math.sin on complex, real part

Average Error: 0.1 → 0.1

Time: 5.9s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

3. Applied egg-rr 0.1

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

4. Taylor expanded in im around inf 0.1

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

5. Simplified 0.0

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

6. Taylor expanded in re around inf 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022202 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))