math.cos on complex, imaginary part

Percentage Accurate: 65.7% → 99.5%

Time: 10.7s

Alternatives: 12

Speedup: 2.7×

• 49.5% 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: 65.7% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-5)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-5)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e-5)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-5):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-5))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-5)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 5.00000000000000024e-5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 5.00000000000000024e-5

   1. Initial program 25.1%

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

   2. Taylor expanded in im around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 93.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -2.3 \cdot 10^{+15} \lor \neg \left(im \leq 0.155\right) \land im \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -8.2e+97)
   (* -0.16666666666666666 (* (sin re) (pow im 3.0)))
   (if (or (<= im -2.3e+15) (and (not (<= im 0.155)) (<= im 3.5e+94)))
     (* 0.5 (* (- (exp (- im)) (exp im)) re))
     (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -8.2e+97) {
		tmp = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	} else if ((im <= -2.3e+15) || (!(im <= 0.155) && (im <= 3.5e+94))) {
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-8.2d+97)) then
        tmp = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    else if ((im <= (-2.3d+15)) .or. (.not. (im <= 0.155d0)) .and. (im <= 3.5d+94)) then
        tmp = 0.5d0 * ((exp(-im) - exp(im)) * re)
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -8.2e+97) {
		tmp = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	} else if ((im <= -2.3e+15) || (!(im <= 0.155) && (im <= 3.5e+94))) {
		tmp = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	} else {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -8.2e+97:
		tmp = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	elif (im <= -2.3e+15) or (not (im <= 0.155) and (im <= 3.5e+94)):
		tmp = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	else:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -8.2e+97)
		tmp = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)));
	elseif ((im <= -2.3e+15) || (!(im <= 0.155) && (im <= 3.5e+94)))
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re));
	else
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -8.2e+97)
		tmp = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	elseif ((im <= -2.3e+15) || (~((im <= 0.155)) && (im <= 3.5e+94)))
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	else
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -8.2e+97], N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -2.3e+15], And[N[Not[LessEqual[im, 0.155]], $MachinePrecision], LessEqual[im, 3.5e+94]]], N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.19999999999999977e97

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*l* 97.9%

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

      5. distribute-lft-out-- 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in im around inf 97.9%

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

   if -8.19999999999999977e97 < im < -2.3e15 or 0.154999999999999999 < im < 3.4999999999999997e94

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 75.9%

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

   if -2.3e15 < im < 0.154999999999999999 or 3.4999999999999997e94 < im

   1. Initial program 45.8%

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

   2. Taylor expanded in im around 0 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-*l* 97.6%

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

      5. distribute-lft-out-- 97.6%

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

   4. Simplified 97.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -2.3 \cdot 10^{+15} \lor \neg \left(im \leq 0.155\right) \land im \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00027:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -8.2e+97)
     t_1
     (if (<= im -1.7e-5)
       t_0
       (if (<= im 0.00027)
         (* im (- (sin re)))
         (if (<= im 3.5e+94) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	double t_1 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -8.2e+97) {
		tmp = t_1;
	} else if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 0.00027) {
		tmp = im * -sin(re);
	} else if (im <= 3.5e+94) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * ((exp(-im) - exp(im)) * re)
    t_1 = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    if (im <= (-8.2d+97)) then
        tmp = t_1
    else if (im <= (-1.7d-5)) then
        tmp = t_0
    else if (im <= 0.00027d0) then
        tmp = im * -sin(re)
    else if (im <= 3.5d+94) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	double t_1 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -8.2e+97) {
		tmp = t_1;
	} else if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 0.00027) {
		tmp = im * -Math.sin(re);
	} else if (im <= 3.5e+94) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	t_1 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -8.2e+97:
		tmp = t_1
	elif im <= -1.7e-5:
		tmp = t_0
	elif im <= 0.00027:
		tmp = im * -math.sin(re)
	elif im <= 3.5e+94:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -8.2e+97)
		tmp = t_1;
	elseif (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 0.00027)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 3.5e+94)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((exp(-im) - exp(im)) * re);
	t_1 = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -8.2e+97)
		tmp = t_1;
	elseif (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 0.00027)
		tmp = im * -sin(re);
	elseif (im <= 3.5e+94)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -8.2e+97], t$95$1, If[LessEqual[im, -1.7e-5], t$95$0, If[LessEqual[im, 0.00027], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 3.5e+94], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -8.19999999999999977e97 or 3.4999999999999997e94 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 95.8%

