Average Error: 0.0 → 0.2
Time: 6.7s
Precision: binary64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \log \left({e}^{\left(\frac{0.5}{e^{im}}\right)}\right)\right) \]
(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) (log (pow E (/ 0.5 (exp im)))))))
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), log(pow(((double) M_E), (0.5 / exp(im)))));
}

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), log((exp(1) ^ Float64(0.5 / exp(im))))))
end

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[Log[N[Power[E, N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\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}, \log \left({e}^{\left(\frac{0.5}{e^{im}}\right)}\right)\right)

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Applied egg-rr0.2

    \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.5}{e^{im}}\right)\right)}\right) \]

  4. Applied egg-rr0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022145 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))