math.sin on complex, real part

?

Percentage Accurate: 100.0% → 100.0%
Time: 16.9s
Precision: binary64
Cost: 19712

?

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[\sin re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right) \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (* (sin re) (* 0.5 (+ (exp (- im)) (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) * (0.5 * (exp(-im) + exp(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) * (0.5d0 * (exp(-im) + exp(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) * (0.5 * (Math.exp(-im) + Math.exp(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) * (0.5 * (math.exp(-im) + math.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) * Float64(0.5 * Float64(exp(Float64(-im)) + exp(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) * (0.5 * (exp(-im) + 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[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 22 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 100.0%

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

  2. Simplified100.0%

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

    [Start]100.0%

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

    *-commutative [=>]100.0%

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

    associate-*l* [=>]100.0%

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

    sub0-neg [=>]100.0%

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

  3. Final simplification100.0%

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

Alternatives

Alternative 1
Accuracy100.0%
Cost19712
\[\sin re \cdot \left(0.5 \cdot \left(e^{-im} + e^{im}\right)\right) \]
Alternative 2
Accuracy93.4%
Cost19913
\[\begin{array}{l} \mathbf{if}\;im \leq -360 \lor \neg \left(im \leq 360\right):\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\sin re \cdot im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \end{array} \]
Alternative 3
Accuracy95.9%
Cost13842
\[\begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+82} \lor \neg \left(im \leq -0.7 \lor \neg \left(im \leq 960000000\right) \land im \leq 2.5 \cdot 10^{+77}\right):\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(re \cdot 0.5\right)\\ \end{array} \]
Alternative 4
Accuracy85.1%
Cost7696
\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -330000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 550:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \left(\left(im \cdot im\right) \cdot \left(re + -0.16666666666666666 \cdot {re}^{3}\right)\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Accuracy86.5%
Cost7508
\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -195:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+78}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Accuracy84.7%
Cost7508
\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -800:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;{re}^{3} \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Accuracy84.9%
Cost7508
\[\begin{array}{l} t_0 := 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ t_1 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.4 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 480:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(0.5 \cdot im\right) + 1\right)\\ \mathbf{elif}\;im \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;{re}^{3} \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Accuracy80.8%
Cost7376
\[\begin{array}{l} t_0 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -0.00092:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 480:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+78}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;im \leq 7.3 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]
Alternative 9
Accuracy87.9%
Cost7360
\[\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right) \]
Alternative 10
Accuracy80.3%
Cost6728
\[\begin{array}{l} t_0 := re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \mathbf{if}\;im \leq -195:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 480:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+78}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Accuracy55.8%
Cost1092
\[\begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+180}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
Alternative 12
Accuracy48.3%
Cost964
\[\begin{array}{l} \mathbf{if}\;re \leq -1.5 \cdot 10^{+180}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]
Alternative 13
Accuracy47.5%
Cost713
\[\begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{-8} \lor \neg \left(im \leq 3.6 \cdot 10^{-8}\right):\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
Alternative 14
Accuracy47.9%
Cost576
\[re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \]
Alternative 15
Accuracy29.6%
Cost452
\[\begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \]
Alternative 16
Accuracy3.8%
Cost64
\[-512 \]
Alternative 17
Accuracy4.7%
Cost64
\[-1 \]
Alternative 18
Accuracy4.6%
Cost64
\[-0.5 \]
Alternative 19
Accuracy4.5%
Cost64
\[0.25 \]
Alternative 20
Accuracy4.7%
Cost64
\[0.5 \]
Alternative 21
Accuracy4.8%
Cost64
\[1 \]
Alternative 22
Accuracy27.4%
Cost64
\[re \]

Reproduce?

herbie shell --seed 2023277 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))