math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 7.3s

Precision: binary64

Cost: 12992

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

↓

\[\cos re \cdot \cosh im \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

↓

(FPCore (re im) :precision binary64 (* (cos re) (cosh im)))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

↓

double code(double re, double im) {
	return cos(re) * cosh(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cos(re) * cosh(im)
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

↓

public static double code(double re, double im) {
	return Math.cos(re) * Math.cosh(im);
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

↓

def code(re, im):
	return math.cos(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

↓

function code(re, im)
	return Float64(cos(re) * cosh(im))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

↓

function tmp = code(re, im)
	tmp = cos(re) * cosh(im);
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(N[Cos[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\cos re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   Proof
   (*.f64 (cos.f64 re) (fma.f64 1/2 (exp.f64 im) (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 1/2 (exp.f64 im) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 1/2 (exp.f64 im) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (fma.f64 1/2 (exp.f64 im) (*.f64 1/2 (Rewrite<= exp-neg_binary64 (exp.f64 (neg.f64 im)))))): 1 points increase in error, 1 points decrease in error
   (*.f64 (cos.f64 re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (exp.f64 im)) (*.f64 1/2 (exp.f64 (neg.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 (exp.f64 im) (exp.f64 (neg.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 re) (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 re) 1/2) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (cos.f64 re))) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in re around inf 0.0

   \[\leadsto \color{blue}{\cos re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]

4. Applied egg-rr 0.0

   \[\leadsto \color{blue}{0 + 0.5 \cdot \left(\left(2 \cdot \cosh im\right) \cdot \cos re\right)} \]

5. Simplified 0.0

   \[\leadsto \color{blue}{\cos re \cdot \cosh im} \]
   Proof
   (*.f64 (cos.f64 re) (cosh.f64 im)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (cosh.f64 im) (cos.f64 re))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 (cosh.f64 im) (cos.f64 re)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 2)) (*.f64 (cosh.f64 im) (cos.f64 re))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 2 (*.f64 (cosh.f64 im) (cos.f64 re))))): 0 points increase in error, 0 points decrease in error
   (*.f64 1/2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (cosh.f64 im)) (cos.f64 re)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 1/2 (*.f64 (*.f64 2 (cosh.f64 im)) (cos.f64 re))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

   \[\leadsto \cos re \cdot \cosh im \]

Alternatives

Alternative 1
Error	0.9
Cost	6976

\[\cos re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 2
Error	1.2
Cost	6464

\[\cos re \]

Alternative 3
Error	28.9
Cost	448

\[0.5 \cdot \left(im \cdot im\right) + 1 \]

Alternative 4
Error	29.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022328 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))