Average Error: 0.5 → 0.5
Time: 16.2s
Precision: binary64
Cost: 32896
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
↓
\[\mathsf{fma}\left(\left(a1 \cdot a1\right) \cdot \cos th, {2}^{-0.5}, \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\right) \]
(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
  (* (* a1 a1) (cos th))
  (pow 2.0 -0.5)
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))
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(((a1 * a1) * cos(th)), pow(2.0, -0.5), ((cos(th) / sqrt(2.0)) * (a2 * a2)));
}

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(Float64(Float64(a1 * a1) * cos(th)), (2.0 ^ -0.5), Float64(Float64(cos(th) / sqrt(2.0)) * Float64(a2 * a2)))
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[(N[(a1 * a1), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision] + N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $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(\left(a1 \cdot a1\right) \cdot \cos th, {2}^{-0.5}, \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\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

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error14.3
Cost19780
\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.99999999995:\\ \;\;\;\;a2 \cdot \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]
Alternative 2
Error20.1
Cost13644
\[\begin{array}{l} t_1 := a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ t_2 := \sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{if}\;a2 \leq 2.45 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a2 \leq 2.7 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 7.4 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error20.1
Cost13644
\[\begin{array}{l} t_1 := a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{if}\;a2 \leq 10^{-146}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 4.5 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a2 \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error20.1
Cost13644
\[\begin{array}{l} \mathbf{if}\;a2 \leq 10^{-146}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 3.7 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]
Alternative 5
Error20.1
Cost13644
\[\begin{array}{l} \mathbf{if}\;a2 \leq 10^{-146}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;a2 \leq 2.7 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]
Alternative 6
Error20.1
Cost13644
\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.6 \cdot 10^{-147}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 2.3 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\cos th \cdot a2}{\sqrt{2}}}{\frac{1}{a2}}\\ \mathbf{elif}\;a2 \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a1 \cdot a1\right) \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]
Alternative 7
Error14.2
Cost13512
\[\begin{array}{l} \mathbf{if}\;th \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)\\ \mathbf{elif}\;th \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right)\\ \end{array} \]
Alternative 8
Error0.5
Cost13504
\[\cos th \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \]
Alternative 9
Error0.5
Cost13504
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
Alternative 10
Error0.5
Cost13504
\[\frac{\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}} \]
Alternative 11
Error25.6
Cost7104
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}} \]
Alternative 12
Error25.5
Cost6976
\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]
Alternative 13
Error36.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.05 \cdot 10^{-121}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 14
Error36.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.05 \cdot 10^{-121}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
Alternative 15
Error36.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -7.7 \cdot 10^{-122}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]
Alternative 16
Error36.4
Cost6852
\[\begin{array}{l} \mathbf{if}\;a1 \leq -9.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
Alternative 17
Error40.1
Cost6720
\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]
Alternative 18
Error40.1
Cost6720
\[a1 \cdot \left(a1 \cdot \sqrt{0.5}\right) \]

Error

Reproduce

herbie shell --seed 2022328 
(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))))