math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.7%

Time: 16.7s

Alternatives: 15

Speedup: 2.8×

• 49.0% 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: 65.5% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.02\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (- (exp (- im)) (exp im))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.02)))
     (* t_1 t_0)
     (*
      t_0
      (+
       (* im -2.0)
       (+
        (* -0.3333333333333333 (pow im 3.0))
        (* -0.016666666666666666 (pow im 5.0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = exp(-im) - exp(im);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.3333333333333333 * pow(im, 3.0)) + (-0.016666666666666666 * pow(im, 5.0))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double t_1 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.02)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + (-0.016666666666666666 * Math.pow(im, 5.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.02):
		tmp = t_1 * t_0
	else:
		tmp = t_0 * ((im * -2.0) + ((-0.3333333333333333 * math.pow(im, 3.0)) + (-0.016666666666666666 * math.pow(im, 5.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.02))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(im * -2.0) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(-0.016666666666666666 * (im ^ 5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.02)))
		tmp = t_1 * t_0;
	else
		tmp = t_0 * ((im * -2.0) + ((-0.3333333333333333 * (im ^ 3.0)) + (-0.016666666666666666 * (im ^ 5.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.02]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 0.0200000000000000004 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.2%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0200000000000000004

   1. Initial program 32.5%

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

   2. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.02\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-15)))
     (* 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 <= 2e-15)) {
		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 <= 2e-15)) {
		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 <= 2e-15):
		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 <= 2e-15))
		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 <= 2e-15)))
		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, 2e-15]], $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 2.0000000000000002e-15 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.2%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.0000000000000002e-15

   1. Initial program 32.0%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-rgt-out-- 99.8%

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

      6. *-commutative 99.8%

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

   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 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ t_1 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.14:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- (exp (- im)) (exp im)) (* 0.5 re)))
        (t_1 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -2.1e+98)
     t_1
     (if (<= im -0.058)
       t_0
       (if (<= im 0.14)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 4.4e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	double t_1 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -0.058) {
		tmp = t_0;
	} else if (im <= 0.14) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.4e+61) {
		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 = (exp(-im) - exp(im)) * (0.5d0 * re)
    t_1 = (-0.008333333333333333d0) * (sin(re) * (im ** 5.0d0))
    if (im <= (-2.1d+98)) then
        tmp = t_1
    else if (im <= (-0.058d0)) then
        tmp = t_0
    else if (im <= 0.14d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 4.4d+61) 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 = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	double t_1 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -0.058) {
		tmp = t_0;
	} else if (im <= 0.14) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	t_1 = -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))
	tmp = 0
	if im <= -2.1e+98:
		tmp = t_1
	elif im <= -0.058:
		tmp = t_0
	elif im <= 0.14:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 4.4e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re))
	t_1 = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.1e+98)
		tmp = t_1;
	elseif (im <= -0.058)
		tmp = t_0;
	elseif (im <= 0.14)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (exp(-im) - exp(im)) * (0.5 * re);
	t_1 = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
	tmp = 0.0;
	if (im <= -2.1e+98)
		tmp = t_1;
	elseif (im <= -0.058)
		tmp = t_0;
	elseif (im <= 0.14)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+98], t$95$1, If[LessEqual[im, -0.058], t$95$0, If[LessEqual[im, 0.14], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.4e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000004e98 or 4.4000000000000001e61 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   if -2.10000000000000004e98 < im < -0.0580000000000000029 or 0.14000000000000001 < im < 4.4000000000000001e61

   1. Initial program 97.1%

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

   2. Taylor expanded in re around 0 84.8%

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

      1. associate-*r* 84.8%

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

      2. *-commutative 84.8%

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

   4. Simplified 84.8%

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

   if -0.0580000000000000029 < im < 0.14000000000000001

   1. Initial program 32.5%

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

   2. Taylor expanded in im around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out-- 99.6%

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

      6. *-commutative 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -0.058:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{elif}\;im \leq 0.14:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 4: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ t_1 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -64000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 90000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (cbrt (* (pow im 15.0) -5.787037037037037e-7))))
        (t_1 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -2.1e+98)
     t_1
     (if (<= im -64000.0)
       t_0
       (if (<= im 90000.0)
         (* im (- (sin re)))
         (if (<= im 4.4e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = re * cbrt((pow(im, 15.0) * -5.787037037037037e-7));
	double t_1 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -64000.0) {
		tmp = t_0;
	} else if (im <= 90000.0) {
		tmp = im * -sin(re);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = re * Math.cbrt((Math.pow(im, 15.0) * -5.787037037037037e-7));
	double t_1 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -64000.0) {
		tmp = t_0;
	} else if (im <= 90000.0) {
		tmp = im * -Math.sin(re);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(re * cbrt(Float64((im ^ 15.0) * -5.787037037037037e-7)))
	t_1 = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.1e+98)
		tmp = t_1;
	elseif (im <= -64000.0)
		tmp = t_0;
	elseif (im <= 90000.0)
		tmp = Float64(im * Float64(-sin(re)));
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[Power[N[(N[Power[im, 15.0], $MachinePrecision] * -5.787037037037037e-7), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+98], t$95$1, If[LessEqual[im, -64000.0], t$95$0, If[LessEqual[im, 90000.0], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 4.4e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000004e98 or 4.4000000000000001e61 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   if -2.10000000000000004e98 < im < -64000 or 9e4 < im < 4.4000000000000001e61

