math.cos on complex, imaginary part

Percentage Accurate: 65.2% → 99.1%

Time: 9.7s

Alternatives: 14

Speedup: 2.8×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-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(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{-im\_m} - e^{im\_m} \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\_m\right) - e^{im\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (- (exp (- im_m)) (exp im_m)) -2e+64)
    (* (* 0.5 (sin re)) (- (- 1.0 im_m) (exp im_m)))
    (* im_m (- (sin re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(-im_m) - exp(im_m)) <= -2e+64) {
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	} else {
		tmp = im_m * -sin(re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(-im_m) - exp(im_m)) <= (-2d+64)) then
        tmp = (0.5d0 * sin(re)) * ((1.0d0 - im_m) - exp(im_m))
    else
        tmp = im_m * -sin(re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(-im_m) - Math.exp(im_m)) <= -2e+64) {
		tmp = (0.5 * Math.sin(re)) * ((1.0 - im_m) - Math.exp(im_m));
	} else {
		tmp = im_m * -Math.sin(re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(-im_m) - math.exp(im_m)) <= -2e+64:
		tmp = (0.5 * math.sin(re)) * ((1.0 - im_m) - math.exp(im_m))
	else:
		tmp = im_m * -math.sin(re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(Float64(-im_m)) - exp(im_m)) <= -2e+64)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(1.0 - im_m) - exp(im_m)));
	else
		tmp = Float64(im_m * Float64(-sin(re)));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(-im_m) - exp(im_m)) <= -2e+64)
		tmp = (0.5 * sin(re)) * ((1.0 - im_m) - exp(im_m));
	else
		tmp = im_m * -sin(re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision], -2e+64], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - im$95$m), $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -2.00000000000000004e64

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   if -2.00000000000000004e64 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(1 - im\right) - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.5:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 10^{+103}:\\ \;\;\;\;\left(\left(-im\_m\right) - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5.5)
    (* im_m (- (sin re)))
    (if (<= im_m 1e+103)
      (* (- (- im_m) (expm1 im_m)) (* 0.5 re))
      (*
       (* 0.5 (sin re))
       (* im_m (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.5) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1e+103) {
		tmp = (-im_m - expm1(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.5) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1e+103) {
		tmp = (-im_m - Math.expm1(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.5:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1e+103:
		tmp = (-im_m - math.expm1(im_m)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.5)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1e+103)
		tmp = Float64(Float64(Float64(-im_m) - expm1(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.5], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1e+103], N[(N[((-im$95$m) - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.5

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

   if 5.5 < im < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 83.3%

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

      1. associate-*r* 83.3%

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

      2. *-commutative 83.3%

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

      3. +-commutative 83.3%

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

      4. associate--r+ 83.3%

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

      5. sub-neg 83.3%

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

      6. +-commutative 83.3%

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

      7. neg-sub0 83.3%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 83.3%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 83.3%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 83.3%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

   if 1e103 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.5:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 10^{+103}:\\ \;\;\;\;\left(\left(-im\right) - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 5.6:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(-im\_m\right) - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im\_m \cdot \left(im\_m \cdot -0.5 - 2\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 5.6)
    (* im_m (- (sin re)))
    (if (<= im_m 1.9e+154)
      (* (- (- im_m) (expm1 im_m)) (* 0.5 re))
      (* (* 0.5 (sin re)) (* im_m (- (* im_m -0.5) 2.0)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.6) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (-im_m - expm1(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (im_m * ((im_m * -0.5) - 2.0));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 5.6) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (-im_m - Math.expm1(im_m)) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (im_m * ((im_m * -0.5) - 2.0));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 5.6:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.9e+154:
		tmp = (-im_m - math.expm1(im_m)) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (im_m * ((im_m * -0.5) - 2.0))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 5.6)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.9e+154)
		tmp = Float64(Float64(Float64(-im_m) - expm1(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im_m * Float64(Float64(im_m * -0.5) - 2.0)));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 5.6], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.9e+154], N[(N[((-im$95$m) - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im$95$m * N[(N[(im$95$m * -0.5), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 5.5999999999999996

