Average Error: 0.0 → 0.0
Time: 15.4s
Precision: binary64
Cost: 32448
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right) \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (* 0.5 (fma (sin re) (exp im) (/ (sin re) (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 0.5 * fma(sin(re), exp(im), (sin(re) / 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(0.5 * fma(sin(re), exp(im), Float64(sin(re) / 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[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[Exp[im], $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(\sin re, e^{im}, \frac{\sin re}{e^{im}}\right)

Error

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Applied egg-rr0.0

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

Alternatives

Alternative 1
Error0.0
Cost19712
\[0.5 \cdot \left(\sin re \cdot \left(e^{-im} + e^{im}\right)\right) \]
Alternative 2
Error0.7
Cost7616
\[0.5 \cdot \left(\sin re \cdot \left(\left(2 + im \cdot im\right) + \left(im \cdot im\right) \cdot \left(0.08333333333333333 \cdot \left(im \cdot im\right)\right)\right)\right) \]
Alternative 3
Error0.9
Cost6976
\[0.5 \cdot \left(\sin re \cdot \left(2 + im \cdot im\right)\right) \]
Alternative 4
Error1.3
Cost6720
\[0.5 \cdot \left(\sin re \cdot 2\right) \]
Alternative 5
Error31.4
Cost704
\[0.5 \cdot \left(\left(re + re\right) + re \cdot \left(im \cdot im\right)\right) \]
Alternative 6
Error31.6
Cost320
\[0.5 \cdot \left(2 \cdot re\right) \]

Error

Reproduce

herbie shell --seed 2023010 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))