math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 15

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 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-rgt-in 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. Final simplification 100.0%

   \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \]
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 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. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \left(e^{im} + e^{-im}\right) \]
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 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

   1. fma-undefine 74.8%

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

7. Applied egg-rr 74.8%

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

8. Final simplification 74.8%

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

Alternative 4: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 580:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 580.0)
   (* re (+ 0.5 (* 0.5 (exp im_m))))
   (* (sin re) (+ 1.0 (* im_m (+ 0.5 (* im_m 0.25)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 580.0) {
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	} else {
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * 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 (re <= 580.0d0) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im_m)))
    else
        tmp = sin(re) * (1.0d0 + (im_m * (0.5d0 + (im_m * 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 (re <= 580.0) {
		tmp = re * (0.5 + (0.5 * Math.exp(im_m)));
	} else {
		tmp = Math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.25))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 580.0:
		tmp = re * (0.5 + (0.5 * math.exp(im_m)))
	else:
		tmp = math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.25))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 580.0)
		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 * 0.25)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 580.0)
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	else
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 580.0], 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 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 580

   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. +-commutative 100.0%

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

      2. distribute-rgt-in 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 75.1%

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

   6. Taylor expanded in re around 0 55.3%

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

   if 580 < re

   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. +-commutative 100.0%

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

      2. distribute-rgt-in 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 73.9%

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

   6. Taylor expanded in im around 0 75.7%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 580:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 550:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 550.0)
   (* re (+ 0.5 (* 0.5 (exp im_m))))
   (* (sin re) (+ 1.0 (* im_m (+ 0.5 (* im_m 0.3333333333333333)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 550.0) {
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	} else {
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.3333333333333333))));
	}
	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 (re <= 550.0d0) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im_m)))
    else
        tmp = sin(re) * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.3333333333333333d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 550.0) {
		tmp = re * (0.5 + (0.5 * Math.exp(im_m)));
	} else {
		tmp = Math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.3333333333333333))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 550.0:
		tmp = re * (0.5 + (0.5 * math.exp(im_m)))
	else:
		tmp = math.sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.3333333333333333))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 550.0)
		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 * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 550.0)
		tmp = re * (0.5 + (0.5 * exp(im_m)));
	else
		tmp = sin(re) * (1.0 + (im_m * (0.5 + (im_m * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 550.0], 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 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 550

   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. +-commutative 100.0%

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

      2. distribute-rgt-in 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 75.1%

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

   6. Taylor expanded in re around 0 55.3%

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

   if 550 < re

   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. +-commutative 100.0%

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

      2. distribute-rgt-in 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 73.9%

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

   6. Taylor expanded in im around 0 61.2%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 61.2%

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

   8. Simplified 61.2%

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

   10. Applied egg-rr 75.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 550:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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) * (1.0d0 + (im_m * (0.5d0 + (im_m * (0.25d0 + (im_m * 0.08333333333333333d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * N[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in im around 0 66.3%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.3%

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

8. Simplified 66.3%

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

9. Final simplification 66.3%

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

Alternative 7: 71.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im\_m}^{2}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 6.8e+14) (sin re) (* re (* 0.5 (pow im_m 2.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 6.8e+14) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 * pow(im_m, 2.0));
	}
	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 <= 6.8d+14) then
        tmp = sin(re)
    else
        tmp = re * (0.5d0 * (im_m ** 2.0d0))
    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 <= 6.8e+14) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.5 * Math.pow(im_m, 2.0));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 6.8e+14:
		tmp = math.sin(re)
	else:
		tmp = re * (0.5 * math.pow(im_m, 2.0))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 6.8e+14)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 * (im_m ^ 2.0)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 6.8e+14)
		tmp = sin(re);
	else
		tmp = re * (0.5 * (im_m ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 6.8e+14], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 * N[Power[im$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.8e14

   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. +-commutative 100.0%

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

      2. distribute-rgt-in 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 68.3%

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

   if 6.8e14 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 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. Add Preprocessing

   5. Taylor expanded in im around 0 47.7%

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

      1. +-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. fma-define 47.7%

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

   7. Simplified 47.7%

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

   8. Taylor expanded in re around 0 52.0%

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

      1. *-commutative 52.0%

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

      2. *-commutative 52.0%

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

      3. associate-*l* 52.0%

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

      4. +-commutative 52.0%

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

      5. unpow2 52.0%

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

      6. fma-undefine 52.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(re \cdot 0.5\right) \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5\right)} \]

