math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 10.8s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (sin re) (+ (/ 0.5 (exp im)) (* 0.5 (exp 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) * ((0.5 / exp(im)) + (0.5 * exp(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) * ((0.5d0 / exp(im)) + (0.5d0 * exp(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) * ((0.5 / Math.exp(im)) + (0.5 * Math.exp(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) * ((0.5 / math.exp(im)) + (0.5 * math.exp(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) * Float64(Float64(0.5 / exp(im)) + Float64(0.5 * exp(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) * ((0.5 / exp(im)) + (0.5 * exp(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[(N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Exp[im], $MachinePrecision]), $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

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	19712

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

Alternative 2
Error	0.6
Cost	13696

\[\left(\sin re \cdot 0.5\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \]

Alternative 3
Error	0.7
Cost	13376

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

Alternative 4
Error	0.7
Cost	6976

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

Alternative 5
Error	1.1
Cost	6464

\[\sin re \]

Alternative 6
Error	31.5
Cost	576

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

Alternative 7
Error	31.7
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023125 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))