math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 10.4s

Precision: binary64

Cost: 19712

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (+ (exp im) (exp (- im))) (* 0.5 (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 (exp(im) + exp(-im)) * (0.5 * 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 = (exp(im) + exp(-im)) * (0.5d0 * 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 (Math.exp(im) + Math.exp(-im)) * (0.5 * 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 (math.exp(im) + math.exp(-im)) * (0.5 * 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(Float64(exp(im) + exp(Float64(-im))) * Float64(0.5 * 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 = (exp(im) + exp(-im)) * (0.5 * 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[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(0.5 * 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. Taylor expanded in re around inf 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]
   Proof
   (*.f64 (+.f64 (exp.f64 im) (exp.f64 (neg.f64 im))) (*.f64 1/2 (sin.f64 re))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/2 (sin.f64 re)) (+.f64 (exp.f64 im) (exp.f64 (neg.f64 im))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (sin.f64 re) (+.f64 (exp.f64 im) (exp.f64 (neg.f64 im)))))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13248

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

Alternative 2
Error	0.8
Cost	6976

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

Alternative 3
Error	1.1
Cost	6464

\[\sin re \]

Alternative 4
Error	31.3
Cost	576

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

Alternative 5
Error	31.3
Cost	576

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

Alternative 6
Error	31.4
Cost	64

\[re \]

Reproduce

herbie shell --seed 2022294 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))