math.sin on complex, real part

Average Accuracy: 49.2% → 50.0%

Time: 10.6s

Precision: binary64

Cost: 45440

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{0.5 \cdot \sin re}\\ \mathsf{fma}\left({\left(im \cdot t_0\right)}^{2}, t_0, \sin re\right) \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (cbrt (* 0.5 (sin re)))))
   (fma (pow (* im t_0) 2.0) t_0 (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) {
	double t_0 = cbrt((0.5 * sin(re)));
	return fma(pow((im * t_0), 2.0), t_0, 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)
	t_0 = cbrt(Float64(0.5 * sin(re)))
	return fma((Float64(im * t_0) ^ 2.0), t_0, 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_] := Block[{t$95$0 = N[Power[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(im * t$95$0), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.1%

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

2. Simplified 53.1%

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

   [Start] 53.1	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   sub0-neg [=>] 53.1	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Taylor expanded in im around 0 53.9%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

4. Simplified 53.9%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]
   Proof

   [Start] 53.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(2 + {im}^{2}\right) \]
   unpow2 [=>] 53.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

5. Applied egg-rr 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(im \cdot \sqrt[3]{0.5 \cdot \sin re}\right)}^{2}, \sqrt[3]{0.5 \cdot \sin re}, \sin re\right)} \]
   Proof

   [Start] 53.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right) \]
   +-commutative [=>] 53.9	\[ \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot im + 2\right)} \]
   distribute-rgt-in [=>] 53.9	\[ \color{blue}{\left(im \cdot im\right) \cdot \left(0.5 \cdot \sin re\right) + 2 \cdot \left(0.5 \cdot \sin re\right)} \]
   add-cube-cbrt [=>] 53.9	\[ \left(im \cdot im\right) \cdot \color{blue}{\left(\left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right) \cdot \sqrt[3]{0.5 \cdot \sin re}\right)} + 2 \cdot \left(0.5 \cdot \sin re\right) \]
   associate-*r* [=>] 53.9	\[ \color{blue}{\left(\left(im \cdot im\right) \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right)\right) \cdot \sqrt[3]{0.5 \cdot \sin re}} + 2 \cdot \left(0.5 \cdot \sin re\right) \]
   associate-*r* [=>] 53.9	\[ \left(\left(im \cdot im\right) \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right)\right) \cdot \sqrt[3]{0.5 \cdot \sin re} + \color{blue}{\left(2 \cdot 0.5\right) \cdot \sin re} \]
   metadata-eval [=>] 53.9	\[ \left(\left(im \cdot im\right) \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right)\right) \cdot \sqrt[3]{0.5 \cdot \sin re} + \color{blue}{1} \cdot \sin re \]
   *-un-lft-identity [<=] 53.9	\[ \left(\left(im \cdot im\right) \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right)\right) \cdot \sqrt[3]{0.5 \cdot \sin re} + \color{blue}{\sin re} \]
   fma-def [=>] 53.9	\[ \color{blue}{\mathsf{fma}\left(\left(im \cdot im\right) \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right), \sqrt[3]{0.5 \cdot \sin re}, \sin re\right)} \]
   pow2 [=>] 53.9	\[ \mathsf{fma}\left(\color{blue}{{im}^{2}} \cdot \left(\sqrt[3]{0.5 \cdot \sin re} \cdot \sqrt[3]{0.5 \cdot \sin re}\right), \sqrt[3]{0.5 \cdot \sin re}, \sin re\right) \]
   pow2 [=>] 53.9	\[ \mathsf{fma}\left({im}^{2} \cdot \color{blue}{{\left(\sqrt[3]{0.5 \cdot \sin re}\right)}^{2}}, \sqrt[3]{0.5 \cdot \sin re}, \sin re\right) \]
   pow-prod-down [=>] 54.2	\[ \mathsf{fma}\left(\color{blue}{{\left(im \cdot \sqrt[3]{0.5 \cdot \sin re}\right)}^{2}}, \sqrt[3]{0.5 \cdot \sin re}, \sin re\right) \]

6. Final simplification 54.2%

   \[\leadsto \mathsf{fma}\left({\left(im \cdot \sqrt[3]{0.5 \cdot \sin re}\right)}^{2}, \sqrt[3]{0.5 \cdot \sin re}, \sin re\right) \]

Alternatives

Alternative 1
Accuracy	49.7%
Cost	6464

\[\sin re \]

Alternative 2
Accuracy	26.1%
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023157 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))