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

      1. mul-1-neg 95.8%

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

      2. unsub-neg 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*l* 95.8%

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

      5. distribute-lft-out-- 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in im around inf 95.8%

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

   if -8.19999999999999977e97 < im < -1.7e-5 or 2.70000000000000003e-4 < im < 3.4999999999999997e94

   1. Initial program 99.6%

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

   2. Taylor expanded in re around 0 74.6%

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

   if -1.7e-5 < im < 2.70000000000000003e-4

   1. Initial program 24.6%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 0.00027:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{+94}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -680:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 310000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* (sin re) (pow im 3.0)))))
   (if (<= im -7.2e+102)
     t_0
     (if (<= im -680.0)
       (log1p (expm1 (* im re)))
       (if (<= im 310000000000.0)
         (* im (- (sin re)))
         (if (<= im 5.7e+102)
           (log (- (- 1.0 (* im re)) (* (* (* im im) -0.5) (* re re))))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	double tmp;
	if (im <= -7.2e+102) {
		tmp = t_0;
	} else if (im <= -680.0) {
		tmp = log1p(expm1((im * re)));
	} else if (im <= 310000000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 5.7e+102) {
		tmp = log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -7.2e+102) {
		tmp = t_0;
	} else if (im <= -680.0) {
		tmp = Math.log1p(Math.expm1((im * re)));
	} else if (im <= 310000000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 5.7e+102) {
		tmp = Math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -7.2e+102:
		tmp = t_0
	elif im <= -680.0:
		tmp = math.log1p(math.expm1((im * re)))
	elif im <= 310000000000.0:
		tmp = im * -math.sin(re)
	elif im <= 5.7e+102:
		tmp = math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -7.2e+102)
		tmp = t_0;
	elseif (im <= -680.0)
		tmp = log1p(expm1(Float64(im * re)));
	elseif (im <= 310000000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 5.7e+102)
		tmp = log(Float64(Float64(1.0 - Float64(im * re)) - Float64(Float64(Float64(im * im) * -0.5) * Float64(re * re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.2e+102], t$95$0, If[LessEqual[im, -680.0], N[Log[1 + N[(Exp[N[(im * re), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 310000000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5.7e+102], N[Log[N[(N[(1.0 - N[(im * re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.2000000000000003e102 or 5.6999999999999999e102 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

   if -7.2000000000000003e102 < im < -680

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. *-commutative 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   4. Simplified 3.0%

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

      1. distribute-rgt-neg-out 3.0%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 1.7%

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

      4. sqr-neg 1.7%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.8%

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

      7. log1p-expm1-u 0.5%

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

      8. log1p-udef 0.8%

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

      9. neg-log 0.8%

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

      10. add-sqr-sqrt 0.4%

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

      11. sqrt-unprod 23.0%

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

      12. sqr-neg 23.0%

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

      13. sqrt-unprod 22.6%

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

      14. add-sqr-sqrt 49.0%

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

   6. Applied egg-rr 49.0%

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

   7. Taylor expanded in re around 0 19.3%

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

      1. add-sqr-sqrt 11.9%

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

      2. sqrt-unprod 23.0%

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

      3. neg-log 23.0%

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

      4. log1p-udef 23.0%

         \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot im\right)\right)}\right) \cdot \log \left(\frac{1}{1 + \mathsf{expm1}\left(re \cdot im\right)}\right)} \]

