2.0) + (12.0 * (b * b))) - 1.0;
	}
	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) * (3.0 + a))))) - 1.0

↓

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-23:
		tmp = (math.pow(a, 4.0) + ((1.0 - a) * (4.0 * (a * a)))) - 1.0
	else:
		tmp = (math.pow(((a * a) + (b * b)), 2.0) + (12.0 * (b * b))) - 1.0
	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(3.0 + a))))) - 1.0)
end

↓

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-23)
		tmp = Float64(Float64((a ^ 4.0) + Float64(Float64(1.0 - a) * Float64(4.0 * Float64(a * a)))) - 1.0);
	else
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(12.0 * Float64(b * b))) - 1.0);
	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) * (3.0 + a))))) - 1.0;
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-23)
		tmp = ((a ^ 4.0) + ((1.0 - a) * (4.0 * (a * a)))) - 1.0;
	else
		tmp = ((((a * a) + (b * b)) ^ 2.0) + (12.0 * (b * b))) - 1.0;
	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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-23], N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(1.0 - a), $MachinePrecision] * N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(12.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999992e-23

   1. Initial program 0.1

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

   2. Taylor expanded in b around 0 0.1

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

   3. Simplified 0.1

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

   4. Taylor expanded in b around 0 0.1

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

   5. Simplified 0.0

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

   if 1.99999999999999992e-23 < (*.f64 b b)

   1. Initial program 0.5

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

   2. Taylor expanded in a around 0 0.8

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

   3. Simplified 0.8

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + \color{blue}{12 \cdot \left(b \cdot b\right)}\right) - 1 \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.2
Cost	21568

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \left(\mathsf{fma}\left(t_0, b \cdot b, \left(t_0 \cdot a\right) \cdot a\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

Alternative 2
Error	0.2
Cost	8192

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

Alternative 3
Error	1.2
Cost	7680

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

Alternative 4
Error	1.6
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\left({a}^{4} + \left(1 - a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} - \left(1 + \left(-12 \cdot b\right) \cdot b\right)\\ \end{array} \]

Alternative 5
Error	2.8
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-21}:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + \left(12 \cdot b\right) \cdot b\right) - 1\\ \end{array} \]

Alternative 6
Error	2.8
Cost	7300

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-21}:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} - \left(1 + \left(-12 \cdot b\right) \cdot b\right)\\ \end{array} \]

Alternative 7
Error	2.9
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-21}:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) - 1\\ \end{array} \]

Alternative 8
Error	11.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-21}:\\ \;\;\;\;4 \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) - 1\\ \end{array} \]

Alternative 9
Error	22.9
Cost	448

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

Alternative 10
Error	23.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))