math.sin on complex, real part

Average Accuracy: 99.9% → 99.9%

Time: 9.6s

Precision: binary64

Cost: 12992

Math

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

↓

\[\sin re \cdot \cosh im \]

FPCore

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

↓

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

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 sin(re) * cosh(im);
}

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 = sin(re) * cosh(im)
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 Math.sin(re) * Math.cosh(im);
}

Python

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

↓

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

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(sin(re) * cosh(im))
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 = sin(re) * cosh(im);
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[(N[Sin[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   distribute-lft-in [=>] 99.9	\[ \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{0 - im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{im} + \left(0.5 \cdot \sin re\right) \cdot e^{0 - im}} \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(\sin re \cdot 0.5\right)} \cdot e^{im} + \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} \]
   associate-*l* [=>] 99.9	\[ \color{blue}{\sin re \cdot \left(0.5 \cdot e^{im}\right)} + \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(0.5 \cdot e^{im}\right) \cdot \sin re} + \left(0.5 \cdot \sin re\right) \cdot e^{0 - im} \]
   *-commutative [<=] 99.9	\[ \left(0.5 \cdot e^{im}\right) \cdot \sin re + \color{blue}{e^{0 - im} \cdot \left(0.5 \cdot \sin re\right)} \]
   associate-*r* [=>] 99.9	\[ \left(0.5 \cdot e^{im}\right) \cdot \sin re + \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot \sin re} \]
   distribute-rgt-in [<=] 99.9	\[ \color{blue}{\sin re \cdot \left(0.5 \cdot e^{im} + e^{0 - im} \cdot 0.5\right)} \]
   fma-def [=>] 99.9	\[ \sin re \cdot \color{blue}{\mathsf{fma}\left(0.5, e^{im}, e^{0 - im} \cdot 0.5\right)} \]
   exp-diff [=>] 99.9	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{e^{0}}{e^{im}}} \cdot 0.5\right) \]
   associate-*l/ [=>] 99.9	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{e^{0} \cdot 0.5}{e^{im}}}\right) \]
   exp-0 [=>] 99.9	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{1} \cdot 0.5}{e^{im}}\right) \]
   metadata-eval [=>] 99.9	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{0.5}}{e^{im}}\right) \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]
   add-log-exp [=>] 54.7	\[ \color{blue}{\log \left(e^{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\right)} \]
   *-un-lft-identity [=>] 54.7	\[ \log \color{blue}{\left(1 \cdot e^{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\right)} \]
   log-prod [=>] 54.7	\[ \color{blue}{\log 1 + \log \left(e^{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\right)} \]
   metadata-eval [=>] 54.7	\[ \color{blue}{0} + \log \left(e^{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\right) \]
   add-log-exp [<=] 99.9	\[ 0 + \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
   fma-udef [=>] 99.9	\[ 0 + \sin re \cdot \color{blue}{\left(0.5 \cdot e^{im} + \frac{0.5}{e^{im}}\right)} \]
   div-inv [=>] 99.9	\[ 0 + \sin re \cdot \left(0.5 \cdot e^{im} + \color{blue}{0.5 \cdot \frac{1}{e^{im}}}\right) \]
   distribute-lft-out [=>] 99.9	\[ 0 + \sin re \cdot \color{blue}{\left(0.5 \cdot \left(e^{im} + \frac{1}{e^{im}}\right)\right)} \]
   rec-exp [=>] 99.9	\[ 0 + \sin re \cdot \left(0.5 \cdot \left(e^{im} + \color{blue}{e^{-im}}\right)\right) \]
   cosh-undef [=>] 99.9	\[ 0 + \sin re \cdot \left(0.5 \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

4. Simplified 99.9%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
   Proof

   [Start] 99.9	\[ 0 + \sin re \cdot \left(0.5 \cdot \left(2 \cdot \cosh im\right)\right) \]
   +-lft-identity [=>] 99.9	\[ \color{blue}{\sin re \cdot \left(0.5 \cdot \left(2 \cdot \cosh im\right)\right)} \]
   associate-*r* [=>] 99.9	\[ \sin re \cdot \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot \cosh im\right)} \]
   metadata-eval [=>] 99.9	\[ \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]
   *-lft-identity [=>] 99.9	\[ \sin re \cdot \color{blue}{\cosh im} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	6976

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

Alternative 2
Accuracy	98.1%
Cost	6464

\[\sin re \]

Alternative 3
Accuracy	49.2%
Cost	576

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

Alternative 4
Accuracy	49.0%
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023151 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))