math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 11

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + 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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + 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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return sin(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\sin re \cdot 0.5\right) \cdot \left(e^{im} + e^{-im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (sin(re) * 0.5) * (exp(im) + exp(-im));
}

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(sin(re) * 0.5) * Float64(exp(im) + exp(Float64(-im))))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left(0.5 + 0.5 \cdot e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return sin(re) * (0.5 + (0.5 * exp(im)));
}

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return Float64(sin(re) * Float64(0.5 + Float64(0.5 * exp(im))))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 77.5%

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

5. Taylor expanded in re around inf 77.5%

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

   1. *-commutative 77.5%

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

7. Simplified 77.5%

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

8. Final simplification 77.5%

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

Alternative 4: 67.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 750.0)
   (sin re)
   (if (<= im 1.35e+154) (pow re -3.0) (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = pow(re, -3.0);
	} else {
		tmp = sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 750.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = re ** (-3.0d0)
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 750.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = Math.pow(re, -3.0);
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 750.0:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = math.pow(re, -3.0)
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 750.0)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = re ^ -3.0;
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 750.0)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = re ^ -3.0;
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 750.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[Power[re, -3.0], $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 750

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.7%

      \[\leadsto \color{blue}{\sin re} \]

   if 750 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 82.4%

         \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{e^{im}} + e^{im}\right)\right)} \]

      2. +-commutative 82.4%

         \[\leadsto re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + \frac{1}{e^{im}}\right)}\right) \]

      3. rec-exp 82.4%

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

   6. Simplified 82.4%

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

   8. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{{re}^{-3}} \]

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \sin re \cdot \color{blue}{\left(0.5 \cdot {im}^{2}\right)} \]

      4. unpow2 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 750:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 5: 72.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4)
   (sin re)
   (if (<= im 1.35e+154)
     (* re (+ 0.5 (* 0.5 (exp im))))
     (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.4d0) then
        tmp = sin(re)
    else if (im <= 1.35d+154) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.4:
		tmp = math.sin(re)
	elif im <= 1.35e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.4)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.4)
		tmp = sin(re);
	elseif (im <= 1.35e+154)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.4], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.4000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.7%

      \[\leadsto \color{blue}{\sin re} \]

   if 4.4000000000000004 < im < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 82.4%

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

   if 1.35000000000000003e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \sin re \cdot \color{blue}{\left(0.5 \cdot {im}^{2}\right)} \]

      4. unpow2 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 6: 85.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 4.4)
   (* (* (sin re) 0.5) (+ 2.0 (* im im)))
   (if (<= im 1e+154)
     (* re (+ 0.5 (* 0.5 (exp im))))
     (* (sin re) (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.4d0) then
        tmp = (sin(re) * 0.5d0) * (2.0d0 + (im * im))
    else if (im <= 1d+154) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = sin(re) * (0.5d0 * (im * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 4.4) {
		tmp = (Math.sin(re) * 0.5) * (2.0 + (im * im));
	} else if (im <= 1e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = Math.sin(re) * (0.5 * (im * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 4.4:
		tmp = (math.sin(re) * 0.5) * (2.0 + (im * im))
	elif im <= 1e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = math.sin(re) * (0.5 * (im * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 4.4)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = Float64(sin(re) * Float64(0.5 * Float64(im * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.4)
		tmp = (sin(re) * 0.5) * (2.0 + (im * im));
	elseif (im <= 1e+154)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = sin(re) * (0.5 * (im * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 4.4], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.4000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 86.8%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   6. Simplified 86.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 4.4000000000000004 < im < 1.00000000000000004e154

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

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

   5. Taylor expanded in re around 0 82.4%

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

   if 1.00000000000000004e154 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(2 + \color{blue}{im \cdot im}\right) \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot {im}^{2}\right) \cdot 0.5} \]

      2. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left({im}^{2} \cdot 0.5\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \sin re \cdot \color{blue}{\left(0.5 \cdot {im}^{2}\right)} \]

      4. unpow2 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.4:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 7: 63.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;{re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 800.0)
   (sin re)
   (if (<= im 6.8e+153) (pow re -3.0) (* re (+ (* 0.5 (* im im)) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 800.0) {
		tmp = sin(re);
	} else if (im <= 6.8e+153) {
		tmp = pow(re, -3.0);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 800.0d0) then
        tmp = sin(re)
    else if (im <= 6.8d+153) then
        tmp = re ** (-3.0d0)
    else
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 800.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.8e+153) {
		tmp = Math.pow(re, -3.0);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 800.0:
		tmp = math.sin(re)
	elif im <= 6.8e+153:
		tmp = math.pow(re, -3.0)
	else:
		tmp = re * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 800.0)
		tmp = sin(re);
	elseif (im <= 6.8e+153)
		tmp = re ^ -3.0;
	else
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 800.0)
		tmp = sin(re);
	elseif (im <= 6.8e+153)
		tmp = re ^ -3.0;
	else
		tmp = re * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 800.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.8e+153], N[Power[re, -3.0], $MachinePrecision], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 800

