Average Error: 0.0 → 0.0
Time: 9.9s
Precision: binary64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\\ t_1 := \sqrt{t_0}\\ t_1 \cdot \left(\left(\sin re \cdot t_0\right) \cdot t_1\right) \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (fma 0.5 (exp im) (/ 0.5 (exp im))))) (t_1 (sqrt t_0)))
   (* t_1 (* (* (sin re) t_0) t_1))))
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 = sqrt(fma(0.5, exp(im), (0.5 / exp(im))));
	double t_1 = sqrt(t_0);
	return t_1 * ((sin(re) * t_0) * t_1);
}

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 = sqrt(fma(0.5, exp(im), Float64(0.5 / exp(im))))
	t_1 = sqrt(t_0)
	return Float64(t_1 * Float64(Float64(sin(re) * t_0) * t_1))
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_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(t$95$1 * N[(N[(N[Sin[re], $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)}\\
t_1 := \sqrt{t_0}\\
t_1 \cdot \left(\left(\sin re \cdot t_0\right) \cdot t_1\right)
\end{array}

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 add-sqr-sqrt_binary640.0

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

  4. Applied associate-*r*_binary640.0

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

  5. Applied add-sqr-sqrt_binary640.0

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

  6. Applied associate-*r*_binary640.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022130 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))