math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

Alternatives: 11

Speedup: 1.0×

• 49.4% 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

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im_m) (/ 0.5 (exp im_m)))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im$95$m], $MachinePrecision] + N[(0.5 / N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (* (sin re) 0.5) (+ (exp im_m) (exp (- im_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. cancel-sign-sub 99.6%

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

   3. distribute-rgt-out-- 99.6%

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

   4. sub-neg 99.6%

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

   5. neg-sub0 99.6%

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

   6. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 98.7% accurate, 1.5× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (sin re) (+ 0.5 (* 0.5 (exp im_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in im around 0 76.2%

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

   1. fma-undefine 76.2%

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

7. Applied egg-rr 76.2%

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

8. Final simplification 76.2%

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

Alternative 4: 95.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.7:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.7)
   (sin re)
   (if (<= im_m 1.3e+103)
     (* re (+ 0.5 (* 0.5 (exp im_m))))
     (*
      (sin re)
      (+
       1.0
       (* im_m (+ 0.5 (* im_m (+ (* im_m 0.08333333333333333) 0.25)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = sin(re);
	} else if (im_m <= 1.3e+103) {
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	} else {
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.7d0) then
        tmp = sin(re)
    else if (im_m <= 1.3d+103) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im_m)))
    else
        tmp = sin(re) * (1.0d0 + (im_m * (0.5d0 + (im_m * ((im_m * 0.08333333333333333d0) + 0.25d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.7) {
		tmp = Math.sin(re);
	} else if (im_m <= 1.3e+103) {
		tmp = re * (0.5 + (0.5 * Math.exp(im_m)));
	} else {
		tmp = Math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.7:
		tmp = math.sin(re)
	elif im_m <= 1.3e+103:
		tmp = re * (0.5 + (0.5 * math.exp(im_m)))
	else:
		tmp = math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.7)
		tmp = sin(re);
	elseif (im_m <= 1.3e+103)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im_m))));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(Float64(im_m * 0.08333333333333333) + 0.25))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.7)
		tmp = sin(re);
	elseif (im_m <= 1.3e+103)
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	else
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.7], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 1.3e+103], N[(re * N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(N[(im$95$m * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.7000000000000002

   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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 70.1%

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

   if 3.7000000000000002 < im < 1.3000000000000001e103

   1. Initial program 95.7%

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

      1. distribute-rgt-in 95.7%

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

      2. +-commutative 95.7%

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

      3. associate-*r* 95.7%

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

      4. associate-*r* 95.7%

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

      5. distribute-rgt-out 95.7%

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

      6. distribute-rgt-in 95.7%

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

      7. distribute-lft-in 95.7%

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

      8. *-commutative 95.7%

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

      9. fma-define 95.7%

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

      10. exp-diff 95.7%

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

      11. associate-*l/ 95.7%

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

      12. exp-0 95.7%

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

      13. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in im around 0 95.7%

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

   6. Taylor expanded in re around 0 57.6%

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

   if 1.3000000000000001e103 < 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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

      1. fma-undefine 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\sin re + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot \sin re\right) + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(\sin re + 0.08333333333333333 \cdot \left({im}^{3} \cdot \sin re\right)\right) + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\sin re + \color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot \sin re}\right) + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      3. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot {im}^{3} + 1\right) \cdot \sin re} + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      4. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} \cdot \sin re + \left(0.25 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) \cdot \sin re + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \left(im \cdot \sin re\right)\right) \]

      6. associate-*r* 100.0%

         \[\leadsto \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) \cdot \sin re + \left(\left(0.25 \cdot {im}^{2}\right) \cdot \sin re + \color{blue}{\left(0.5 \cdot im\right) \cdot \sin re}\right) \]

      7. distribute-rgt-out 100.0%

         \[\leadsto \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) \cdot \sin re + \color{blue}{\sin re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      8. *-commutative 100.0%

         \[\leadsto \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) \cdot \sin re + \color{blue}{\left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right) \cdot \sin re} \]

      9. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      10. associate-+r+ 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(0.08333333333333333 \cdot {im}^{3} + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)\right)} \]

