math.sin on complex, real part

Average Accuracy: 99.9% → 99.9%

Time: 9.7s

Precision: binary64

Cost: 26304

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \frac{t_0}{e^{im}} + t_0 \cdot e^{im} \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))) (+ (/ t_0 (exp im)) (* t_0 (exp im)))))

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 = 0.5 * sin(re);
	return (t_0 / exp(im)) + (t_0 * exp(im));
}

Fortran

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
    real(8) :: t_0
    t_0 = 0.5d0 * sin(re)
    code = (t_0 / exp(im)) + (t_0 * exp(im))
end function

Java

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) {
	double t_0 = 0.5 * Math.sin(re);
	return (t_0 / Math.exp(im)) + (t_0 * Math.exp(im));
}

Python

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

↓

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	return (t_0 / math.exp(im)) + (t_0 * math.exp(im))

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 = Float64(0.5 * sin(re))
	return Float64(Float64(t_0 / exp(im)) + Float64(t_0 * exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

↓

function tmp = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = (t_0 / exp(im)) + (t_0 * exp(im));
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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   sub0-neg [=>] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   distribute-lft-in [=>] 99.9	\[ \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{-im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

4. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot e^{-im} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]
   exp-neg [=>] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\frac{1}{e^{im}}} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]
   un-div-inv [=>] 99.9	\[ \color{blue}{\frac{0.5 \cdot \sin re}{e^{im}}} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	12992

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

Alternative 2
Accuracy	98.7%
Cost	6976

\[\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 3
Accuracy	98.1%
Cost	6464

\[\sin re \]

Alternative 4
Accuracy	50.5%
Cost	576

\[re \cdot \left(1 + im \cdot \left(0.5 \cdot im\right)\right) \]

Alternative 5
Accuracy	50.5%
Cost	576

\[re + 0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right) \]

Alternative 6
Accuracy	50.2%
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023136 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))