Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 12.4s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

(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
 (/ (+ (* a1 a1) (* a2 a2)) (/ (sqrt 2.0) (cos th))))

C

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 ((a1 * a1) + (a2 * a2)) / (sqrt(2.0) / cos(th));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((cos(th) / sqrt(2.0d0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function

↓

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a1 * a1) + (a2 * a2)) / (sqrt(2.0d0) / cos(th))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.sqrt(2.0)) * (a1 * a1)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}

↓

public static double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) / (Math.sqrt(2.0) / Math.cos(th));
}

Python

def code(a1, a2, th):
	return ((math.cos(th) / math.sqrt(2.0)) * (a1 * a1)) + ((math.cos(th) / math.sqrt(2.0)) * (a2 * a2))

↓

def code(a1, a2, th):
	return ((a1 * a1) + (a2 * a2)) / (math.sqrt(2.0) / math.cos(th))

Julia

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(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / Float64(sqrt(2.0) / cos(th)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2));
end

↓

function tmp = code(a1, a2, th)
	tmp = ((a1 * a1) + (a2 * a2)) / (sqrt(2.0) / cos(th));
end

Wolfram

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[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   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)))): 1 points increase in error, 3 points decrease in error

3. Taylor expanded in th around inf 0.5

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

4. Simplified 0.5

   \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]
   Proof
   (/.f64 (+.f64 (*.f64 a1 a1) (*.f64 a2 a2)) (/.f64 (sqrt.f64 2) (cos.f64 th))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 a2 a2) (*.f64 a1 a1))) (/.f64 (sqrt.f64 2) (cos.f64 th))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 a2 2)) (*.f64 a1 a1)) (/.f64 (sqrt.f64 2) (cos.f64 th))): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (pow.f64 a2 2) (Rewrite<= unpow2_binary64 (pow.f64 a1 2))) (/.f64 (sqrt.f64 2) (cos.f64 th))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (pow.f64 a2 2) (pow.f64 a1 2)) (cos.f64 th)) (sqrt.f64 2))): 31 points increase in error, 17 points decrease in error

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	14.9
Cost	13512

\[\begin{array}{l} t_1 := a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \mathbf{if}\;th \leq -0.23:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(\left(th \cdot th\right) \cdot -0.5 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.9
Cost	13512

\[\begin{array}{l} t_1 := \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\right)\\ \mathbf{if}\;th \leq -0.24:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 3.7 \cdot 10^{+28}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \left(\left(th \cdot th\right) \cdot -0.5 + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13504

\[\sqrt{0.5} \cdot \left(\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \cos th\right) \]

Alternative 4
Error	0.5
Cost	13504

\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \]

Alternative 5
Error	21.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.95 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 \cdot \cos th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 6
Error	21.2
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.95 \cdot 10^{-124}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 7
Error	21.2
Cost	13380

\[\begin{array}{l} t_1 := \frac{\sqrt{2}}{\cos th}\\ \mathbf{if}\;a2 \leq 2.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{a1 \cdot a1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{t_1}\\ \end{array} \]

Alternative 8
Error	25.9
Cost	6976

\[\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5} \]

Alternative 9
Error	25.9
Cost	6976

\[\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]

Alternative 10
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 11
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 1.7 \cdot 10^{-135}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Error	36.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.5 \cdot 10^{-135}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13
Error	40.5
Cost	6720

\[\left(a1 \cdot a1\right) \cdot \sqrt{0.5} \]

Reproduce

herbie shell --seed 2022331 
(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))))