math.cos on complex, imaginary part

Percentage Accurate: 65.8% → 99.3%

Time: 11.2s

Alternatives: 9

Speedup: 2.7×

• 49.7% 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: 65.8% 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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{-im\_m} - e^{im\_m}\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (- (exp (- im_m)) (exp im_m))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* t_0 (* 0.5 (sin re)))
      (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))))))

C

im_m = fabs(im);
im_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(-im_m) - exp(im_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	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 t_0 = Math.exp(-im_m) - Math.exp(im_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	return im_s * tmp;
}

Python

im_m = math.fabs(im)
im_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = math.exp(-im_m) - math.exp(im_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	return im_s * tmp

Julia

im_m = abs(im)
im_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(Float64(-im_m)) - exp(im_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	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)
	t_0 = exp(-im_m) - exp(im_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	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_] := Block[{t$95$0 = N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 88.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 88.4%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 88.4%

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

      3. associate-*r* 88.4%

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

      4. distribute-rgt-out 88.4%

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

      5. *-commutative 88.4%

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

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. +-commutative 88.4%

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

      2. unsub-neg 88.4%

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

   7. Applied egg-rr 88.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.085:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{-im\_m} - e^{im\_m}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 0.085)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 1.4e+100)
      (* (- (exp (- im_m)) (exp im_m)) (* 0.5 re))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

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 <= 0.085) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.4e+100) {
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 0.085d0) then
        tmp = sin(re) * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    else if (im_m <= 1.4d+100) then
        tmp = (exp(-im_m) - exp(im_m)) * (0.5d0 * re)
    else
        tmp = (im_m ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    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 <= 0.085) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.4e+100) {
		tmp = (Math.exp(-im_m) - Math.exp(im_m)) * (0.5 * re);
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 0.085:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 1.4e+100:
		tmp = (math.exp(-im_m) - math.exp(im_m)) * (0.5 * re)
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	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 <= 0.085)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 1.4e+100)
		tmp = Float64(Float64(exp(Float64(-im_m)) - exp(im_m)) * Float64(0.5 * re));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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 <= 0.085)
		tmp = sin(re) * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	elseif (im_m <= 1.4e+100)
		tmp = (exp(-im_m) - exp(im_m)) * (0.5 * re);
	else
		tmp = (im_m ^ 3.0) * (sin(re) * -0.16666666666666666);
	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, 0.085], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+100], N[(N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 0.0850000000000000061

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 88.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 88.4%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 88.4%

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

      3. associate-*r* 88.4%

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

      4. distribute-rgt-out 88.4%

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

      5. *-commutative 88.4%

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

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. +-commutative 88.4%

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

      2. unsub-neg 88.4%

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

   7. Applied egg-rr 88.4%

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

   if 0.0850000000000000061 < im < 1.3999999999999999e100

   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 re around 0 83.3%

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

      1. associate-*r* 83.3%

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

      2. *-commutative 83.3%

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

   5. Simplified 83.3%

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

   if 1.3999999999999999e100 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.085:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 7 \cdot 10^{+17}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im\_m \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 7e+17)
    (* (- im_m) (sin re))
    (if (<= im_m 1.4e+100)
      (log (/ 1.0 (+ 1.0 (expm1 (* im_m re)))))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

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 <= 7e+17) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.4e+100) {
		tmp = log((1.0 / (1.0 + expm1((im_m * re)))));
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 7e+17) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.4e+100) {
		tmp = Math.log((1.0 / (1.0 + Math.expm1((im_m * re)))));
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 7e+17:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.4e+100:
		tmp = math.log((1.0 / (1.0 + math.expm1((im_m * re)))))
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	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 <= 7e+17)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.4e+100)
		tmp = log(Float64(1.0 / Float64(1.0 + expm1(Float64(im_m * re)))));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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, 7e+17], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+100], N[Log[N[(1.0 / N[(1.0 + N[(Exp[N[(im$95$m * re), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 7e17

   1. Initial program 56.4%

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

   3. Taylor expanded in im around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. neg-mul-1 66.7%

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

   5. Simplified 66.7%

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

   if 7e17 < im < 1.3999999999999999e100

   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 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

   5. Simplified 3.0%

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

      1. distribute-lft-neg-out 3.0%

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

      2. add-sqr-sqrt 3.0%

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

      3. sqrt-unprod 3.0%

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

      4. sqr-neg 3.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.5%

         \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)} \]