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

   if 5.5999999999999996 < im < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. *-commutative 75.9%

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

      3. +-commutative 75.9%

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

      4. associate--r+ 75.9%

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

      5. sub-neg 75.9%

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

      6. +-commutative 75.9%

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

      7. neg-sub0 75.9%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 75.9%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 75.9%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 75.9%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 75.9%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

   if 1.8999999999999999e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around 0 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(-im\right) - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot \left(im \cdot -0.5 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 66000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im\_m \leq 1.05 \cdot 10^{+98}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im\_m \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(re \cdot -0.25 + im\_m \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im\_m \cdot re\right)\right)\right) - re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 66000000.0)
    (* im_m (- (sin re)))
    (if (<= im_m 1.05e+98)
      (* 0.16666666666666666 (* im_m (pow re 3.0)))
      (*
       im_m
       (-
        (*
         im_m
         (+
          (* re -0.25)
          (*
           im_m
           (+
            (* re -0.08333333333333333)
            (* -0.020833333333333332 (* im_m re))))))
        re))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 66000000.0) {
		tmp = im_m * -sin(re);
	} else if (im_m <= 1.05e+98) {
		tmp = 0.16666666666666666 * (im_m * pow(re, 3.0));
	} else {
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 66000000.0d0) then
        tmp = im_m * -sin(re)
    else if (im_m <= 1.05d+98) then
        tmp = 0.16666666666666666d0 * (im_m * (re ** 3.0d0))
    else
        tmp = im_m * ((im_m * ((re * (-0.25d0)) + (im_m * ((re * (-0.08333333333333333d0)) + ((-0.020833333333333332d0) * (im_m * re)))))) - re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 66000000.0) {
		tmp = im_m * -Math.sin(re);
	} else if (im_m <= 1.05e+98) {
		tmp = 0.16666666666666666 * (im_m * Math.pow(re, 3.0));
	} else {
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 66000000.0:
		tmp = im_m * -math.sin(re)
	elif im_m <= 1.05e+98:
		tmp = 0.16666666666666666 * (im_m * math.pow(re, 3.0))
	else:
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 66000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	elseif (im_m <= 1.05e+98)
		tmp = Float64(0.16666666666666666 * Float64(im_m * (re ^ 3.0)));
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(Float64(re * -0.25) + Float64(im_m * Float64(Float64(re * -0.08333333333333333) + Float64(-0.020833333333333332 * Float64(im_m * re)))))) - re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 66000000.0)
		tmp = im_m * -sin(re);
	elseif (im_m <= 1.05e+98)
		tmp = 0.16666666666666666 * (im_m * (re ^ 3.0));
	else
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 66000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im$95$m, 1.05e+98], N[(0.16666666666666666 * N[(im$95$m * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * N[(N[(re * -0.25), $MachinePrecision] + N[(im$95$m * N[(N[(re * -0.08333333333333333), $MachinePrecision] + N[(-0.020833333333333332 * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 6.6e7

   1. Initial program 52.6%

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

   3. Taylor expanded in im around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. neg-mul-1 66.2%

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

   5. Simplified 66.2%

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

   if 6.6e7 < im < 1.05000000000000002e98

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 2.8%

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

      1. associate-*r* 2.8%

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

      2. neg-mul-1 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in re around 0 21.7%

      \[\leadsto \color{blue}{re \cdot \left(-1 \cdot im + 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 21.7%

         \[\leadsto re \cdot \left(\color{blue}{\left(-im\right)} + 0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \]

      2. +-commutative 21.7%

         \[\leadsto re \cdot \color{blue}{\left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right) + \left(-im\right)\right)} \]

      3. distribute-lft-in 15.0%

         \[\leadsto \color{blue}{re \cdot \left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) + re \cdot \left(-im\right)} \]

      4. *-commutative 15.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot re} + re \cdot \left(-im\right) \]

      5. distribute-rgt-neg-in 15.0%

         \[\leadsto \left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot re + \color{blue}{\left(-re \cdot im\right)} \]

