Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.0

Time: 4.3s

Precision: binary64

Cost: 14016

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double a, double b) {
	return (pow(a, 4.0) + (((4.0 + (2.0 * (a * a))) * (b * b)) + pow(b, 4.0))) + -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 * (b * b))) - 1.0d0
end function

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(a, b)
	tmp = ((a ^ 4.0) + (((4.0 + (2.0 * (a * a))) * (b * b)) + (b ^ 4.0))) + -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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

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

Derivation

1. Initial program 0.2

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

2. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \left(b \cdot b\right), {b}^{4} + {a}^{4}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   Proof
   (fma.f64 2 (*.f64 (*.f64 a a) (*.f64 b b)) (+.f64 (pow.f64 b 4) (pow.f64 a 4))): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 b b)) (+.f64 (pow.f64 b 4) (pow.f64 a 4))): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (+.f64 (pow.f64 b 4) (pow.f64 a 4))): 0 points increase in error, 0 points decrease in error
   (fma.f64 2 (*.f64 (pow.f64 a 2) (pow.f64 b 2)) (Rewrite=> +-commutative_binary64 (+.f64 (pow.f64 a 4) (pow.f64 b 4)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (*.f64 (pow.f64 a 2) (pow.f64 b 2))) (+.f64 (pow.f64 a 4) (pow.f64 b 4)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.0

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

5. Taylor expanded in b around 0 0.0

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

6. Simplified 0.0

   \[\leadsto \color{blue}{\left({a}^{4} + \left(\left(4 + 2 \cdot \left(a \cdot a\right)\right) \cdot \left(b \cdot b\right) + {b}^{4}\right)\right)} - 1 \]
   Proof
   (+.f64 (pow.f64 a 4) (+.f64 (*.f64 (+.f64 4 (*.f64 2 (*.f64 a a))) (*.f64 b b)) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (*.f64 (+.f64 4 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (*.f64 b b)) (pow.f64 b 4))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (pow.f64 b 4))): 1 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 4) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (Rewrite=> unpow2_binary64 (*.f64 b b))))): 0 points increase in error, 1 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (pow.f64 a 2))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 4 (*.f64 b b)) (*.f64 (*.f64 2 (pow.f64 a 2)) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 b b) 4)) (*.f64 (*.f64 2 (pow.f64 a 2)) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 b (*.f64 b 4))) (*.f64 (*.f64 2 (pow.f64 a 2)) (*.f64 b b))))): 3 points increase in error, 1 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (*.f64 b (*.f64 b 4)) (*.f64 (*.f64 2 (Rewrite=> unpow2_binary64 (*.f64 a a))) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (*.f64 b (*.f64 b 4)) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (*.f64 a a) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (*.f64 b (*.f64 b 4)) (*.f64 2 (Rewrite=> unswap-sqr_binary64 (*.f64 (*.f64 a b) (*.f64 a b))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (+.f64 (*.f64 b (*.f64 b 4)) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 a b) 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 b 4) (*.f64 b (*.f64 b 4))) (*.f64 2 (pow.f64 (*.f64 a b) 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (+.f64 (pow.f64 b 4) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 b b) 4))) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 1 points increase in error, 3 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (+.f64 (pow.f64 b 4) (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 b b)))) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 a 4) (+.f64 (+.f64 (pow.f64 b 4) (*.f64 4 (Rewrite<= unpow2_binary64 (pow.f64 b 2)))) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 a 4) (+.f64 (pow.f64 b 4) (*.f64 4 (pow.f64 b 2)))) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (pow.f64 a 4) (pow.f64 b 4)) (*.f64 4 (pow.f64 b 2)))) (*.f64 2 (pow.f64 (*.f64 a b) 2))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 4) (pow.f64 a 4))) (*.f64 4 (pow.f64 b 2))) (*.f64 2 (pow.f64 (*.f64 a b) 2))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (*.f64 4 (Rewrite=> unpow2_binary64 (*.f64 b b)))) (*.f64 2 (pow.f64 (*.f64 a b) 2))): 0 points increase in error, 1 points decrease in error
   (+.f64 (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 b b) 4))) (*.f64 2 (pow.f64 (*.f64 a b) 2))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (Rewrite<= associate-*r*_binary64 (*.f64 b (*.f64 b 4)))) (*.f64 2 (pow.f64 (*.f64 a b) 2))): 3 points increase in error, 1 points decrease in error
   (Rewrite=> associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (*.f64 b (*.f64 b 4)) (*.f64 2 (pow.f64 (*.f64 a b) 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 b b) 4)) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 1 points increase in error, 3 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 b b))) (*.f64 2 (pow.f64 (*.f64 a b) 2)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (*.f64 4 (*.f64 b b)) (*.f64 2 (Rewrite=> unpow2_binary64 (*.f64 (*.f64 a b) (*.f64 a b)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (*.f64 4 (*.f64 b b)) (*.f64 2 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 a a) (*.f64 b b)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (*.f64 4 (*.f64 b b)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 (*.f64 a a)) (*.f64 b b))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (+.f64 (*.f64 4 (*.f64 b b)) (*.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (*.f64 b b)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (Rewrite<= distribute-rgt-in_binary64 (*.f64 (*.f64 b b) (+.f64 4 (*.f64 2 (pow.f64 a 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (*.f64 b b)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (pow.f64 b 4) (pow.f64 a 4)) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (Rewrite<= unpow2_binary64 (pow.f64 b 2)))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (pow.f64 b 4) (+.f64 (pow.f64 a 4) (*.f64 (+.f64 4 (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.2
Cost	7424

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

Alternative 2
Error	1.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00085:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{elif}\;a \leq 0.00021:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \end{array} \]

Alternative 3
Error	1.5
Cost	7304

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq -0.0031:\\ \;\;\;\;{a}^{4} - 1\\ \mathbf{elif}\;a \leq 0.0023:\\ \;\;\;\;\left({b}^{4} + t_0\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + t_0\right) + -1\\ \end{array} \]

Alternative 4
Error	1.5
Cost	7240

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.6
Cost	6920

\[\begin{array}{l} t_0 := {a}^{4} - 1\\ \mathbf{if}\;a \leq -0.0002:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.00072:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.8
Cost	960

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

Alternative 7
Error	11.8
Cost	704

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

Alternative 8
Error	12.7
Cost	576

\[\left(b \cdot b\right) \cdot \left(b \cdot b\right) + -1 \]

Alternative 9
Error	22.3
Cost	448

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

Reproduce

herbie shell --seed 2022338 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))