      8. log1p-udef 0.8%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)} \]

      9. neg-log 0.8%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \sin re\right)}\right) \]

      11. sqrt-unprod 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \sin re\right)}\right) \]

      12. sqr-neg 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \sin re\right)}\right) \]

      13. sqrt-unprod 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sin re\right)}\right) \]

      14. add-sqr-sqrt 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \sin re\right)}\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sin re\right)}\right)} \]

   8. Taylor expanded in re around 0 30.2%

      \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im \cdot re}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 30.2%

         \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{re \cdot im}\right)}\right) \]

   10. Simplified 30.2%

       \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{re \cdot im}\right)}\right) \]

   if 1.3999999999999999e100 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 7 \cdot 10^{+17}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\sin re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \mathbf{elif}\;im\_m \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im\_m \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 1.95e+17)
    (* (sin re) (- (* (pow im_m 3.0) -0.16666666666666666) im_m))
    (if (<= im_m 1.4e+100)
      (log (/ 1.0 (+ 1.0 (expm1 (* im_m re)))))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

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 <= 1.95e+17) {
		tmp = sin(re) * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.4e+100) {
		tmp = log((1.0 / (1.0 + expm1((im_m * re)))));
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 1.95e+17) {
		tmp = Math.sin(re) * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	} else if (im_m <= 1.4e+100) {
		tmp = Math.log((1.0 / (1.0 + Math.expm1((im_m * re)))));
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 1.95e+17:
		tmp = math.sin(re) * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	elif im_m <= 1.4e+100:
		tmp = math.log((1.0 / (1.0 + math.expm1((im_m * re)))))
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	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 <= 1.95e+17)
		tmp = Float64(sin(re) * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	elseif (im_m <= 1.4e+100)
		tmp = log(Float64(1.0 / Float64(1.0 + expm1(Float64(im_m * re)))));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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, 1.95e+17], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.4e+100], N[Log[N[(1.0 / N[(1.0 + N[(Exp[N[(im$95$m * re), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 1.95e17

   1. Initial program 56.4%

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

   3. Taylor expanded in im around 0 88.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 88.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 88.0%

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

      3. associate-*r* 88.0%

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

      4. distribute-rgt-out 88.0%

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

      5. *-commutative 88.0%

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

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{3} \cdot -0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. +-commutative 88.0%

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

      2. unsub-neg 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 1.95e17 < im < 1.3999999999999999e100

   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 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

   5. Simplified 3.0%

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

      1. distribute-lft-neg-out 3.0%

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

      2. add-sqr-sqrt 3.0%

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

      3. sqrt-unprod 3.0%

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

      4. sqr-neg 3.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.5%

         \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)} \]

      8. log1p-udef 0.8%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)\right)} \]

      9. neg-log 0.8%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \sin re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \sin re\right)}\right) \]

      11. sqrt-unprod 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \sin re\right)}\right) \]

      12. sqr-neg 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \sin re\right)}\right) \]

      13. sqrt-unprod 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \sin re\right)}\right) \]

      14. add-sqr-sqrt 47.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \sin re\right)}\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \sin re\right)}\right)} \]

   8. Taylor expanded in re around 0 30.2%

      \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im \cdot re}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 30.2%

         \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{re \cdot im}\right)}\right) \]

   10. Simplified 30.2%

       \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{re \cdot im}\right)}\right) \]

   if 1.3999999999999999e100 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 480000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{re}^{3} \cdot \left(im\_m \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;{im\_m}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 480000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 5.5e+102)
      (* (pow re 3.0) (* im_m 0.16666666666666666))
      (* (pow im_m 3.0) (* (sin re) -0.16666666666666666))))))