      6. *-commutative 15.0%

         \[\leadsto \left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot re + \left(-\color{blue}{im \cdot re}\right) \]

      7. unsub-neg 15.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(im \cdot {re}^{2}\right)\right) \cdot re - im \cdot re} \]

      8. associate-*r* 15.0%

         \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot im\right) \cdot {re}^{2}\right)} \cdot re - im \cdot re \]

      9. associate-*l* 15.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot im\right) \cdot \left({re}^{2} \cdot re\right)} - im \cdot re \]

      10. *-commutative 15.0%

          \[\leadsto \color{blue}{\left(im \cdot 0.16666666666666666\right)} \cdot \left({re}^{2} \cdot re\right) - im \cdot re \]

      11. pow-plus 15.0%

          \[\leadsto \left(im \cdot 0.16666666666666666\right) \cdot \color{blue}{{re}^{\left(2 + 1\right)}} - im \cdot re \]

      12. metadata-eval 15.0%

          \[\leadsto \left(im \cdot 0.16666666666666666\right) \cdot {re}^{\color{blue}{3}} - im \cdot re \]

   8. Simplified 15.0%

      \[\leadsto \color{blue}{\left(im \cdot 0.16666666666666666\right) \cdot {re}^{3} - im \cdot re} \]

   9. Taylor expanded in re around inf 20.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)} \]

   if 1.05000000000000002e98 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. *-commutative 69.8%

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

      3. +-commutative 69.8%

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

      4. associate--r+ 69.8%

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

      5. sub-neg 69.8%

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

      6. +-commutative 69.8%

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

      7. neg-sub0 69.8%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 69.8%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 69.8%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 69.8%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 69.8%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

   9. Taylor expanded in im around 0 64.5%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + im \cdot \left(-0.25 \cdot re + im \cdot \left(-0.08333333333333333 \cdot re + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 66000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+98}:\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot -0.25 + im \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right) - re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(-im\_m\right) - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(\sin re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= re 4e-8)
    (* (- (- im_m) (expm1 im_m)) (* 0.5 re))
    (* im_m (* (sin re) (+ -1.0 (* im_m -0.25)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 4e-8) {
		tmp = (-im_m - expm1(im_m)) * (0.5 * re);
	} else {
		tmp = im_m * (sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (re <= 4e-8) {
		tmp = (-im_m - Math.expm1(im_m)) * (0.5 * re);
	} else {
		tmp = im_m * (Math.sin(re) * (-1.0 + (im_m * -0.25)));
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if re <= 4e-8:
		tmp = (-im_m - math.expm1(im_m)) * (0.5 * re)
	else:
		tmp = im_m * (math.sin(re) * (-1.0 + (im_m * -0.25)))
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (re <= 4e-8)
		tmp = Float64(Float64(Float64(-im_m) - expm1(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64(im_m * Float64(sin(re) * Float64(-1.0 + Float64(im_m * -0.25))));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[re, 4e-8], N[(N[((-im$95$m) - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(im$95$m * N[(N[Sin[re], $MachinePrecision] * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if re < 4.0000000000000001e-8

   1. Initial program 71.2%

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

   3. Taylor expanded in im around 0 42.8%

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

      1. neg-mul-1 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in re around 0 39.3%

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

      1. associate-*r* 39.3%

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

      2. *-commutative 39.3%

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

      3. +-commutative 39.3%

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

      4. associate--r+ 40.6%

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

      5. sub-neg 40.6%

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

      6. +-commutative 40.6%

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

      7. neg-sub0 40.6%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 40.6%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 51.4%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 51.4%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 51.4%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

