Average Error: 0.2 → 0.0
Time: 13.4s
Precision: binary64
Cost: 27584
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \left(a \cdot a\right)\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + \left(1 + a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\right) + -1 \]
(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
 (+
  (+
   (fma (fma 4.0 (+ 1.0 (* a -3.0)) (* 2.0 (* a a))) (* b b) (pow b 4.0))
   (+ (pow a 4.0) (* (+ 1.0 a) (* 4.0 (* a a)))))
  -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 (fma(fma(4.0, (1.0 + (a * -3.0)), (2.0 * (a * a))), (b * b), pow(b, 4.0)) + (pow(a, 4.0) + ((1.0 + a) * (4.0 * (a * a))))) + -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(fma(fma(4.0, Float64(1.0 + Float64(a * -3.0)), Float64(2.0 * Float64(a * a))), Float64(b * b), (b ^ 4.0)) + Float64((a ^ 4.0) + Float64(Float64(1.0 + a) * Float64(4.0 * Float64(a * a))))) + -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[(N[(4.0 * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(N[(1.0 + a), $MachinePrecision] * N[(4.0 * N[(a * a), $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(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \left(a \cdot a\right)\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + \left(1 + a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right)\right) + -1

Error

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. Taylor expanded in b around 0 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(4, 1 + a \cdot -3, 2 \cdot \left(a \cdot a\right)\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + \left(4 \cdot \left(a \cdot a\right)\right) \cdot \left(1 + a\right)\right)\right)} - 1 \]
    Proof
    (-.f64 (+.f64 (fma.f64 (fma.f64 4 (+.f64 1 (*.f64 a -3)) (*.f64 2 (*.f64 a a))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (fma.f64 (fma.f64 4 (+.f64 1 (*.f64 a -3)) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 a 2)))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 4 (+.f64 1 (*.f64 a -3))) (*.f64 2 (pow.f64 a 2)))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (fma.f64 (+.f64 (*.f64 4 (+.f64 1 (Rewrite=> *-commutative_binary64 (*.f64 -3 a)))) (*.f64 2 (pow.f64 a 2))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (fma.f64 (+.f64 (*.f64 4 (+.f64 1 (*.f64 (Rewrite<= metadata-eval (neg.f64 3)) a))) (*.f64 2 (pow.f64 a 2))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (fma.f64 (+.f64 (*.f64 4 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 3 a)))) (*.f64 2 (pow.f64 a 2))) (*.f64 b b) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 (*.f64 4 (-.f64 1 (*.f64 3 a))) (*.f64 2 (pow.f64 a 2))) (*.f64 b b)) (pow.f64 b 4))) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 4 (-.f64 1 (*.f64 3 a))) (*.f64 2 (pow.f64 a 2))) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (*.f64 a a)) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 4 (-.f64 1 (*.f64 3 a))) (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2)) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (*.f64 (*.f64 4 (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (+.f64 1 a)))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 4 (-.f64 1 (*.f64 3 a))) (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2)) (pow.f64 b 4)) (+.f64 (pow.f64 a 4) (Rewrite<= associate-*r*_binary64 (*.f64 4 (*.f64 (pow.f64 a 2) (+.f64 1 a)))))) 1): 0 points increase in error, 0 points decrease in error
    (-.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (+.f64 (*.f64 4 (-.f64 1 (*.f64 3 a))) (*.f64 2 (pow.f64 a 2))) (pow.f64 b 2)) (+.f64 (pow.f64 b 4) (+.f64 (pow.f64 a 4) (*.f64 4 (*.f64 (pow.f64 a 2) (+.f64 1 a))))))) 1): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.1
Cost14720
\[\left(\left({a}^{4} + \left(1 + a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4 + a \cdot \left(a \cdot 2 + -12\right)\right)\right) + -1 \]
Alternative 2
Error0.2
Cost8704
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\frac{1 - a \cdot a}{\frac{1 - a}{a \cdot a}} + \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right)\right) + -1 \]
Alternative 3
Error0.2
Cost8320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right) + \left(a \cdot a\right) \cdot \left(1 + a\right)\right)\right) + -1 \]
Alternative 4
Error0.2
Cost8320
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b + \frac{1 - a \cdot a}{\frac{1 - a}{a \cdot a}}\right)\right) + -1 \]
Alternative 5
Error0.2
Cost7936
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot \left(1 + a\right)\right)\right) + -1 \]
Alternative 6
Error0.2
Cost7689
\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-7} \lor \neg \left(b \leq 1.3 \cdot 10^{-10}\right):\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left(1 + a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \end{array} \]
Alternative 7
Error1.9
Cost7560
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 16.5:\\ \;\;\;\;\left({a}^{3} \cdot \left(4 + a\right) + a \cdot \left(4 \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
Alternative 8
Error1.9
Cost7560
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 37:\\ \;\;\;\;\left({a}^{4} + \left(1 + a\right) \cdot \left(4 \cdot \left(a \cdot a\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
Alternative 9
Error2.0
Cost6792
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{elif}\;b \leq 840:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right) + 4 \cdot \left(a \cdot a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
Alternative 10
Error1.8
Cost1481
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \lor \neg \left(b \leq 12.5\right):\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(4 + b \cdot b\right)\right) + b \cdot \left(b \cdot \left(a \cdot -12\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right) + 4 \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]
Alternative 11
Error1.8
Cost1353
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \lor \neg \left(b \leq 12.5\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(a \cdot \left(4 + a\right)\right)\right) + 4 \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]
Alternative 12
Error1.9
Cost1225
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \lor \neg \left(b \leq 51\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + \frac{a}{\frac{1}{4 + a}}\right) + -1\\ \end{array} \]
Alternative 13
Error1.8
Cost1097
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \lor \neg \left(b \leq 12.5\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(4 + a\right)\right) + -1\\ \end{array} \]
Alternative 14
Error2.7
Cost969
\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \lor \neg \left(b \leq 12.5\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(4 + a\right)\right) + -1\\ \end{array} \]
Alternative 15
Error13.1
Cost704
\[\left(a \cdot a\right) \cdot \left(a \cdot \left(4 + a\right)\right) + -1 \]
Alternative 16
Error13.5
Cost576
\[\left(a \cdot a\right) \cdot \left(a \cdot a\right) + -1 \]
Alternative 17
Error22.8
Cost448
\[4 \cdot \left(a \cdot a\right) + -1 \]
Alternative 18
Error23.7
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022342 
(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))