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 <= 480000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 5.5e+102) {
		tmp = pow(re, 3.0) * (im_m * 0.16666666666666666);
	} else {
		tmp = pow(im_m, 3.0) * (sin(re) * -0.16666666666666666);
	}
	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 <= 480000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 5.5d+102) then
        tmp = (re ** 3.0d0) * (im_m * 0.16666666666666666d0)
    else
        tmp = (im_m ** 3.0d0) * (sin(re) * (-0.16666666666666666d0))
    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 <= 480000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 5.5e+102) {
		tmp = Math.pow(re, 3.0) * (im_m * 0.16666666666666666);
	} else {
		tmp = Math.pow(im_m, 3.0) * (Math.sin(re) * -0.16666666666666666);
	}
	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 <= 480000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 5.5e+102:
		tmp = math.pow(re, 3.0) * (im_m * 0.16666666666666666)
	else:
		tmp = math.pow(im_m, 3.0) * (math.sin(re) * -0.16666666666666666)
	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 <= 480000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 5.5e+102)
		tmp = Float64((re ^ 3.0) * Float64(im_m * 0.16666666666666666));
	else
		tmp = Float64((im_m ^ 3.0) * Float64(sin(re) * -0.16666666666666666));
	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 <= 480000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 5.5e+102)
		tmp = (re ^ 3.0) * (im_m * 0.16666666666666666);
	else
		tmp = (im_m ^ 3.0) * (sin(re) * -0.16666666666666666);
	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, 480000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 5.5e+102], N[(N[Power[re, 3.0], $MachinePrecision] * N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[Power[im$95$m, 3.0], $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.8e5

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   if 4.8e5 < im < 5.49999999999999981e102

   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 3.0%

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

      1. associate-*r* 3.0%

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

      2. neg-mul-1 3.0%

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

   5. Simplified 3.0%

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

   6. Taylor expanded in re around 0 1.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right) + 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 1.7%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right) + -1 \cdot \left(im \cdot re\right)} \]

      2. mul-1-neg 1.7%

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

      3. unsub-neg 1.7%

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

      4. *-commutative 1.7%

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

      5. associate-*l* 1.7%

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

   8. Simplified 1.7%

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

   9. Taylor expanded in re around inf 12.3%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 12.3%

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

       2. *-commutative 12.3%

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

       3. *-commutative 12.3%

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

   11. Simplified 12.3%

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

   if 5.49999999999999981e102 < 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%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;{re}^{3} \cdot \left(im \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot \left(\sin re \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 490000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{elif}\;im\_m \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;{re}^{3} \cdot \left(im\_m \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 490000.0)
    (* (- im_m) (sin re))
    (if (<= im_m 1.75e+157)
      (* (pow re 3.0) (* im_m 0.16666666666666666))
      (* (- im_m) 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 <= 490000.0) {
		tmp = -im_m * sin(re);
	} else if (im_m <= 1.75e+157) {
		tmp = pow(re, 3.0) * (im_m * 0.16666666666666666);
	} else {
		tmp = -im_m * 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 <= 490000.0d0) then
        tmp = -im_m * sin(re)
    else if (im_m <= 1.75d+157) then
        tmp = (re ** 3.0d0) * (im_m * 0.16666666666666666d0)
    else
        tmp = -im_m * 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 <= 490000.0) {
		tmp = -im_m * Math.sin(re);
	} else if (im_m <= 1.75e+157) {
		tmp = Math.pow(re, 3.0) * (im_m * 0.16666666666666666);
	} else {
		tmp = -im_m * 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 <= 490000.0:
		tmp = -im_m * math.sin(re)
	elif im_m <= 1.75e+157:
		tmp = math.pow(re, 3.0) * (im_m * 0.16666666666666666)
	else:
		tmp = -im_m * 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 <= 490000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	elseif (im_m <= 1.75e+157)
		tmp = Float64((re ^ 3.0) * Float64(im_m * 0.16666666666666666));
	else
		tmp = Float64(Float64(-im_m) * 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 <= 490000.0)
		tmp = -im_m * sin(re);
	elseif (im_m <= 1.75e+157)
		tmp = (re ^ 3.0) * (im_m * 0.16666666666666666);
	else
		tmp = -im_m * 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, 490000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im$95$m, 1.75e+157], N[(N[Power[re, 3.0], $MachinePrecision] * N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if im < 4.9e5

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   if 4.9e5 < im < 1.75000000000000001e157

   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 3.5%

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

      1. associate-*r* 3.5%

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

      2. neg-mul-1 3.5%

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

   5. Simplified 3.5%

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

   6. Taylor expanded in re around 0 2.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot re\right) + 0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 2.0%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right) + -1 \cdot \left(im \cdot re\right)} \]

