math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 9.4s

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 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
   (*.f64 (sin.f64 re) (fma.f64 1/2 (exp.f64 im) (/.f64 1/2 (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (fma.f64 1/2 (exp.f64 im) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (fma.f64 1/2 (exp.f64 im) (/.f64 (*.f64 1/2 (Rewrite<= exp-0_binary64 (exp.f64 0))) (exp.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (fma.f64 1/2 (exp.f64 im) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (exp.f64 0) (exp.f64 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (fma.f64 1/2 (exp.f64 im) (*.f64 1/2 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 1 points decrease in error
   (*.f64 (sin.f64 re) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (exp.f64 im)) (*.f64 1/2 (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (Rewrite<= distribute-lft-in_binary64 (*.f64 1/2 (+.f64 (exp.f64 im) (exp.f64 (-.f64 0 im)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (*.f64 1/2 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 re) 1/2) (+.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)))): 1 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (sin.f64 re))) (+.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Applied egg-rr 29.4

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin re \cdot \left(0.5 \cdot \left(2 \cdot \cosh im\right)\right)\right)} - 1} \]

5. Simplified 0.0

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
   Proof
   (*.f64 (sin.f64 re) (cosh.f64 im)): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (cosh.f64 im)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 2)) (cosh.f64 im))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 re) (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 2 (cosh.f64 im))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 (sin.f64 re) (*.f64 1/2 (*.f64 2 (cosh.f64 im))))))): 21 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 (sin.f64 re) (*.f64 1/2 (*.f64 2 (cosh.f64 im)))))) 1)): 170 points increase in error, 3 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.7
Cost	6976

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

Alternative 2
Error	1.1
Cost	6464

\[\sin re \]

Alternative 3
Error	31.9
Cost	576

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

Alternative 4
Error	32.0
Cost	64

\[re \]

Reproduce

herbie shell --seed 2022330 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))