Average Error: 0.5 → 0.6
Time: 10.8s
Precision: binary64
Cost: 32896
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\mathsf{fma}\left(\cos th, a2 \cdot \left(a2 \cdot {2}^{-0.5}\right), \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\right) \]
(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))
↓
(FPCore (a1 a2 th)
 :precision binary64
 (fma
  (cos th)
  (* a2 (* a2 (pow 2.0 -0.5)))
  (/ (cos th) (/ (sqrt 2.0) (* a1 a1)))))
double code(double a1, double a2, double th) {
	return ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
}
↓
double code(double a1, double a2, double th) {
	return fma(cos(th), (a2 * (a2 * pow(2.0, -0.5))), (cos(th) / (sqrt(2.0) / (a1 * a1))));
}

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a1 * a1)) + Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)))
end
↓
function code(a1, a2, th)
	return fma(cos(th), Float64(a2 * Float64(a2 * (2.0 ^ -0.5))), Float64(cos(th) / Float64(sqrt(2.0) / Float64(a1 * a1))))
end

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\mathsf{fma}\left(\cos th, a2 \cdot \left(a2 \cdot {2}^{-0.5}\right), \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\right)

Error

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
    Proof
    (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (+.f64 (*.f64 a1 a1) (*.f64 a2 a2))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a2 a2)))): 0 points increase in error, 2 points decrease in error

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos th, {2}^{-0.5} \cdot \left(a2 \cdot a2\right), \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\right)} \]

  4. Simplified0.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos th, a2 \cdot \left(a2 \cdot {2}^{-0.5}\right), \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\right)} \]
    Proof
    (fma.f64 (cos.f64 th) (*.f64 a2 (*.f64 a2 (pow.f64 2 -1/2))) (/.f64 (cos.f64 th) (/.f64 (sqrt.f64 2) (*.f64 a1 a1)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (cos.f64 th) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a2 a2) (pow.f64 2 -1/2))) (/.f64 (cos.f64 th) (/.f64 (sqrt.f64 2) (*.f64 a1 a1)))): 21 points increase in error, 11 points decrease in error
    (fma.f64 (cos.f64 th) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 2 -1/2) (*.f64 a2 a2))) (/.f64 (cos.f64 th) (/.f64 (sqrt.f64 2) (*.f64 a1 a1)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (cos.f64 th) (*.f64 (pow.f64 2 -1/2) (*.f64 a2 a2)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (cos.f64 th) (*.f64 a1 a1)) (sqrt.f64 2)))): 15 points increase in error, 18 points decrease in error
    (fma.f64 (cos.f64 th) (*.f64 (pow.f64 2 -1/2) (*.f64 a2 a2)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 2)) (*.f64 a1 a1)))): 18 points increase in error, 8 points decrease in error

  5. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\cos th, a2 \cdot \left(a2 \cdot {2}^{-0.5}\right), \frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\right) \]

Alternatives

Alternative 1
Error15.3
Cost19780
\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.945:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 2
Error0.5
Cost13504
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \]
Alternative 3
Error0.5
Cost13504
\[\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}} \]
Alternative 4
Error21.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.18 \cdot 10^{-154}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]
Alternative 5
Error21.7
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.18 \cdot 10^{-154}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]
Alternative 6
Error21.7
Cost13380
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.18 \cdot 10^{-154}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot \sqrt{0.5}\right) \cdot \left(\cos th \cdot a2\right)\\ \end{array} \]
Alternative 7
Error26.4
Cost6976
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]
Alternative 8
Error36.9
Cost6916
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \]
Alternative 9
Error36.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.5 \cdot 10^{-83}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 10
Error36.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 11
Error36.9
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.2 \cdot 10^{-83}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 12
Error41.1
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Error

Reproduce

herbie shell --seed 2022335 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))