?

Average Accuracy: 99.2% → 99.2%
Time: 17.1s
Precision: binary64
Cost: 19776

?

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]
(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
 (* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0))))
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 cos(th) * (fma(a1, a1, (a2 * a2)) / sqrt(2.0));
}

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 Float64(cos(th) * Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0)))
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[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $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)
↓
\cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}

Error?

Derivation?

  1. Initial program 99.2%

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

  2. Simplified99.2%

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

    [Start]99.2

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

    distribute-lft-out [=>]99.2

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

    associate-*l/ [=>]99.2

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

    associate-*r/ [<=]99.2

    \[ \color{blue}{\cos th \cdot \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]

    fma-def [=>]99.2

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

  3. Final simplification99.2%

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

Alternatives

Alternative 1
Accuracy67.2%
Cost13645
\[\begin{array}{l} \mathbf{if}\;a1 \leq -3 \cdot 10^{-82} \lor \neg \left(a1 \leq -1.2 \cdot 10^{-96}\right) \land a1 \leq -5.2 \cdot 10^{-155}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \end{array} \]
Alternative 2
Accuracy67.1%
Cost13645
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.2 \cdot 10^{-81} \lor \neg \left(a1 \leq -1.3 \cdot 10^{-95}\right) \land a1 \leq -3 \cdot 10^{-155}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 3
Accuracy67.1%
Cost13645
\[\begin{array}{l} t_1 := \cos th \cdot a1\\ \mathbf{if}\;a1 \leq -1.7 \cdot 10^{-81}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot t_1\right)\\ \mathbf{elif}\;a1 \leq -1.45 \cdot 10^{-96} \lor \neg \left(a1 \leq -5.2 \cdot 10^{-155}\right):\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot t_1\right)\\ \end{array} \]
Alternative 4
Accuracy67.2%
Cost13644
\[\begin{array}{l} t_1 := \cos th \cdot a1\\ \mathbf{if}\;a1 \leq -3.2 \cdot 10^{-82}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot t_1\right)\\ \mathbf{elif}\;a1 \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]
Alternative 5
Accuracy67.1%
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -8.6 \cdot 10^{-97}:\\ \;\;\;\;\cos th \cdot \frac{a2}{\frac{\sqrt{2}}{a2}}\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]
Alternative 6
Accuracy67.0%
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -2.15 \cdot 10^{-81}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -1.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot \left(\cos th \cdot a2\right)\right)\\ \end{array} \]
Alternative 7
Accuracy67.0%
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.85 \cdot 10^{-81}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \mathbf{elif}\;a1 \leq -3 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(\cos th \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
Alternative 8
Accuracy67.1%
Cost13644
\[\begin{array}{l} \mathbf{if}\;a1 \leq -7.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \mathbf{elif}\;a1 \leq -5.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
Alternative 9
Accuracy76.8%
Cost13513
\[\begin{array}{l} \mathbf{if}\;th \leq -7.8 \cdot 10^{+15} \lor \neg \left(th \leq 800000\right):\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]
Alternative 10
Accuracy99.2%
Cost13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
Alternative 11
Accuracy59.7%
Cost6976
\[\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}} \]
Alternative 12
Accuracy41.4%
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 13
Accuracy41.5%
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 14
Accuracy41.5%
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
Alternative 15
Accuracy41.5%
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 16
Accuracy41.4%
Cost6852
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.8 \cdot 10^{-189}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 17
Accuracy36.8%
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Error

Reproduce?

herbie shell --seed 2023122 
(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))))