      5. log1p-expm1-u 23.0%

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

      6. neg-log 23.0%

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

      7. log1p-udef 23.0%

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

      8. log1p-expm1-u 16.0%

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

      9. sqr-neg 16.0%

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

      10. sqrt-unprod 4.6%

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

      11. add-sqr-sqrt 6.2%

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

      12. log1p-expm1-u 30.1%

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

   9. Applied egg-rr 30.1%

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

   if -680 < im < 3.1e11

   1. Initial program 27.0%

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

   2. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

   if 3.1e11 < im < 5.6999999999999999e102

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.3%

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

      1. mul-1-neg 3.3%

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

      2. *-commutative 3.3%

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

      3. distribute-rgt-neg-in 3.3%

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

   4. Simplified 3.3%

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

      1. distribute-rgt-neg-out 3.3%

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

      2. add-sqr-sqrt 0.9%

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

      3. sqrt-unprod 1.2%

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

      4. sqr-neg 1.2%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.6%

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

      7. log1p-expm1-u 0.3%

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

      8. log1p-udef 0.4%

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

      9. neg-log 0.4%

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

      10. add-sqr-sqrt 0.2%

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

      11. sqrt-unprod 10.5%

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

      12. sqr-neg 10.5%

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

      13. sqrt-unprod 10.4%

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

      14. add-sqr-sqrt 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in re around 0 41.0%

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

      1. associate-+r+ 41.0%

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

      2. mul-1-neg 41.0%

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

      3. unsub-neg 41.0%

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

      4. +-commutative 41.0%

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

      5. mul-1-neg 41.0%

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

      6. unsub-neg 41.0%

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

      7. *-commutative 41.0%

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

      8. distribute-rgt-out 41.0%

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

      9. unpow2 41.0%

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

      10. metadata-eval 41.0%

          \[\leadsto \log \left(\left(1 - re \cdot im\right) - \left(\left(im \cdot im\right) \cdot \color{blue}{-0.5}\right) \cdot {re}^{2}\right) \]

      11. unpow2 41.0%

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

   9. Simplified 41.0%

      \[\leadsto \log \color{blue}{\left(\left(1 - re \cdot im\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -680:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 310000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 5: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -5.2 \cdot 10^{+109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -610:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1000000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -5.2e+109)
     t_0
     (if (<= im -610.0)
       (log1p (expm1 (* im re)))
       (if (<= im 1000000000000.0)
         (* im (- (sin re)))
         (if (<= im 6.6e+108)
           (log (- (- 1.0 (* im re)) (* (* (* im im) -0.5) (* re re))))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5.2e+109) {
		tmp = t_0;
	} else if (im <= -610.0) {
		tmp = log1p(expm1((im * re)));
	} else if (im <= 1000000000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 6.6e+108) {
		tmp = log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -5.2e+109) {
		tmp = t_0;
	} else if (im <= -610.0) {
		tmp = Math.log1p(Math.expm1((im * re)));
	} else if (im <= 1000000000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 6.6e+108) {
		tmp = Math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -5.2e+109:
		tmp = t_0
	elif im <= -610.0:
		tmp = math.log1p(math.expm1((im * re)))
	elif im <= 1000000000000.0:
		tmp = im * -math.sin(re)
	elif im <= 6.6e+108:
		tmp = math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -5.2e+109)
		tmp = t_0;
	elseif (im <= -610.0)
		tmp = log1p(expm1(Float64(im * re)));
	elseif (im <= 1000000000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 6.6e+108)
		tmp = log(Float64(Float64(1.0 - Float64(im * re)) - Float64(Float64(Float64(im * im) * -0.5) * Float64(re * re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.2e+109], t$95$0, If[LessEqual[im, -610.0], N[Log[1 + N[(Exp[N[(im * re), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1000000000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 6.6e+108], N[Log[N[(N[(1.0 - N[(im * re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -5.1999999999999997e109 or 6.60000000000000038e108 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-lft-out-- 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in re around 0 80.3%

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

   if -5.1999999999999997e109 < im < -610

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. *-commutative 3.0%

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

      3. distribute-rgt-neg-in 3.0%

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

   4. Simplified 3.0%

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

      1. distribute-rgt-neg-out 3.0%

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

      2. add-sqr-sqrt 1.4%

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

      3. sqrt-unprod 1.8%

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

      4. sqr-neg 1.8%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.7%

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

      7. log1p-expm1-u 0.5%

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

      8. log1p-udef 0.8%

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

      9. neg-log 0.8%

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

      10. add-sqr-sqrt 0.4%

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

      11. sqrt-unprod 24.9%

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

      12. sqr-neg 24.9%

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

      13. sqrt-unprod 24.5%

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

      14. add-sqr-sqrt 52.5%

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

   6. Applied egg-rr 52.5%

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

   7. Taylor expanded in re around 0 21.4%

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

      1. add-sqr-sqrt 14.5%

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

      2. sqrt-unprod 24.9%

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

      3. neg-log 24.9%

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

      4. log1p-udef 24.9%

         \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot im\right)\right)}\right) \cdot \log \left(\frac{1}{1 + \mathsf{expm1}\left(re \cdot im\right)}\right)} \]