      11. associate-+r+ 100.0%

          \[\leadsto \sin re \cdot \left(1 + \color{blue}{\left(\left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right) + 0.5 \cdot im\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 860:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 3.15 \cdot 10^{+102}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 860.0)
   (sin re)
   (if (<= im_m 3.15e+102)
     (pow re -2.0)
     (*
      re
      (+
       1.0
       (* im_m (+ 0.5 (* im_m (+ (* im_m 0.08333333333333333) 0.25)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 860.0) {
		tmp = sin(re);
	} else if (im_m <= 3.15e+102) {
		tmp = pow(re, -2.0);
	} else {
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 860.0d0) then
        tmp = sin(re)
    else if (im_m <= 3.15d+102) then
        tmp = re ** (-2.0d0)
    else
        tmp = re * (1.0d0 + (im_m * (0.5d0 + (im_m * ((im_m * 0.08333333333333333d0) + 0.25d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 860.0) {
		tmp = Math.sin(re);
	} else if (im_m <= 3.15e+102) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 860.0:
		tmp = math.sin(re)
	elif im_m <= 3.15e+102:
		tmp = math.pow(re, -2.0)
	else:
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 860.0)
		tmp = sin(re);
	elseif (im_m <= 3.15e+102)
		tmp = re ^ -2.0;
	else
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(Float64(im_m * 0.08333333333333333) + 0.25))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 860.0)
		tmp = sin(re);
	elseif (im_m <= 3.15e+102)
		tmp = re ^ -2.0;
	else
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 860.0], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 3.15e+102], N[Power[re, -2.0], $MachinePrecision], N[(re * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(N[(im$95$m * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 860

   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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 70.1%

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

   if 860 < im < 3.15000000000000015e102

   1. Initial program 95.7%

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

      1. distribute-rgt-in 95.7%

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

      2. +-commutative 95.7%

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

      3. associate-*r* 95.7%

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

      4. associate-*r* 95.7%

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

      5. distribute-rgt-out 95.7%

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

      6. distribute-rgt-in 95.7%

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

      7. distribute-lft-in 95.7%

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

      8. *-commutative 95.7%

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

      9. fma-define 95.7%

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

      10. exp-diff 95.7%

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

      11. associate-*l/ 95.7%

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

      12. exp-0 95.7%

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

      13. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in im around 0 95.7%

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

   6. Taylor expanded in re around 0 57.6%

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

   8. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{{re}^{-2}} \]

   if 3.15000000000000015e102 < 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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 80.5%

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

   7. Taylor expanded in im around 0 80.5%

      \[\leadsto \color{blue}{re + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot re\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 80.5%

         \[\leadsto \color{blue}{\left(re + 0.08333333333333333 \cdot \left({im}^{3} \cdot re\right)\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)} \]

      2. associate-*r* 80.5%

         \[\leadsto \left(re + \color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot re}\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      3. distribute-rgt1-in 80.5%

         \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot {im}^{3} + 1\right) \cdot re} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      4. +-commutative 80.5%

         \[\leadsto \color{blue}{\left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} \cdot re + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      5. *-commutative 80.5%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      6. associate-*r* 80.5%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot re} + 0.5 \cdot \left(im \cdot re\right)\right) \]

      7. associate-*r* 80.5%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\left(0.25 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.5 \cdot im\right) \cdot re}\right) \]

      8. distribute-rgt-out 80.5%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \color{blue}{re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      9. distribute-lft-out 80.5%

         \[\leadsto \color{blue}{re \cdot \left(\left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      10. associate-+r+ 80.5%

          \[\leadsto re \cdot \color{blue}{\left(1 + \left(0.08333333333333333 \cdot {im}^{3} + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)\right)} \]

      11. associate-+r+ 80.5%

          \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right) + 0.5 \cdot im\right)}\right) \]

      12. +-commutative 80.5%

          \[\leadsto re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

      13. *-commutative 80.5%

          \[\leadsto re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

   9. Simplified 80.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 860:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.15 \cdot 10^{+102}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.2:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im\_m}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.2) (sin re) (* re (+ 0.5 (* 0.5 (exp im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.2) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.2d0) then
        tmp = sin(re)
    else
        tmp = re * (0.5d0 + (0.5d0 * exp(im_m)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.2) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.5 + (0.5 * Math.exp(im_m)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.2:
		tmp = math.sin(re)
	else:
		tmp = re * (0.5 + (0.5 * math.exp(im_m)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.2)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.2)
		tmp = sin(re);
	else
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.2], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 + N[(0.5 * N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2000000000000002

   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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 70.1%