   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 28.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 28.1%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   4. Taylor expanded in re around 0 37.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot re\right)} \]
   5. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 37.8%

         \[\leadsto \color{blue}{\left(re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 37.8%

         \[\leadsto \color{blue}{re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

   6. Simplified 37.8%

      \[\leadsto \color{blue}{re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 75.6%

         \[\leadsto re \cdot \color{blue}{\sqrt[3]{\left(\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)}} \]

      2. pow1/3 53.9%

         \[\leadsto re \cdot \color{blue}{{\left(\left(\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right)}^{0.3333333333333333}} \]

      3. pow3 53.9%

         \[\leadsto re \cdot {\color{blue}{\left({\left({im}^{5} \cdot -0.008333333333333333\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 53.9%

         \[\leadsto re \cdot {\color{blue}{\left({\left({im}^{5}\right)}^{3} \cdot {-0.008333333333333333}^{3}\right)}}^{0.3333333333333333} \]

      5. pow3 53.9%

         \[\leadsto re \cdot {\left(\color{blue}{\left(\left({im}^{5} \cdot {im}^{5}\right) \cdot {im}^{5}\right)} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 53.9%

         \[\leadsto re \cdot {\left(\left(\color{blue}{{im}^{\left(5 + 5\right)}} \cdot {im}^{5}\right) \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      7. metadata-eval 53.9%

         \[\leadsto re \cdot {\left(\left({im}^{\color{blue}{10}} \cdot {im}^{5}\right) \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      8. pow-prod-up 53.9%

         \[\leadsto re \cdot {\left(\color{blue}{{im}^{\left(10 + 5\right)}} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      9. metadata-eval 53.9%

         \[\leadsto re \cdot {\left({im}^{\color{blue}{15}} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 53.9%

          \[\leadsto re \cdot {\left({im}^{15} \cdot \color{blue}{-5.787037037037037 \cdot 10^{-7}}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 53.9%

      \[\leadsto re \cdot \color{blue}{{\left({im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 75.6%

         \[\leadsto re \cdot \color{blue}{\sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}} \]

   10. Simplified 75.6%

       \[\leadsto re \cdot \color{blue}{\sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}} \]

   if -64000 < im < 9e4

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 96.4%

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

      1. associate-*r* 96.4%

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

      2. neg-mul-1 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -64000:\\ \;\;\;\;re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ \mathbf{elif}\;im \leq 90000:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 5: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ t_1 := -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -31000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 920:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* re (cbrt (* (pow im 15.0) -5.787037037037037e-7))))
        (t_1 (* -0.008333333333333333 (* (sin re) (pow im 5.0)))))
   (if (<= im -2.1e+98)
     t_1
     (if (<= im -31000.0)
       t_0
       (if (<= im 920.0)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 4.4e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = re * cbrt((pow(im, 15.0) * -5.787037037037037e-7));
	double t_1 = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -31000.0) {
		tmp = t_0;
	} else if (im <= 920.0) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = re * Math.cbrt((Math.pow(im, 15.0) * -5.787037037037037e-7));
	double t_1 = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	double tmp;
	if (im <= -2.1e+98) {
		tmp = t_1;
	} else if (im <= -31000.0) {
		tmp = t_0;
	} else if (im <= 920.0) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.4e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(re * cbrt(Float64((im ^ 15.0) * -5.787037037037037e-7)))
	t_1 = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
	tmp = 0.0
	if (im <= -2.1e+98)
		tmp = t_1;
	elseif (im <= -31000.0)
		tmp = t_0;
	elseif (im <= 920.0)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 4.4e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(re * N[Power[N[(N[Power[im, 15.0], $MachinePrecision] * -5.787037037037037e-7), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+98], t$95$1, If[LessEqual[im, -31000.0], t$95$0, If[LessEqual[im, 920.0], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.4e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.10000000000000004e98 or 4.4000000000000001e61 < 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%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   if -2.10000000000000004e98 < im < -31000 or 920 < im < 4.4000000000000001e61

   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 28.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 28.1%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   4. Taylor expanded in re around 0 37.8%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot re\right)} \]
   5. Step-by-step derivation