   if 4.0000000000000001e-8 < re

   1. Initial program 43.7%

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

   3. Taylor expanded in im around 0 33.6%

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

      1. neg-mul-1 33.6%

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

      2. unsub-neg 33.6%

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

   5. Simplified 33.6%

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

   6. Taylor expanded in im around 0 80.4%

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

      1. associate-*r* 80.4%

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

      2. distribute-rgt-out 80.4%

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

      3. *-commutative 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(-im\right) - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\sin re \cdot \left(-1 + im \cdot -0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 4.8:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-im\_m\right) - \mathsf{expm1}\left(im\_m\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 4.8)
    (* im_m (- (sin re)))
    (* (- (- im_m) (expm1 im_m)) (* 0.5 re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = im_m * -sin(re);
	} else {
		tmp = (-im_m - expm1(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = (-im_m - Math.expm1(im_m)) * (0.5 * re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 4.8:
		tmp = im_m * -math.sin(re)
	else:
		tmp = (-im_m - math.expm1(im_m)) * (0.5 * re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 4.8)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(Float64(Float64(-im_m) - expm1(im_m)) * Float64(0.5 * re));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 4.8], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(N[((-im$95$m) - N[(Exp[im$95$m] - 1), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 4.79999999999999982

   1. Initial program 52.3%

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

   3. Taylor expanded in im around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. neg-mul-1 66.5%

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

   5. Simplified 66.5%

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

   if 4.79999999999999982 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

      3. +-commutative 72.5%

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

      4. associate--r+ 72.5%

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

      5. sub-neg 72.5%

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

      6. +-commutative 72.5%

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

      7. neg-sub0 72.5%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 72.5%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 72.5%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 72.5%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 72.5%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-im\right) - \mathsf{expm1}\left(im\right)\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 150000000:\\ \;\;\;\;im\_m \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(im\_m \cdot \left(re \cdot -0.25 + im\_m \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im\_m \cdot re\right)\right)\right) - re\right)\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 150000000.0)
    (* im_m (- (sin re)))
    (*
     im_m
     (-
      (*
       im_m
       (+
        (* re -0.25)
        (*
         im_m
         (+
          (* re -0.08333333333333333)
          (* -0.020833333333333332 (* im_m re))))))
      re)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 150000000.0) {
		tmp = im_m * -sin(re);
	} else {
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	}
	return im_s * tmp;
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 150000000.0d0) then
        tmp = im_m * -sin(re)
    else
        tmp = im_m * ((im_m * ((re * (-0.25d0)) + (im_m * ((re * (-0.08333333333333333d0)) + ((-0.020833333333333332d0) * (im_m * re)))))) - re)
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if (im_m <= 150000000.0) {
		tmp = im_m * -Math.sin(re);
	} else {
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if im_m <= 150000000.0:
		tmp = im_m * -math.sin(re)
	else:
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re)
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (im_m <= 150000000.0)
		tmp = Float64(im_m * Float64(-sin(re)));
	else
		tmp = Float64(im_m * Float64(Float64(im_m * Float64(Float64(re * -0.25) + Float64(im_m * Float64(Float64(re * -0.08333333333333333) + Float64(-0.020833333333333332 * Float64(im_m * re)))))) - re));
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if (im_m <= 150000000.0)
		tmp = im_m * -sin(re);
	else
		tmp = im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re);
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[im$95$m, 150000000.0], N[(im$95$m * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im$95$m * N[(N[(im$95$m * N[(N[(re * -0.25), $MachinePrecision] + N[(im$95$m * N[(N[(re * -0.08333333333333333), $MachinePrecision] + N[(-0.020833333333333332 * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.5e8

   1. Initial program 52.8%

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

   3. Taylor expanded in im around 0 65.8%

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

      1. associate-*r* 65.8%

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

      2. neg-mul-1 65.8%

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

   5. Simplified 65.8%

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

   if 1.5e8 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. *-commutative 73.1%

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

      3. +-commutative 73.1%

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

      4. associate--r+ 73.1%

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

      5. sub-neg 73.1%

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

      6. +-commutative 73.1%

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

      7. neg-sub0 73.1%

         \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      8. associate-+l- 73.1%