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

   if 3.2000000000000002 < im

   1. Initial program 98.5%

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

      1. distribute-rgt-in 98.5%

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

      2. +-commutative 98.5%

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

      3. associate-*r* 98.5%

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

      4. associate-*r* 98.5%

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

      5. distribute-rgt-out 98.5%

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

      6. distribute-rgt-in 98.5%

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

      7. distribute-lft-in 98.5%

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

      8. *-commutative 98.5%

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

      9. fma-define 98.5%

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

      10. exp-diff 98.5%

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

      11. associate-*l/ 98.5%

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

      12. exp-0 98.5%

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

      13. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in im around 0 98.5%

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

   6. Taylor expanded in re around 0 72.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.2:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.9e+53)
   (sin re)
   (*
    re
    (+ 1.0 (* im_m (+ 0.5 (* im_m (+ (* im_m 0.08333333333333333) 0.25))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.9e+53) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.9d+53) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (im_m * (0.5d0 + (im_m * ((im_m * 0.08333333333333333d0) + 0.25d0)))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.9e+53) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.9e+53:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.9e+53)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(Float64(im_m * 0.08333333333333333) + 0.25))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.9e+53)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.9e+53], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(N[(im$95$m * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.89999999999999999e53

   1. Initial program 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-*r* 99.5%

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

      4. associate-*r* 99.5%

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

      5. distribute-rgt-out 99.5%

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

      6. distribute-rgt-in 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-define 99.5%

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

      10. exp-diff 99.5%

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

      11. associate-*l/ 99.5%

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

      12. exp-0 99.5%

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

      13. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in im around 0 66.5%

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

   if 1.89999999999999999e53 < 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-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. *-commutative 100.0%

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

      9. fma-define 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. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

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

   6. Taylor expanded in re around 0 78.4%

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

   7. Taylor expanded in im around 0 67.3%

      \[\leadsto \color{blue}{re + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot re\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ 67.3%

         \[\leadsto \color{blue}{\left(re + 0.08333333333333333 \cdot \left({im}^{3} \cdot re\right)\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)} \]

      2. associate-*r* 67.3%

         \[\leadsto \left(re + \color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot re}\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      3. distribute-rgt1-in 67.3%

         \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot {im}^{3} + 1\right) \cdot re} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      4. +-commutative 67.3%

         \[\leadsto \color{blue}{\left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} \cdot re + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      5. *-commutative 67.3%

         \[\leadsto \color{blue}{re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

      6. associate-*r* 67.3%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot re} + 0.5 \cdot \left(im \cdot re\right)\right) \]

      7. associate-*r* 67.3%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\left(0.25 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.5 \cdot im\right) \cdot re}\right) \]

      8. distribute-rgt-out 67.3%

         \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \color{blue}{re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

      9. distribute-lft-out 67.3%

         \[\leadsto \color{blue}{re \cdot \left(\left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

      10. associate-+r+ 67.3%

          \[\leadsto re \cdot \color{blue}{\left(1 + \left(0.08333333333333333 \cdot {im}^{3} + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)\right)} \]

      11. associate-+r+ 67.3%

          \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right) + 0.5 \cdot im\right)}\right) \]

      12. +-commutative 67.3%

          \[\leadsto re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

      13. *-commutative 67.3%

          \[\leadsto re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

   9. Simplified 67.3%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.9 \cdot 10^{+53}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot \left(im\_m \cdot 0.08333333333333333 + 0.25\right)\right)\right) \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (*
  re
  (+ 1.0 (* im_m (+ 0.5 (* im_m (+ (* im_m 0.08333333333333333) 0.25)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = re * (1.0d0 + (im_m * (0.5d0 + (im_m * ((im_m * 0.08333333333333333d0) + 0.25d0)))))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(re * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * Float64(Float64(im_m * 0.08333333333333333) + 0.25))))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = re * (1.0 + (im_m * (0.5 + (im_m * ((im_m * 0.08333333333333333) + 0.25)))));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(N[(im$95$m * 0.08333333333333333), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in im around 0 76.2%

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

6. Taylor expanded in re around 0 45.4%

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

7. Taylor expanded in im around 0 41.6%

   \[\leadsto \color{blue}{re + \left(0.08333333333333333 \cdot \left({im}^{3} \cdot re\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)\right)} \]
8. Step-by-step derivation

   1. associate-+r+ 41.6%

      \[\leadsto \color{blue}{\left(re + 0.08333333333333333 \cdot \left({im}^{3} \cdot re\right)\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right)} \]