      5. log1p-expm1-u 24.9%

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

      6. neg-log 24.9%

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

      7. log1p-udef 24.9%

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

      8. log1p-expm1-u 18.3%

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

      9. sqr-neg 18.3%

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

      10. sqrt-unprod 4.2%

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

      11. add-sqr-sqrt 5.9%

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

      12. log1p-expm1-u 31.5%

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

   9. Applied egg-rr 31.5%

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

   if -610 < im < 1e12

   1. Initial program 27.0%

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

   2. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

   if 1e12 < im < 6.60000000000000038e108

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.3%

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

      1. mul-1-neg 3.3%

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

      2. *-commutative 3.3%

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

      3. distribute-rgt-neg-in 3.3%

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

   4. Simplified 3.3%

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

      1. distribute-rgt-neg-out 3.3%

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

      2. add-sqr-sqrt 0.8%

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

      3. sqrt-unprod 1.2%

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

      4. sqr-neg 1.2%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.6%

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

      7. log1p-expm1-u 0.3%

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

      8. log1p-udef 0.4%

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

      9. neg-log 0.4%

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

      10. add-sqr-sqrt 0.1%

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

      11. sqrt-unprod 9.6%

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

      12. sqr-neg 9.6%

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

      13. sqrt-unprod 9.5%

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

      14. add-sqr-sqrt 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in re around 0 43.4%

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

      1. associate-+r+ 43.4%

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

      2. mul-1-neg 43.4%

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

      3. unsub-neg 43.4%

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

      4. +-commutative 43.4%

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

      5. mul-1-neg 43.4%

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

      6. unsub-neg 43.4%

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

      7. *-commutative 43.4%

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

      8. distribute-rgt-out 43.4%

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

      9. unpow2 43.4%

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

      10. metadata-eval 43.4%

          \[\leadsto \log \left(\left(1 - re \cdot im\right) - \left(\left(im \cdot im\right) \cdot \color{blue}{-0.5}\right) \cdot {re}^{2}\right) \]

      11. unpow2 43.4%

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

   9. Simplified 43.4%

      \[\leadsto \log \color{blue}{\left(\left(1 - re \cdot im\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.2 \cdot 10^{+109}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq -610:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot re\right)\right)\\ \mathbf{elif}\;im \leq 1000000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 6: 77.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -1.6e-5)
     t_0
     (if (<= im 9000000000.0)
       (* im (- (sin re)))
       (if (<= im 6.6e+108)
         (log (- (- 1.0 (* im re)) (* (* (* im im) -0.5) (* re re))))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -1.6e-5) {
		tmp = t_0;
	} else if (im <= 9000000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 6.6e+108) {
		tmp = log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-1.6d-5)) then
        tmp = t_0
    else if (im <= 9000000000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 6.6d+108) then
        tmp = log(((1.0d0 - (im * re)) - (((im * im) * (-0.5d0)) * (re * re))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -1.6e-5) {
		tmp = t_0;
	} else if (im <= 9000000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 6.6e+108) {
		tmp = Math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -1.6e-5:
		tmp = t_0
	elif im <= 9000000000.0:
		tmp = im * -math.sin(re)
	elif im <= 6.6e+108:
		tmp = math.log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -1.6e-5)
		tmp = t_0;
	elseif (im <= 9000000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 6.6e+108)
		tmp = log(Float64(Float64(1.0 - Float64(im * re)) - Float64(Float64(Float64(im * im) * -0.5) * Float64(re * re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -1.6e-5)
		tmp = t_0;
	elseif (im <= 9000000000.0)
		tmp = im * -sin(re);
	elseif (im <= 6.6e+108)
		tmp = log(((1.0 - (im * re)) - (((im * im) * -0.5) * (re * re))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.6e-5], t$95$0, If[LessEqual[im, 9000000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 6.6e+108], N[Log[N[(N[(1.0 - N[(im * re), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.59999999999999993e-5 or 6.60000000000000038e108 < im