      2. mul-1-neg 2.0%

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

      3. unsub-neg 2.0%

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

      4. *-commutative 2.0%

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

      5. associate-*l* 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in re around inf 18.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 18.8%

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

       2. *-commutative 18.8%

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

       3. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   if 1.75000000000000001e157 < 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 6.0%

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

      1. associate-*r* 6.0%

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

      2. neg-mul-1 6.0%

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

   5. Simplified 6.0%

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

   6. Taylor expanded in re around 0 18.8%

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

      1. associate-*r* 18.8%

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

      2. mul-1-neg 18.8%

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

   8. Simplified 18.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.75 \cdot 10^{+157}:\\ \;\;\;\;{re}^{3} \cdot \left(im \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 510000000000:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im\_m}^{3} \cdot -0.16666666666666666 - im\_m\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= im_m 510000000000.0)
    (* (- im_m) (sin re))
    (* re (- (* (pow im_m 3.0) -0.16666666666666666) im_m)))))

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 <= 510000000000.0) {
		tmp = -im_m * sin(re);
	} else {
		tmp = re * ((pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	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 <= 510000000000.0d0) then
        tmp = -im_m * sin(re)
    else
        tmp = re * (((im_m ** 3.0d0) * (-0.16666666666666666d0)) - im_m)
    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 <= 510000000000.0) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = re * ((Math.pow(im_m, 3.0) * -0.16666666666666666) - im_m);
	}
	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 <= 510000000000.0:
		tmp = -im_m * math.sin(re)
	else:
		tmp = re * ((math.pow(im_m, 3.0) * -0.16666666666666666) - im_m)
	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 <= 510000000000.0)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(re * Float64(Float64((im_m ^ 3.0) * -0.16666666666666666) - im_m));
	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 <= 510000000000.0)
		tmp = -im_m * sin(re);
	else
		tmp = re * (((im_m ^ 3.0) * -0.16666666666666666) - im_m);
	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, 510000000000.0], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(N[Power[im$95$m, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 5.1e11

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   if 5.1e11 < 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 74.0%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 74.0%

         \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) \]

      2. neg-mul-1 74.0%

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

      3. associate-*r* 74.0%

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

      4. distribute-rgt-out 74.0%

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

      5. *-commutative 74.0%

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

   5. Simplified 74.0%

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

   6. Taylor expanded in re around 0 59.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 510000000000:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ im_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\left(-im\_m\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\_m\right) \cdot re\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
im_s = (copysign.f64 1 im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= im_m 1.7e+16) (* (- im_m) (sin re)) (* (- im_m) 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 <= 1.7e+16) {
		tmp = -im_m * sin(re);
	} else {
		tmp = -im_m * 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 <= 1.7d+16) then
        tmp = -im_m * sin(re)
    else
        tmp = -im_m * 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 <= 1.7e+16) {
		tmp = -im_m * Math.sin(re);
	} else {
		tmp = -im_m * 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 <= 1.7e+16:
		tmp = -im_m * math.sin(re)
	else:
		tmp = -im_m * 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 <= 1.7e+16)
		tmp = Float64(Float64(-im_m) * sin(re));
	else
		tmp = Float64(Float64(-im_m) * 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 <= 1.7e+16)
		tmp = -im_m * sin(re);
	else
		tmp = -im_m * 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, 1.7e+16], N[((-im$95$m) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[((-im$95$m) * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 1.7e16

   1. Initial program 56.2%

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

   3. Taylor expanded in im around 0 67.1%

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

      1. associate-*r* 67.1%

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

      2. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

   if 1.7e16 < 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 4.7%

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

      1. associate-*r* 4.7%

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

      2. neg-mul-1 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in re around 0 15.2%

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

      1. associate-*r* 15.2%

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

      2. mul-1-neg 15.2%

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

   8. Simplified 15.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.7 \cdot 10^{+16}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.8% 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 1 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 67.5%

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

3. Taylor expanded in im around 0 51.0%

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

   1. associate-*r* 51.0%

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

   2. neg-mul-1 51.0%

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

5. Simplified 51.0%

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

6. Taylor expanded in re around 0 39.0%

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

   1. associate-*r* 39.0%

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

   2. mul-1-neg 39.0%

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

8. Simplified 39.0%

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

9. Final simplification 39.0%

   \[\leadsto \left(-im\right) \cdot re \]
10. 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 2024027 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (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))))