   11. Taylor expanded in im around inf 52.0%

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

       1. associate-*r* 52.0%

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

       2. *-commutative 52.0%

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

   13. Simplified 52.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re * N[(0.5 + N[(0.5 * N[Exp[im$95$m], $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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in re around 0 45.0%

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

7. Final simplification 45.0%

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

Alternative 9: 50.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sin(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 = sin(re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sin[re], $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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 55.2%

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

6. Final simplification 55.2%

   \[\leadsto \sin re \]
7. Add Preprocessing

Alternative 10: 52.2% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

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 * (0.25d0 + (im_m * 0.08333333333333333d0))))))
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 * (0.25 + (im_m * 0.08333333333333333))))));
}

Python

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

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(0.25 + Float64(im_m * 0.08333333333333333)))))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = re * (1.0 + (im_m * (0.5 + (im_m * (0.25 + (im_m * 0.08333333333333333))))));
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[(0.25 + N[(im$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in im around 0 66.3%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.3%

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

8. Simplified 66.3%

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

9. Taylor expanded in re around 0 45.8%

   \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 45.8%

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

11. Simplified 45.8%

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

12. Final simplification 45.8%

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

Alternative 11: 32.2% accurate, 44.1× speedup

Math

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

FPCore

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

C

im_m = fabs(im);
double code(double re, double im_m) {
	return re * (1.0 + (0.5 * 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 * (1.0d0 + (0.5d0 * im_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(re * N[(1.0 + N[(0.5 * im$95$m), $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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in im around 0 54.3%

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

7. Taylor expanded in re around 0 35.7%

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

   1. *-commutative 35.7%

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

9. Simplified 35.7%

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

10. Final simplification 35.7%

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

Alternative 12: 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 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-rgt-in 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

6. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.5 \cdot \sin re\right)} - -1} \]
7. Step-by-step derivation

   1. sub-neg 4.3%

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

   2. metadata-eval 4.3%

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

   3. +-commutative 4.3%

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

   4. log1p-undefine 4.3%

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

   5. rem-exp-log 4.3%

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

   6. associate-+r+ 4.3%

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

   7. metadata-eval 4.3%

      \[\leadsto \color{blue}{2} + -0.5 \cdot \sin re \]

8. Simplified 4.3%

   \[\leadsto \color{blue}{2 + -0.5 \cdot \sin re} \]
9. Step-by-step derivation

   1. +-commutative 4.3%

      \[\leadsto \color{blue}{-0.5 \cdot \sin re + 2} \]

   2. *-un-lft-identity 4.3%

      \[\leadsto \color{blue}{1 \cdot \left(-0.5 \cdot \sin re\right)} + 2 \]

   3. fma-define 4.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, -0.5 \cdot \sin re, 2\right)} \]

   4. add-sqr-sqrt 0.7%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\sqrt{-0.5 \cdot \sin re} \cdot \sqrt{-0.5 \cdot \sin re}}, 2\right) \]

   5. sqrt-unprod 4.2%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\sqrt{\left(-0.5 \cdot \sin re\right) \cdot \left(-0.5 \cdot \sin re\right)}}, 2\right) \]

   6. swap-sqr 4.2%

      \[\leadsto \mathsf{fma}\left(1, \sqrt{\color{blue}{\left(-0.5 \cdot -0.5\right) \cdot \left(\sin re \cdot \sin re\right)}}, 2\right) \]

   7. metadata-eval 4.2%

      \[\leadsto \mathsf{fma}\left(1, \sqrt{\color{blue}{0.25} \cdot \left(\sin re \cdot \sin re\right)}, 2\right) \]

   8. metadata-eval 4.2%

      \[\leadsto \mathsf{fma}\left(1, \sqrt{\color{blue}{\left(0.5 \cdot 0.5\right)} \cdot \left(\sin re \cdot \sin re\right)}, 2\right) \]

   9. swap-sqr 4.2%

      \[\leadsto \mathsf{fma}\left(1, \sqrt{\color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(0.5 \cdot \sin re\right)}}, 2\right) \]

   10. sqrt-unprod 3.4%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\sqrt{0.5 \cdot \sin re} \cdot \sqrt{0.5 \cdot \sin re}}, 2\right) \]

   11. add-sqr-sqrt 4.2%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{0.5 \cdot \sin re}, 2\right) \]

   12. *-commutative 4.2%

       \[\leadsto \mathsf{fma}\left(1, \color{blue}{\sin re \cdot 0.5}, 2\right) \]

10. Applied egg-rr 4.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \sin re \cdot 0.5, 2\right)} \]
11. Step-by-step derivation