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. unsub-neg 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-*l* 76.5%

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

      5. distribute-lft-out-- 76.5%

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

   4. Simplified 76.5%

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

   5. Taylor expanded in re around 0 62.8%

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

   if -1.59999999999999993e-5 < im < 9e9

   1. Initial program 26.5%

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

   2. Taylor expanded in im around 0 97.4%

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

      1. mul-1-neg 97.4%

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

      2. *-commutative 97.4%

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

      3. distribute-rgt-neg-in 97.4%

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

   4. Simplified 97.4%

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

   if 9e9 < im < 6.60000000000000038e108

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.3%

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

      1. mul-1-neg 3.3%

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

      2. *-commutative 3.3%

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

      3. distribute-rgt-neg-in 3.3%

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

   4. Simplified 3.3%

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

      1. distribute-rgt-neg-out 3.3%

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

      2. add-sqr-sqrt 0.8%

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

      3. sqrt-unprod 1.2%

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

      4. sqr-neg 1.2%

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

      5. sqrt-unprod 0.4%

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

      6. add-sqr-sqrt 0.6%

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

      7. log1p-expm1-u 0.3%

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

      8. log1p-udef 0.4%

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

      9. neg-log 0.4%

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

      10. add-sqr-sqrt 0.1%

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

      11. sqrt-unprod 9.6%

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

      12. sqr-neg 9.6%

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

      13. sqrt-unprod 9.5%

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

      14. add-sqr-sqrt 70.2%

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

   6. Applied egg-rr 70.2%

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

   7. Taylor expanded in re around 0 43.4%

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

      1. associate-+r+ 43.4%

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

      2. mul-1-neg 43.4%

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

      3. unsub-neg 43.4%

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

      4. +-commutative 43.4%

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

      5. mul-1-neg 43.4%

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

      6. unsub-neg 43.4%

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

      7. *-commutative 43.4%

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

      8. distribute-rgt-out 43.4%

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

      9. unpow2 43.4%

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

      10. metadata-eval 43.4%

          \[\leadsto \log \left(\left(1 - re \cdot im\right) - \left(\left(im \cdot im\right) \cdot \color{blue}{-0.5}\right) \cdot {re}^{2}\right) \]

      11. unpow2 43.4%

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

   9. Simplified 43.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 9000000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;\log \left(\left(1 - im \cdot re\right) - \left(\left(im \cdot im\right) \cdot -0.5\right) \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{if}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 510000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (- (* (pow im 3.0) -0.16666666666666666) im))))
   (if (<= im -1.7e-5)
     t_0
     (if (<= im 510000000.0)
       (* im (- (sin re)))
       (if (<= im 6.6e+108)
         (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 510000000.0) {
		tmp = im * -sin(re);
	} else if (im <= 6.6e+108) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    if (im <= (-1.7d-5)) then
        tmp = t_0
    else if (im <= 510000000.0d0) then
        tmp = im * -sin(re)
    else if (im <= 6.6d+108) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	double tmp;
	if (im <= -1.7e-5) {
		tmp = t_0;
	} else if (im <= 510000000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 6.6e+108) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	tmp = 0
	if im <= -1.7e-5:
		tmp = t_0
	elif im <= 510000000.0:
		tmp = im * -math.sin(re)
	elif im <= 6.6e+108:
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
	tmp = 0.0
	if (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 510000000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 6.6e+108)
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	tmp = 0.0;
	if (im <= -1.7e-5)
		tmp = t_0;
	elseif (im <= 510000000.0)
		tmp = im * -sin(re);
	elseif (im <= 6.6e+108)
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.7e-5], t$95$0, If[LessEqual[im, 510000000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 6.6e+108], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.7e-5 or 6.60000000000000038e108 < im

   1. Initial program 99.8%

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

   2. Taylor expanded in im around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. unsub-neg 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-*l* 76.5%