      1. *-commutative 37.8%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot re\right) \cdot -0.008333333333333333} \]

      2. *-commutative 37.8%

         \[\leadsto \color{blue}{\left(re \cdot {im}^{5}\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 37.8%

         \[\leadsto \color{blue}{re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]

   6. Simplified 37.8%

      \[\leadsto \color{blue}{re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 75.6%

         \[\leadsto re \cdot \color{blue}{\sqrt[3]{\left(\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)}} \]

      2. pow1/3 53.9%

         \[\leadsto re \cdot \color{blue}{{\left(\left(\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right) \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\right)}^{0.3333333333333333}} \]

      3. pow3 53.9%

         \[\leadsto re \cdot {\color{blue}{\left({\left({im}^{5} \cdot -0.008333333333333333\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. unpow-prod-down 53.9%

         \[\leadsto re \cdot {\color{blue}{\left({\left({im}^{5}\right)}^{3} \cdot {-0.008333333333333333}^{3}\right)}}^{0.3333333333333333} \]

      5. pow3 53.9%

         \[\leadsto re \cdot {\left(\color{blue}{\left(\left({im}^{5} \cdot {im}^{5}\right) \cdot {im}^{5}\right)} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 53.9%

         \[\leadsto re \cdot {\left(\left(\color{blue}{{im}^{\left(5 + 5\right)}} \cdot {im}^{5}\right) \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      7. metadata-eval 53.9%

         \[\leadsto re \cdot {\left(\left({im}^{\color{blue}{10}} \cdot {im}^{5}\right) \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      8. pow-prod-up 53.9%

         \[\leadsto re \cdot {\left(\color{blue}{{im}^{\left(10 + 5\right)}} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      9. metadata-eval 53.9%

         \[\leadsto re \cdot {\left({im}^{\color{blue}{15}} \cdot {-0.008333333333333333}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 53.9%

          \[\leadsto re \cdot {\left({im}^{15} \cdot \color{blue}{-5.787037037037037 \cdot 10^{-7}}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 53.9%

      \[\leadsto re \cdot \color{blue}{{\left({im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 75.6%

         \[\leadsto re \cdot \color{blue}{\sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}} \]

   10. Simplified 75.6%

       \[\leadsto re \cdot \color{blue}{\sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}} \]

   if -31000 < im < 920

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 97.2%

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

      1. +-commutative 97.2%

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

      2. mul-1-neg 97.2%

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

      3. unsub-neg 97.2%

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

      4. associate-*r* 97.2%

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

      5. distribute-rgt-out-- 97.2%

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

      6. *-commutative 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{elif}\;im \leq -31000:\\ \;\;\;\;re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ \mathbf{elif}\;im \leq 920:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{+61}:\\ \;\;\;\;re \cdot \sqrt[3]{{im}^{15} \cdot -5.787037037037037 \cdot 10^{-7}}\\ \mathbf{else}:\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \end{array} \]

Alternative 6: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.3) (not (<= im 3.4)))
   (* -0.008333333333333333 (* (sin re) (pow im 5.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.4)) {
		tmp = -0.008333333333333333 * (sin(re) * pow(im, 5.0));
	} 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 <= (-3.3d0)) .or. (.not. (im <= 3.4d0))) then
        tmp = (-0.008333333333333333d0) * (sin(re) * (im ** 5.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.3) || !(im <= 3.4)) {
		tmp = -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.3) or not (im <= 3.4):
		tmp = -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.3) || !(im <= 3.4))
		tmp = Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.3) || ~((im <= 3.4)))
		tmp = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.3], N[Not[LessEqual[im, 3.4]], $MachinePrecision]], N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.2999999999999998 or 3.39999999999999991 < im

   1. Initial program 99.3%

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

   2. Taylor expanded in im around 0 81.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 81.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   if -3.2999999999999998 < im < 3.39999999999999991

   1. Initial program 33.0%

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

   2. Taylor expanded in im around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.3 \lor \neg \left(im \leq 3.4\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 7: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 5.0) -0.008333333333333333) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 5.0) * -0.008333333333333333) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 90.8%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

3. Taylor expanded in im around inf 90.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-2 \cdot im + \color{blue}{-0.016666666666666666 \cdot {im}^{5}}\right) \]

4. Taylor expanded in im around 0 90.2%

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]
5. Step-by-step derivation

   1. associate-*r* 90.2%

      \[\leadsto \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

   2. neg-mul-1 90.2%

      \[\leadsto \color{blue}{\left(-im\right)} \cdot \sin re + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right) \]

   3. associate-*r* 90.2%

      \[\leadsto \left(-im\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re} \]

   4. *-commutative 90.2%

      \[\leadsto \left(-im\right) \cdot \sin re + \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right)} \cdot \sin re \]

