math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 11.1s

Precision: binary64

Cost: 13248

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

↓

\[0.5 \cdot \left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

↓

(FPCore (re im) :precision binary64 (* 0.5 (* (* 2.0 (cosh im)) (sin re))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

↓

double code(double re, double im) {
	return 0.5 * ((2.0 * cosh(im)) * sin(re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

↓

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * ((2.0d0 * cosh(im)) * sin(re))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

↓

public static double code(double re, double im) {
	return 0.5 * ((2.0 * Math.cosh(im)) * Math.sin(re));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

↓

def code(re, im):
	return 0.5 * ((2.0 * math.cosh(im)) * math.sin(re))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

↓

function code(re, im)
	return Float64(0.5 * Float64(Float64(2.0 * cosh(im)) * sin(re)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

↓

function tmp = code(re, im)
	tmp = 0.5 * ((2.0 * cosh(im)) * sin(re));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := N[(0.5 * N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   Proof

   [Start] 0.0	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   *-commutative [=>] 0.0	\[ \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   associate-*l* [=>] 0.0	\[ \color{blue}{\sin re \cdot \left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]
   +-commutative [=>] 0.0	\[ \sin re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]
   distribute-lft-in [=>] 0.0	\[ \sin re \cdot \color{blue}{\left(0.5 \cdot e^{im} + 0.5 \cdot e^{0 - im}\right)} \]
   fma-def [=>] 0.0	\[ \sin re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot e^{0 - im}\right)} \]
   exp-diff [=>] 0.0	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, 0.5 \cdot \color{blue}{\frac{e^{0}}{e^{im}}}\right) \]
   associate-*r/ [=>] 0.0	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{0.5 \cdot e^{0}}{e^{im}}}\right) \]
   exp-0 [=>] 0.0	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5 \cdot \color{blue}{1}}{e^{im}}\right) \]
   metadata-eval [=>] 0.0	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Applied egg-rr 0.0

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

4. Taylor expanded in re around inf 0.0

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

5. Simplified 0.0

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

   [Start] 0.0	\[ 0.5 \cdot \left(e^{im} \cdot \sin re\right) + 0.5 \cdot \frac{\sin re}{e^{im}} \]
   *-commutative [=>] 0.0	\[ \color{blue}{\left(e^{im} \cdot \sin re\right) \cdot 0.5} + 0.5 \cdot \frac{\sin re}{e^{im}} \]
   *-commutative [<=] 0.0	\[ \color{blue}{\left(\sin re \cdot e^{im}\right)} \cdot 0.5 + 0.5 \cdot \frac{\sin re}{e^{im}} \]
   *-commutative [=>] 0.0	\[ \left(\sin re \cdot e^{im}\right) \cdot 0.5 + \color{blue}{\frac{\sin re}{e^{im}} \cdot 0.5} \]
   distribute-rgt-in [<=] 0.0	\[ \color{blue}{0.5 \cdot \left(\sin re \cdot e^{im} + \frac{\sin re}{e^{im}}\right)} \]
   *-commutative [=>] 0.0	\[ 0.5 \cdot \left(\color{blue}{e^{im} \cdot \sin re} + \frac{\sin re}{e^{im}}\right) \]

6. Applied egg-rr 10.5

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re \cdot \left(e^{im} + e^{-im}\right)\right)\right)} \]

7. Applied egg-rr 0.0

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

8. Final simplification 0.0

   \[\leadsto 0.5 \cdot \left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	19656

\[\begin{array}{l} \mathbf{if}\;\sin re \leq -0.0005:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;\sin re \leq 10^{-43}:\\ \;\;\;\;\cosh im \cdot re\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 2
Error	0.7
Cost	6976

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

Alternative 3
Error	1.0
Cost	6464

\[\sin re \]

Alternative 4
Error	31.4
Cost	576

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

Alternative 5
Error	31.6
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023016 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))