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

      5. distribute-lft-out-- 76.5%

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

   4. Simplified 76.5%

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

   5. Taylor expanded in re around 0 62.8%

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

   if -1.7e-5 < im < 5.1e8

   1. Initial program 25.9%

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

   2. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-rgt-neg-in 98.2%

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

   4. Simplified 98.2%

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

   if 5.1e8 < im < 6.60000000000000038e108

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 3.3%

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

      1. mul-1-neg 3.3%

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

      2. *-commutative 3.3%

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

      3. distribute-rgt-neg-in 3.3%

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

   4. Simplified 3.3%

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

   5. Taylor expanded in re around 0 25.2%

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

      1. +-commutative 25.2%

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

      2. mul-1-neg 25.2%

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

      3. unsub-neg 25.2%

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

      4. associate-*r* 25.2%

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

      5. distribute-rgt-out-- 31.1%

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

   7. Simplified 31.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 510000000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 6.6 \cdot 10^{+108}:\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 8: 60.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -620 \lor \neg \left(im \leq 510000000\right):\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -620.0) (not (<= im 510000000.0)))
   (* im (- (* 0.16666666666666666 (pow re 3.0)) re))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -620.0) || !(im <= 510000000.0)) {
		tmp = im * ((0.16666666666666666 * pow(re, 3.0)) - re);
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-620.0d0)) .or. (.not. (im <= 510000000.0d0))) then
        tmp = im * ((0.16666666666666666d0 * (re ** 3.0d0)) - re)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -620.0) || !(im <= 510000000.0)) {
		tmp = im * ((0.16666666666666666 * Math.pow(re, 3.0)) - re);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -620.0) or not (im <= 510000000.0):
		tmp = im * ((0.16666666666666666 * math.pow(re, 3.0)) - re)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -620.0) || !(im <= 510000000.0))
		tmp = Float64(im * Float64(Float64(0.16666666666666666 * (re ^ 3.0)) - re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -620.0) || ~((im <= 510000000.0)))
		tmp = im * ((0.16666666666666666 * (re ^ 3.0)) - re);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -620.0], N[Not[LessEqual[im, 510000000.0]], $MachinePrecision]], N[(im * N[(N[(0.16666666666666666 * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -620 or 5.1e8 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. *-commutative 4.1%

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

      3. distribute-rgt-neg-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in re around 0 14.3%

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

      1. +-commutative 14.3%

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

      2. mul-1-neg 14.3%

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

      3. unsub-neg 14.3%

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

      4. associate-*r* 14.3%

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

      5. distribute-rgt-out-- 25.1%

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

   7. Simplified 25.1%

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

   if -620 < im < 5.1e8

   1. Initial program 26.4%

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

   2. Taylor expanded in im around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. distribute-rgt-neg-in 97.9%

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

   4. Simplified 97.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -620 \lor \neg \left(im \leq 510000000\right):\\ \;\;\;\;im \cdot \left(0.16666666666666666 \cdot {re}^{3} - re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 9: 60.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -650 \lor \neg \left(im \leq 9000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -650.0) (not (<= im 9000000000.0)))
   (* 0.16666666666666666 (* im (pow re 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -650.0) || !(im <= 9000000000.0)) {
		tmp = 0.16666666666666666 * (im * pow(re, 3.0));
	} else {
		tmp = im * -sin(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-650.0d0)) .or. (.not. (im <= 9000000000.0d0))) then
        tmp = 0.16666666666666666d0 * (im * (re ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -650.0) || !(im <= 9000000000.0)) {
		tmp = 0.16666666666666666 * (im * Math.pow(re, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -650.0) or not (im <= 9000000000.0):
		tmp = 0.16666666666666666 * (im * math.pow(re, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -650.0) || !(im <= 9000000000.0))
		tmp = Float64(0.16666666666666666 * Float64(im * (re ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -650.0) || ~((im <= 9000000000.0)))
		tmp = 0.16666666666666666 * (im * (re ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -650.0], N[Not[LessEqual[im, 9000000000.0]], $MachinePrecision]], N[(0.16666666666666666 * N[(im * N[Power[re, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -650 or 9e9 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.1%