         \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

      9. expm1-undefine 73.1%

         \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

      10. sub0-neg 73.1%

          \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. Simplified 73.1%

      \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

   9. Taylor expanded in im around 0 53.2%

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + im \cdot \left(-0.25 \cdot re + im \cdot \left(-0.08333333333333333 \cdot re + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 150000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(im \cdot \left(re \cdot -0.25 + im \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right) - re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(im\_m \cdot \left(re \cdot -0.25 + im\_m \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im\_m \cdot re\right)\right)\right) - re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   im_m
   (-
    (*
     im_m
     (+
      (* re -0.25)
      (*
       im_m
       (+ (* re -0.08333333333333333) (* -0.020833333333333332 (* im_m re))))))
    re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * ((im_m * ((re * (-0.25d0)) + (im_m * ((re * (-0.08333333333333333d0)) + ((-0.020833333333333332d0) * (im_m * re)))))) - re))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(Float64(im_m * Float64(Float64(re * -0.25) + Float64(im_m * Float64(Float64(re * -0.08333333333333333) + Float64(-0.020833333333333332 * Float64(im_m * re)))))) - re)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * ((im_m * ((re * -0.25) + (im_m * ((re * -0.08333333333333333) + (-0.020833333333333332 * (im_m * re)))))) - re));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(N[(im$95$m * N[(N[(re * -0.25), $MachinePrecision] + N[(im$95$m * N[(N[(re * -0.08333333333333333), $MachinePrecision] + N[(-0.020833333333333332 * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 40.8%

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

   1. neg-mul-1 40.8%

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

   2. unsub-neg 40.8%

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

5. Simplified 40.8%

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

6. Taylor expanded in re around 0 35.0%

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

   1. associate-*r* 35.0%

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

   2. *-commutative 35.0%

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

   3. +-commutative 35.0%

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

   4. associate--r+ 35.9%

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

   5. sub-neg 35.9%

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

   6. +-commutative 35.9%

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

   7. neg-sub0 35.9%

      \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. associate-+l- 35.9%

      \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   9. expm1-undefine 44.7%

      \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   10. sub0-neg 44.7%

       \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

8. Simplified 44.7%

   \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

9. Taylor expanded in im around 0 40.8%

   \[\leadsto \color{blue}{im \cdot \left(-1 \cdot re + im \cdot \left(-0.25 \cdot re + im \cdot \left(-0.08333333333333333 \cdot re + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right)\right)} \]

10. Final simplification 40.8%

    \[\leadsto im \cdot \left(im \cdot \left(re \cdot -0.25 + im \cdot \left(re \cdot -0.08333333333333333 + -0.020833333333333332 \cdot \left(im \cdot re\right)\right)\right) - re\right) \]
11. Add Preprocessing

Alternative 9: 49.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0.5 \cdot \left(im\_m \cdot \left(re \cdot \left(im\_m \cdot \left(im\_m \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (*
   0.5
   (* im_m (* re (- (* im_m (- (* im_m -0.16666666666666666) 0.5)) 2.0))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (0.5d0 * (im_m * (re * ((im_m * ((im_m * (-0.16666666666666666d0)) - 0.5d0)) - 2.0d0))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(0.5 * Float64(im_m * Float64(re * Float64(Float64(im_m * Float64(Float64(im_m * -0.16666666666666666) - 0.5)) - 2.0)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (0.5 * (im_m * (re * ((im_m * ((im_m * -0.16666666666666666) - 0.5)) - 2.0))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(0.5 * N[(im$95$m * N[(re * N[(N[(im$95$m * N[(N[(im$95$m * -0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 40.8%

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

   1. neg-mul-1 40.8%

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

   2. unsub-neg 40.8%

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

5. Simplified 40.8%

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

6. Taylor expanded in im around 0 82.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)} \]

7. Taylor expanded in re around 0 46.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(-0.16666666666666666 \cdot im - 0.5\right) - 2\right)\right)\right)} \]

8. Final simplification 46.3%

   \[\leadsto 0.5 \cdot \left(im \cdot \left(re \cdot \left(im \cdot \left(im \cdot -0.16666666666666666 - 0.5\right) - 2\right)\right)\right) \]
9. Add Preprocessing

