Average Error: 0.2 → 0.0
Time: 7.5s
Precision: binary64
Cost: 14400
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right)\right) + -1 \]
(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
 (+
  (+
   (pow (hypot b a) 4.0)
   (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (+ 1.0 (* a -3.0))))))
  -1.0))
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) {
	return (pow(hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (a * -3.0)))))) + -1.0;
}

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) {
	return (Math.pow(Math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (a * -3.0)))))) + -1.0;
}

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):
	return (math.pow(math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (a * -3.0)))))) + -1.0

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)
	return Float64(Float64((hypot(b, a) ^ 4.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 + Float64(a * -3.0)))))) + -1.0)
end

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 = code(a, b)
	tmp = ((hypot(b, a) ^ 4.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 + (a * -3.0)))))) + -1.0;
end

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_] := N[(N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $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] + -1.0), $MachinePrecision]

\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
↓
\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{4} + 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) + -1

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Applied egg-rr2.1

    \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{2}\right)} - 1\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

  3. Simplified0.0

    \[\leadsto \left(\color{blue}{{\left(\mathsf{hypot}\left(b, a\right)\right)}^{4}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
    Proof
    (pow.f64 (hypot.f64 b a) 4): 0 points increase in error, 0 points decrease in error
    (pow.f64 (hypot.f64 b a) (Rewrite<= metadata-eval (*.f64 2 2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (hypot.f64 b a) 2) (pow.f64 (hypot.f64 b a) 2))): 55 points increase in error, 25 points decrease in error
    (Rewrite<= unpow2_binary64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2)))): 69 points increase in error, 7 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (pow.f64 (pow.f64 (hypot.f64 b a) 2) 2))) 1)): 6 points increase in error, 80 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.2
Cost8320
\[\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + -1 \]
Alternative 2
Error0.2
Cost8196
\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\\ \end{array} \]
Alternative 3
Error0.3
Cost7936
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) + -1 \]
Alternative 4
Error0.3
Cost7812
\[\begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;\left(t_0 + \left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right)\right) + -1\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + b \cdot \left(b \cdot 4\right)\right) + -1\\ \end{array} \]
Alternative 5
Error0.3
Cost7688
\[\begin{array}{l} t_0 := \left({\left(a \cdot a + b \cdot b\right)}^{2} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \mathbf{if}\;b \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error2.5
Cost7560
\[\begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right) + {b}^{4}\right) + -1\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-5}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error1.6
Cost7560
\[\begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot \left(4 + a \cdot -12\right) + {b}^{4}\right) + -1\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error2.6
Cost7240
\[\begin{array}{l} t_0 := -1 + b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 0.00019:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error3.7
Cost7176
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 120000000000:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
Alternative 10
Error4.1
Cost6920
\[\begin{array}{l} \mathbf{if}\;b \leq -0.65:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;b \leq 120000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
Alternative 11
Error12.4
Cost6792
\[\begin{array}{l} \mathbf{if}\;a \leq -2.4:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 0.415:\\ \;\;\;\;-1 + 4 \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
Alternative 12
Error23.2
Cost448
\[-1 + 4 \cdot \left(b \cdot b\right) \]
Alternative 13
Error24.2
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022328 
(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))