    1. fma-undefine 4.2%

       \[\leadsto \color{blue}{1 \cdot \left(\sin re \cdot 0.5\right) + 2} \]

    2. *-lft-identity 4.2%

       \[\leadsto \color{blue}{\sin re \cdot 0.5} + 2 \]

    3. +-commutative 4.2%

       \[\leadsto \color{blue}{2 + \sin re \cdot 0.5} \]

    4. *-commutative 4.2%

       \[\leadsto 2 + \color{blue}{0.5 \cdot \sin re} \]

12. Simplified 4.2%

    \[\leadsto \color{blue}{2 + 0.5 \cdot \sin re} \]

14. Applied egg-rr 3.0%

    \[\leadsto \color{blue}{\sin re - \sin re} \]
15. Step-by-step derivation

    1. +-inverses 3.0%

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

16. Simplified 3.0%

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

17. Final simplification 3.0%

    \[\leadsto 0 \]
18. Add Preprocessing

Alternative 13: 3.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.0005787037037037037 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 0.0005787037037037037)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.0005787037037037037;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.0005787037037037037d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.0005787037037037037;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.0005787037037037037

Julia

im_m = abs(im)
function code(re, im_m)
	return 0.0005787037037037037
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.0005787037037037037;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 0.0005787037037037037

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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in im around 0 66.3%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.3%

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

8. Simplified 66.3%

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

10. Applied egg-rr 3.8%

    \[\leadsto \color{blue}{\frac{\sin re \cdot 0.0005787037037037037}{\sin re + \left(\sin re \cdot 0.0005787037037037037 - \sin re \cdot 0.0005787037037037037\right)}} \]
11. Step-by-step derivation

    1. *-commutative 3.8%

       \[\leadsto \frac{\color{blue}{0.0005787037037037037 \cdot \sin re}}{\sin re + \left(\sin re \cdot 0.0005787037037037037 - \sin re \cdot 0.0005787037037037037\right)} \]

    2. +-inverses 3.8%

       \[\leadsto \frac{0.0005787037037037037 \cdot \sin re}{\sin re + \color{blue}{0}} \]

    3. +-rgt-identity 3.8%

       \[\leadsto \frac{0.0005787037037037037 \cdot \sin re}{\color{blue}{\sin re}} \]

    4. associate-/l* 3.8%

       \[\leadsto \color{blue}{0.0005787037037037037 \cdot \frac{\sin re}{\sin re}} \]

    5. *-inverses 3.8%

       \[\leadsto 0.0005787037037037037 \cdot \color{blue}{1} \]

    6. metadata-eval 3.8%

       \[\leadsto \color{blue}{0.0005787037037037037} \]

12. Simplified 3.8%

    \[\leadsto \color{blue}{0.0005787037037037037} \]

13. Final simplification 3.8%

    \[\leadsto 0.0005787037037037037 \]
14. Add Preprocessing

Alternative 14: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 1.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 1.0

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. +-commutative 100.0%

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

   2. distribute-rgt-in 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 74.8%

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

6. Taylor expanded in im around 0 66.3%

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot \left(0.25 + 0.08333333333333333 \cdot im\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.3%

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

8. Simplified 66.3%

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

10. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{\frac{\sin re - \sin re \cdot 0.0005787037037037037}{\sin re - \sin re \cdot 0.0005787037037037037}} \]
11. Step-by-step derivation

    1. *-inverses 4.6%

       \[\leadsto \color{blue}{1} \]

12. Simplified 4.6%

    \[\leadsto \color{blue}{1} \]

13. Final simplification 4.6%

    \[\leadsto 1 \]
14. Add Preprocessing

Alternative 15: 26.6% 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 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. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 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. Add Preprocessing

5. Taylor expanded in im around 0 77.3%

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

   1. +-commutative 77.3%

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

   2. unpow2 77.3%

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

   3. fma-define 77.3%

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

7. Simplified 77.3%

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

8. Taylor expanded in re around 0 51.0%

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

   1. *-commutative 51.0%

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

   2. *-commutative 51.0%

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

   3. associate-*l* 51.0%

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

   4. +-commutative 51.0%

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

   5. unpow2 51.0%

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

   6. fma-undefine 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \cdot \left(re \cdot 0.5\right) \]

10. Simplified 51.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(im, im, 2\right) \cdot \left(re \cdot 0.5\right)} \]

11. Taylor expanded in im around 0 30.3%

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

12. Final simplification 30.3%

    \[\leadsto re \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))