Alternative 10: 48.9% accurate, 23.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(im\_m \cdot \left(re \cdot \left(-0.25 + im\_m \cdot -0.08333333333333333\right)\right) - re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (* im_m (- (* im_m (* re (+ -0.25 (* im_m -0.08333333333333333)))) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * (re * (-0.25 + (im_m * -0.08333333333333333)))) - re));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * ((im_m * (re * ((-0.25d0) + (im_m * (-0.08333333333333333d0))))) - re))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * ((im_m * (re * (-0.25 + (im_m * -0.08333333333333333)))) - re));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * ((im_m * (re * (-0.25 + (im_m * -0.08333333333333333)))) - re))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(Float64(im_m * Float64(re * Float64(-0.25 + Float64(im_m * -0.08333333333333333)))) - re)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * ((im_m * (re * (-0.25 + (im_m * -0.08333333333333333)))) - re));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(N[(im$95$m * N[(re * N[(-0.25 + N[(im$95$m * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 40.8%

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

   1. neg-mul-1 40.8%

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

   2. unsub-neg 40.8%

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

5. Simplified 40.8%

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

6. Taylor expanded in re around 0 35.0%

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

   1. associate-*r* 35.0%

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

   2. *-commutative 35.0%

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

   3. +-commutative 35.0%

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

   4. associate--r+ 35.9%

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

   5. sub-neg 35.9%

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

   6. +-commutative 35.9%

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

   7. neg-sub0 35.9%

      \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. associate-+l- 35.9%

      \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   9. expm1-undefine 44.7%

      \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   10. sub0-neg 44.7%

       \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

8. Simplified 44.7%

   \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

9. Taylor expanded in im around 0 45.9%

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

    1. +-commutative 45.9%

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

    2. mul-1-neg 45.9%

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

    3. unsub-neg 45.9%

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

    4. +-commutative 45.9%

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

    5. associate-*r* 45.9%

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

    6. distribute-rgt-out 45.9%

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

    7. *-commutative 45.9%

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

11. Simplified 45.9%

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

12. Final simplification 45.9%

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

Alternative 11: 41.9% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(-1 + im\_m \cdot -0.25\right) \cdot \left(im\_m \cdot re\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* (+ -1.0 (* im_m -0.25)) (* im_m re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * ((-1.0 + (im_m * -0.25)) * (im_m * re));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (((-1.0d0) + (im_m * (-0.25d0))) * (im_m * re))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * ((-1.0 + (im_m * -0.25)) * (im_m * re));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * ((-1.0 + (im_m * -0.25)) * (im_m * re))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(-1.0 + Float64(im_m * -0.25)) * Float64(im_m * re)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * ((-1.0 + (im_m * -0.25)) * (im_m * re));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 40.8%

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

   1. neg-mul-1 40.8%

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

   2. unsub-neg 40.8%

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

5. Simplified 40.8%

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

6. Taylor expanded in re around 0 35.0%

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

   1. associate-*r* 35.0%

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

   2. *-commutative 35.0%

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

   3. +-commutative 35.0%

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

   4. associate--r+ 35.9%

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

   5. sub-neg 35.9%

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

   6. +-commutative 35.9%

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

   7. neg-sub0 35.9%

      \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. associate-+l- 35.9%

      \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   9. expm1-undefine 44.7%

      \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   10. sub0-neg 44.7%

       \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

8. Simplified 44.7%

   \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

9. Taylor expanded in im around 0 36.7%

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

    1. distribute-lft-in 34.0%

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

    2. mul-1-neg 34.0%

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

    3. distribute-rgt-neg-in 34.0%

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

    4. mul-1-neg 34.0%

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

    5. associate-*r* 34.0%

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

    6. metadata-eval 34.0%

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

    7. associate-*l* 34.0%

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

    8. distribute-rgt-out 36.7%

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

    9. associate-*l* 36.7%

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

    10. metadata-eval 36.7%

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

11. Simplified 36.7%

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

12. Final simplification 36.7%

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

Alternative 12: 41.9% accurate, 34.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(im\_m \cdot \left(re \cdot \left(-1 + im\_m \cdot -0.25\right)\right)\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (* im_m (* re (+ -1.0 (* im_m -0.25))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * (-1.0 + (im_m * -0.25))));
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (im_m * (re * ((-1.0d0) + (im_m * (-0.25d0)))))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (im_m * (re * (-1.0 + (im_m * -0.25))));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (im_m * (re * (-1.0 + (im_m * -0.25))))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(im_m * Float64(re * Float64(-1.0 + Float64(im_m * -0.25)))))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (im_m * (re * (-1.0 + (im_m * -0.25))));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * N[(re * N[(-1.0 + N[(im$95$m * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 40.8%