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.7%

      \[\leadsto \color{blue}{\sin re} \]

   if 800 < im < 6.7999999999999995e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 82.4%

      \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 82.4%

         \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{e^{im}} + e^{im}\right)\right)} \]

      2. +-commutative 82.4%

         \[\leadsto re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + \frac{1}{e^{im}}\right)}\right) \]

      3. rec-exp 82.4%

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

   6. Simplified 82.4%

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

   8. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{{re}^{-3}} \]

   if 6.7999999999999995e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 63.4%

      \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]

   5. Taylor expanded in im around 0 63.4%

      \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   7. Simplified 63.4%

      \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 800:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+153}:\\ \;\;\;\;{re}^{-3}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 8: 61.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00095:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00095) (sin re) (* re (+ (* 0.5 (* im im)) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00095) {
		tmp = sin(re);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00095d0) then
        tmp = sin(re)
    else
        tmp = re * ((0.5d0 * (im * im)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00095) {
		tmp = Math.sin(re);
	} else {
		tmp = re * ((0.5 * (im * im)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00095:
		tmp = math.sin(re)
	else:
		tmp = re * ((0.5 * (im * im)) + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00095)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00095)
		tmp = sin(re);
	else
		tmp = re * ((0.5 * (im * im)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00095], N[Sin[re], $MachinePrecision], N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 9.49999999999999998e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.9%

      \[\leadsto \color{blue}{\sin re} \]

   if 9.49999999999999998e-4 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*r* 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. fma-def 100.0%

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

      10. exp-diff 100.0%

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

      11. associate-*l/ 100.0%

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

      12. exp-0 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 69.5%

      \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]

   5. Taylor expanded in im around 0 47.6%

      \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 47.6%

         \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

   7. Simplified 47.6%

      \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00095:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right)\\ \end{array} \]

Alternative 9: 47.9% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (+ (* 0.5 (* im im)) 1.0)))

C

double code(double re, double im) {
	return re * ((0.5 * (im * im)) + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * ((0.5d0 * (im * im)) + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return re * ((0.5 * (im * im)) + 1.0);
}

Python

def code(re, im):
	return re * ((0.5 * (im * im)) + 1.0)

Julia

function code(re, im)
	return Float64(re * Float64(Float64(0.5 * Float64(im * im)) + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = re * ((0.5 * (im * im)) + 1.0);
end

Wolfram

code[re_, im_] := N[(re * N[(N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 61.0%

   \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]

5. Taylor expanded in im around 0 49.1%

   \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot {im}^{2}\right)} \]
6. Step-by-step derivation

   1. unpow2 49.1%

      \[\leadsto re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(im \cdot im\right)}\right) \]

7. Simplified 49.1%

   \[\leadsto re \cdot \color{blue}{\left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]

8. Final simplification 49.1%

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

Alternative 10: 32.1% accurate, 44.1× speedup

Math

\[\begin{array}{l} \\ re + 0.5 \cdot \left(re \cdot im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ re (* 0.5 (* re im))))

C

double code(double re, double im) {
	return re + (0.5 * (re * im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re + (0.5d0 * (re * im))
end function

Java

public static double code(double re, double im) {
	return re + (0.5 * (re * im));
}

Python

def code(re, im):
	return re + (0.5 * (re * im))

Julia

function code(re, im)
	return Float64(re + Float64(0.5 * Float64(re * im)))
end

MATLAB

function tmp = code(re, im)
	tmp = re + (0.5 * (re * im));
end

Wolfram

code[re_, im_] := N[(re + N[(0.5 * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 77.5%

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

5. Taylor expanded in re around 0 44.5%

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

6. Taylor expanded in im around 0 35.9%

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

7. Final simplification 35.9%

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

Alternative 11: 26.2% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r* 100.0%

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

   8. distribute-rgt-in 100.0%

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

   9. fma-def 100.0%

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

   10. exp-diff 100.0%

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

   11. associate-*l/ 100.0%

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

   12. exp-0 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 61.0%

   \[\leadsto \color{blue}{re \cdot \left(0.5 \cdot \frac{1}{e^{im}} + 0.5 \cdot e^{im}\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 61.0%

      \[\leadsto re \cdot \color{blue}{\left(0.5 \cdot \left(\frac{1}{e^{im}} + e^{im}\right)\right)} \]

   2. +-commutative 61.0%

      \[\leadsto re \cdot \left(0.5 \cdot \color{blue}{\left(e^{im} + \frac{1}{e^{im}}\right)}\right) \]

   3. rec-exp 61.0%

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

6. Simplified 61.0%

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

7. Taylor expanded in im around 0 29.7%

   \[\leadsto \color{blue}{re} \]

8. Final simplification 29.7%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))