Average Error: 0.0 → 0.0
Time: 7.1s
Precision: binary64
Cost: 12992
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[\sin re \cdot \cosh im \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))
↓
(FPCore (re im) :precision binary64 (* (sin re) (cosh 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) * cosh(im);
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function
↓
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * cosh(im)
end function

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}
↓
public static double code(double re, double im) {
	return Math.sin(re) * Math.cosh(im);
}

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))
↓
def code(re, im):
	return math.sin(re) * math.cosh(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) * cosh(im))
end

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end
↓
function tmp = code(re, im)
	tmp = sin(re) * cosh(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[Cosh[im], $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \]
    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, 1 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. Applied egg-rr0.0

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

  4. Applied egg-rr0.1

    \[\leadsto \sin re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \left(2 \cdot \cosh im\right)\right)} - 1\right)} \]

  5. Simplified0.0

    \[\leadsto \sin re \cdot \color{blue}{\cosh im} \]
    Proof
    (cosh.f64 im): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-lft-identity_binary64 (*.f64 1 (cosh.f64 im))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= metadata-eval (*.f64 1/2 2)) (cosh.f64 im)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 2 (cosh.f64 im)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 1/2 (*.f64 2 (cosh.f64 im)))))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 1/2 (*.f64 2 (cosh.f64 im))))) 1)): 1 points increase in error, 3 points decrease in error

  6. Final simplification0.0

    \[\leadsto \sin re \cdot \cosh im \]

Reproduce

herbie shell --seed 2022334 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))