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

      1. mul-1-neg 4.1%

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

      2. *-commutative 4.1%

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

      3. distribute-rgt-neg-in 4.1%

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

   4. Simplified 4.1%

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

   5. Taylor expanded in re around 0 14.4%

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

   6. Taylor expanded in re around inf 24.1%

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

   if -650 < im < 9e9

   1. Initial program 27.0%

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

   2. Taylor expanded in im around 0 97.1%

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

      1. mul-1-neg 97.1%

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

      2. *-commutative 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -650 \lor \neg \left(im \leq 9000000000\right):\\ \;\;\;\;0.16666666666666666 \cdot \left(im \cdot {re}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9.8 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+40}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -9.8e+30)
   (* im (- re))
   (if (<= im 7.4e+40) (* im (- (sin re))) (* im re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -9.8e+30) {
		tmp = im * -re;
	} else if (im <= 7.4e+40) {
		tmp = im * -sin(re);
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-9.8d+30)) then
        tmp = im * -re
    else if (im <= 7.4d+40) then
        tmp = im * -sin(re)
    else
        tmp = im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -9.8e+30) {
		tmp = im * -re;
	} else if (im <= 7.4e+40) {
		tmp = im * -Math.sin(re);
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -9.8e+30:
		tmp = im * -re
	elif im <= 7.4e+40:
		tmp = im * -math.sin(re)
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -9.8e+30)
		tmp = Float64(im * Float64(-re));
	elseif (im <= 7.4e+40)
		tmp = Float64(im * Float64(-sin(re)));
	else
		tmp = Float64(im * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -9.8e+30)
		tmp = im * -re;
	elseif (im <= 7.4e+40)
		tmp = im * -sin(re);
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -9.8e+30], N[(im * (-re)), $MachinePrecision], If[LessEqual[im, 7.4e+40], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], N[(im * re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.79999999999999969e30

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. *-commutative 4.2%

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

      3. distribute-rgt-neg-in 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 13.6%

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

      1. mul-1-neg 13.6%

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

      2. distribute-rgt-neg-in 13.6%

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

   7. Simplified 13.6%

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

   if -9.79999999999999969e30 < im < 7.4e40

   1. Initial program 34.7%

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

   2. Taylor expanded in im around 0 87.1%

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

      1. mul-1-neg 87.1%

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

      2. *-commutative 87.1%

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

      3. distribute-rgt-neg-in 87.1%

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

   4. Simplified 87.1%

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

   if 7.4e40 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 4.4%

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

      1. mul-1-neg 4.4%

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

      2. *-commutative 4.4%

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

      3. distribute-rgt-neg-in 4.4%

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

   4. Simplified 4.4%

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

      1. distribute-rgt-neg-out 4.4%

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

      2. add-sqr-sqrt 1.7%

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

      3. sqrt-unprod 1.9%

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

      4. sqr-neg 1.9%

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

      5. sqrt-unprod 0.2%

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

      6. add-sqr-sqrt 0.5%

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

      7. log1p-expm1-u 0.1%

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

      8. log1p-udef 0.2%

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

      9. neg-log 0.2%

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

      10. add-sqr-sqrt 0.1%

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

      11. sqrt-unprod 25.0%

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

      12. sqr-neg 25.0%

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

      13. sqrt-unprod 29.7%

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

      14. add-sqr-sqrt 83.9%

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

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in re around 0 54.4%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 54.3%

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

      3. neg-log 54.3%

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

      4. log1p-udef 54.3%

         \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot im\right)\right)}\right) \cdot \log \left(\frac{1}{1 + \mathsf{expm1}\left(re \cdot im\right)}\right)} \]

      5. log1p-expm1-u 54.3%

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

      6. neg-log 54.3%

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

      7. log1p-udef 54.3%

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

      8. log1p-expm1-u 38.6%

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

      9. sqr-neg 38.6%

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

      10. sqrt-unprod 9.3%

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

      11. add-sqr-sqrt 16.0%

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

      12. *-commutative 16.0%

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

   9. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{im \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.8 \cdot 10^{+30}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{elif}\;im \leq 7.4 \cdot 10^{+40}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]

Alternative 11: 33.5% accurate, 50.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+167}:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.8e+167) (* im re) (* im (- re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+167) {
		tmp = im * re;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.8d+167)) then
        tmp = im * re
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.8e+167) {
		tmp = im * re;
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.8e+167:
		tmp = im * re
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.8e+167)
		tmp = Float64(im * re);
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.8e+167)
		tmp = im * re;
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.8e+167], N[(im * re), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.7999999999999999e167