   2. associate-*r* 41.6%

      \[\leadsto \left(re + \color{blue}{\left(0.08333333333333333 \cdot {im}^{3}\right) \cdot re}\right) + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

   3. distribute-rgt1-in 41.6%

      \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot {im}^{3} + 1\right) \cdot re} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

   4. +-commutative 41.6%

      \[\leadsto \color{blue}{\left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} \cdot re + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

   5. *-commutative 41.6%

      \[\leadsto \color{blue}{re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right)} + \left(0.25 \cdot \left({im}^{2} \cdot re\right) + 0.5 \cdot \left(im \cdot re\right)\right) \]

   6. associate-*r* 41.6%

      \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\color{blue}{\left(0.25 \cdot {im}^{2}\right) \cdot re} + 0.5 \cdot \left(im \cdot re\right)\right) \]

   7. associate-*r* 41.6%

      \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(\left(0.25 \cdot {im}^{2}\right) \cdot re + \color{blue}{\left(0.5 \cdot im\right) \cdot re}\right) \]

   8. distribute-rgt-out 41.6%

      \[\leadsto re \cdot \left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \color{blue}{re \cdot \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)} \]

   9. distribute-lft-out 42.8%

      \[\leadsto \color{blue}{re \cdot \left(\left(1 + 0.08333333333333333 \cdot {im}^{3}\right) + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \]

   10. associate-+r+ 42.8%

       \[\leadsto re \cdot \color{blue}{\left(1 + \left(0.08333333333333333 \cdot {im}^{3} + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)\right)} \]

   11. associate-+r+ 42.8%

       \[\leadsto re \cdot \left(1 + \color{blue}{\left(\left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right) + 0.5 \cdot im\right)}\right) \]

   12. +-commutative 42.8%

       \[\leadsto re \cdot \left(1 + \color{blue}{\left(0.5 \cdot im + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)}\right) \]

   13. *-commutative 42.8%

       \[\leadsto re \cdot \left(1 + \left(\color{blue}{im \cdot 0.5} + \left(0.08333333333333333 \cdot {im}^{3} + 0.25 \cdot {im}^{2}\right)\right)\right) \]

9. Simplified 44.8%

   \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right)} \]

10. Final simplification 44.8%

    \[\leadsto re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(im \cdot 0.08333333333333333 + 0.25\right)\right)\right) \]
11. Add Preprocessing

Alternative 9: 32.2% accurate, 44.1× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (+ re (* 0.5 (* re im_m))))

C

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

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re + (0.5 * (re * im_m));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re + (0.5 * (re * im_m))

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re + N[(0.5 * N[(re * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in im around 0 76.2%

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

6. Taylor expanded in re around 0 45.4%

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

7. Taylor expanded in im around 0 31.8%

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

   1. *-commutative 31.8%

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

9. Simplified 31.8%

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

10. Final simplification 31.8%

    \[\leadsto re + 0.5 \cdot \left(re \cdot im\right) \]
11. Add Preprocessing

Alternative 10: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.0

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 3.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sin re \cdot -0.5, \sin re \cdot -0.5\right)} \]
7. Step-by-step derivation

   1. fma-undefine 3.2%

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

   2. neg-mul-1 3.2%

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

   3. +-commutative 3.2%

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

   4. sub-neg 3.2%

      \[\leadsto \color{blue}{\sin re \cdot -0.5 - \sin re \cdot -0.5} \]

   5. +-inverses 3.2%

      \[\leadsto \color{blue}{0} \]

8. Simplified 3.2%

   \[\leadsto \color{blue}{0} \]

9. Final simplification 3.2%

   \[\leadsto 0 \]
10. Add Preprocessing

Alternative 11: 26.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ re \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 re)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re;
}

Fortran

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

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return re;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re

Julia

im_m = abs(im)
function code(re, im_m)
	return re
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := re

Derivation

1. Initial program 99.6%

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

   1. distribute-rgt-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. distribute-rgt-out 99.6%

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

   6. distribute-rgt-in 99.6%

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

   7. distribute-lft-in 99.6%

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

   8. *-commutative 99.6%

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

   9. fma-define 99.6%

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

   10. exp-diff 99.6%

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

   11. associate-*l/ 99.6%

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

   12. exp-0 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in im around 0 76.2%

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

6. Taylor expanded in re around 0 45.4%

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

7. Taylor expanded in im around 0 28.8%

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

8. Final simplification 28.8%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))