math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 9.9s

Precision: binary64

Cost: 26048

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 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) * fma(0.5, exp(im), (0.5 / 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) * fma(0.5, exp(im), Float64(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[(0.5 * N[Exp[im], $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
   (*.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)))))): 1 points increase in error, 0 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)))): 0 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. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.3
Cost	20040

\[\begin{array}{l} t_0 := \sin re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right)\\ \mathbf{if}\;\sin re \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin re \leq 2 \cdot 10^{-41}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	19912

\[\begin{array}{l} \mathbf{if}\;\sin re \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;\sin re \leq 10^{-12}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(2 \cdot \cosh im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re\\ \end{array} \]

Alternative 3
Error	0.0
Cost	19712

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

Alternative 4
Error	0.0
Cost	13248

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

Alternative 5
Error	1.1
Cost	6464

\[\sin re \]

Alternative 6
Error	31.9
Cost	64

\[re \]

Reproduce

herbie shell --seed 2022316 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))