   1. Initial program 50.1%

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

   2. Taylor expanded in im around 0 56.7%

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

      1. mul-1-neg 56.7%

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

      2. *-commutative 56.7%

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

      3. distribute-rgt-neg-in 56.7%

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

   4. Simplified 56.7%

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

      1. distribute-rgt-neg-out 56.7%

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

      2. add-sqr-sqrt 34.7%

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

      3. sqrt-unprod 36.0%

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

      4. sqr-neg 36.0%

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

      5. sqrt-unprod 1.1%

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

      6. add-sqr-sqrt 2.4%

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

      7. log1p-expm1-u 2.2%

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

      8. log1p-udef 3.0%

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

      9. neg-log 3.0%

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

      10. add-sqr-sqrt 1.4%

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

      11. sqrt-unprod 35.0%

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

      12. sqr-neg 35.0%

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

      13. sqrt-unprod 33.6%

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

      14. add-sqr-sqrt 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in re around 0 17.1%

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

      1. add-sqr-sqrt 13.7%

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

      2. sqrt-unprod 32.1%

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

      3. neg-log 32.1%

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

      4. log1p-udef 32.1%

         \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot im\right)\right)}\right) \cdot \log \left(\frac{1}{1 + \mathsf{expm1}\left(re \cdot im\right)}\right)} \]

      5. log1p-expm1-u 32.1%

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

      6. neg-log 32.1%

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

      7. log1p-udef 31.3%

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

      8. log1p-expm1-u 31.5%

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

      9. sqr-neg 31.5%

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

      10. sqrt-unprod 13.4%

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

      11. add-sqr-sqrt 26.5%

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

      12. *-commutative 26.5%

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

   9. Applied egg-rr 26.5%

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

   if -2.7999999999999999e167 < re

   1. Initial program 68.7%

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

   2. Taylor expanded in im around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. *-commutative 45.6%

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

      3. distribute-rgt-neg-in 45.6%

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

   4. Simplified 45.6%

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

   5. Taylor expanded in re around 0 28.9%

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

      1. mul-1-neg 28.9%

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

      2. distribute-rgt-neg-in 28.9%

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

   7. Simplified 28.9%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8 \cdot 10^{+167}:\\ \;\;\;\;im \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 12: 20.1% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in im around 0 47.0%

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

   1. mul-1-neg 47.0%

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

   2. *-commutative 47.0%

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

   3. distribute-rgt-neg-in 47.0%

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

4. Simplified 47.0%

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

   1. distribute-rgt-neg-out 47.0%

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

   2. add-sqr-sqrt 27.1%

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

   3. sqrt-unprod 30.5%

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

   4. sqr-neg 30.5%

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

   5. sqrt-unprod 3.6%

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

   6. add-sqr-sqrt 9.4%

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

   7. log1p-expm1-u 9.2%

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

   8. log1p-udef 9.9%

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

   9. neg-log 9.9%

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

   10. add-sqr-sqrt 3.8%

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

   11. sqrt-unprod 25.8%

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

   12. sqr-neg 25.8%

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

   13. sqrt-unprod 24.3%

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

   14. add-sqr-sqrt 51.3%

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

6. Applied egg-rr 51.3%

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

7. Taylor expanded in re around 0 35.8%

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

   1. add-sqr-sqrt 25.9%

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

   2. sqrt-unprod 35.0%

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

   3. neg-log 35.0%

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

   4. log1p-udef 35.0%

      \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re \cdot im\right)\right)}\right) \cdot \log \left(\frac{1}{1 + \mathsf{expm1}\left(re \cdot im\right)}\right)} \]

   5. log1p-expm1-u 35.0%

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

   6. neg-log 35.0%

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

   7. log1p-udef 35.4%

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

   8. log1p-expm1-u 27.6%

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

   9. sqr-neg 27.6%

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

   10. sqrt-unprod 12.3%

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

   11. add-sqr-sqrt 15.9%

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

   12. *-commutative 15.9%

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

9. Applied egg-rr 15.9%

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

10. Final simplification 15.9%

    \[\leadsto im \cdot re \]

Developer target: 99.8% accurate, 0.8× 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 2023195 
(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))))