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

   1. neg-mul-1 40.8%

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

   2. unsub-neg 40.8%

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

5. Simplified 40.8%

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

6. Taylor expanded in re around 0 35.0%

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

   1. associate-*r* 35.0%

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

   2. *-commutative 35.0%

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

   3. +-commutative 35.0%

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

   4. associate--r+ 35.9%

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

   5. sub-neg 35.9%

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

   6. +-commutative 35.9%

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

   7. neg-sub0 35.9%

      \[\leadsto \left(\left(\color{blue}{\left(0 - e^{im}\right)} + 1\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   8. associate-+l- 35.9%

      \[\leadsto \left(\color{blue}{\left(0 - \left(e^{im} - 1\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

   9. expm1-undefine 44.7%

      \[\leadsto \left(\left(0 - \color{blue}{\mathsf{expm1}\left(im\right)}\right) - im\right) \cdot \left(0.5 \cdot re\right) \]

   10. sub0-neg 44.7%

       \[\leadsto \left(\color{blue}{\left(-\mathsf{expm1}\left(im\right)\right)} - im\right) \cdot \left(0.5 \cdot re\right) \]

8. Simplified 44.7%

   \[\leadsto \color{blue}{\left(\left(-\mathsf{expm1}\left(im\right)\right) - im\right) \cdot \left(0.5 \cdot re\right)} \]

9. Taylor expanded in im around 0 36.7%

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

    1. associate-*r* 36.7%

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

    2. distribute-rgt-out 36.7%

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

    3. *-commutative 36.7%

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

11. Simplified 36.7%

    \[\leadsto \color{blue}{im \cdot \left(re \cdot \left(-1 + im \cdot -0.25\right)\right)} \]
12. Add Preprocessing

Alternative 13: 33.4% accurate, 77.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(\left(-im\_m\right) \cdot re\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (- im_m) re)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (-im_m * re);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (-im_m * re)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (-im_m * re);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (-im_m * re)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(Float64(-im_m) * re))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (-im_m * re);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[((-im$95$m) * re), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in im around 0 49.8%

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

   1. associate-*r* 49.8%

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

   2. neg-mul-1 49.8%

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

5. Simplified 49.8%

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

6. Taylor expanded in re around 0 28.8%

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

   1. associate-*r* 28.8%

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

   2. mul-1-neg 28.8%

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

8. Simplified 28.8%

   \[\leadsto \color{blue}{\left(-im\right) \cdot re} \]
9. Add Preprocessing

Alternative 14: 2.8% accurate, 102.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot 4\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* re 4.0)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (re * 4.0);
}

Fortran

im\_m = abs(im)
im\_s = copysign(1.0d0, im)
real(8) function code(im_s, re, im_m)
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (re * 4.0d0)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (re * 4.0);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (re * 4.0)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(re * 4.0))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (re * 4.0);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(re * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in re around 0 52.7%

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

   1. associate-*r* 52.7%

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

   2. *-commutative 52.7%

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

5. Simplified 52.7%

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

7. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{8} \cdot \left(0.5 \cdot re\right) \]

8. Taylor expanded in re around 0 3.0%

   \[\leadsto \color{blue}{4 \cdot re} \]
9. Step-by-step derivation

   1. *-commutative 3.0%

      \[\leadsto \color{blue}{re \cdot 4} \]

10. Simplified 3.0%

    \[\leadsto \color{blue}{re \cdot 4} \]
11. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024111 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))