   5. distribute-rgt-out 90.2%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{5} \cdot -0.008333333333333333\right)} \]

6. Simplified 90.2%

   \[\leadsto \color{blue}{\sin re \cdot \left(\left(-im\right) + {im}^{5} \cdot -0.008333333333333333\right)} \]

7. Taylor expanded in re around inf 90.2%

   \[\leadsto \color{blue}{\sin re \cdot \left(-0.008333333333333333 \cdot {im}^{5} - im\right)} \]

8. Final simplification 90.2%

   \[\leadsto \sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 - im\right) \]

Alternative 8: 81.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -47000 \lor \neg \left(im \leq 7500\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -47000.0) (not (<= im 7500.0)))
   (* -0.008333333333333333 (* re (pow im 5.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -47000.0) || !(im <= 7500.0)) {
		tmp = -0.008333333333333333 * (re * pow(im, 5.0));
	} 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 <= (-47000.0d0)) .or. (.not. (im <= 7500.0d0))) then
        tmp = (-0.008333333333333333d0) * (re * (im ** 5.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -47000.0) || !(im <= 7500.0)) {
		tmp = -0.008333333333333333 * (re * Math.pow(im, 5.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -47000.0) or not (im <= 7500.0):
		tmp = -0.008333333333333333 * (re * math.pow(im, 5.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -47000.0) || !(im <= 7500.0))
		tmp = Float64(-0.008333333333333333 * Float64(re * (im ^ 5.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -47000.0) || ~((im <= 7500.0)))
		tmp = -0.008333333333333333 * (re * (im ^ 5.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -47000.0], N[Not[LessEqual[im, 7500.0]], $MachinePrecision]], N[(-0.008333333333333333 * N[(re * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -47000 or 7500 < 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 82.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

   3. Taylor expanded in im around inf 82.9%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

   4. Taylor expanded in re around 0 66.6%

      \[\leadsto -0.008333333333333333 \cdot \color{blue}{\left({im}^{5} \cdot re\right)} \]

   if -47000 < im < 7500

   1. Initial program 33.8%

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

   2. Taylor expanded in im around 0 96.4%

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

      1. associate-*r* 96.4%

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

      2. neg-mul-1 96.4%

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

   4. Simplified 96.4%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -47000 \lor \neg \left(im \leq 7500\right):\\ \;\;\;\;-0.008333333333333333 \cdot \left(re \cdot {im}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 9: 57.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+57} \lor \neg \left(im \leq 1900000000\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -9.5e+57) (not (<= im 1900000000.0)))
   (* im (- re))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -9.5e+57) || !(im <= 1900000000.0)) {
		tmp = im * -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 <= (-9.5d+57)) .or. (.not. (im <= 1900000000.0d0))) then
        tmp = im * -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 <= -9.5e+57) || !(im <= 1900000000.0)) {
		tmp = im * -re;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -9.5e+57) or not (im <= 1900000000.0):
		tmp = im * -re
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -9.5e+57) || !(im <= 1900000000.0))
		tmp = Float64(im * Float64(-re));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -9.5e+57) || ~((im <= 1900000000.0)))
		tmp = im * -re;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -9.5e+57], N[Not[LessEqual[im, 1900000000.0]], $MachinePrecision]], N[(im * (-re)), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -9.4999999999999997e57 or 1.9e9 < 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 4.2%

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

      1. associate-*r* 4.2%

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

      2. neg-mul-1 4.2%

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

   4. Simplified 4.2%

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

   5. Taylor expanded in re around 0 19.2%

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

      1. associate-*r* 19.2%

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

      2. neg-mul-1 19.2%

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

   7. Simplified 19.2%

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

   if -9.4999999999999997e57 < im < 1.9e9

   1. Initial program 38.7%

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

   2. Taylor expanded in im around 0 89.5%

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

      1. associate-*r* 89.5%

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

      2. neg-mul-1 89.5%

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

   4. Simplified 89.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -9.5 \cdot 10^{+57} \lor \neg \left(im \leq 1900000000\right):\\ \;\;\;\;im \cdot \left(-re\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 32.8% accurate, 77.0× speedup

Math

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

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 * Float64(-re))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

5. Taylor expanded in re around 0 32.6%

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

   1. associate-*r* 32.6%

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

   2. neg-mul-1 32.6%

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

7. Simplified 32.6%

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

8. Final simplification 32.6%

   \[\leadsto im \cdot \left(-re\right) \]

Alternative 11: 2.7% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -512 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -512.0)

C

double code(double re, double im) {
	return -512.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -512.0d0
end function

Java

public static double code(double re, double im) {
	return -512.0;
}

Python

def code(re, im):
	return -512.0

Julia

function code(re, im)
	return -512.0
end

MATLAB

function tmp = code(re, im)
	tmp = -512.0;
end

Wolfram

code[re_, im_] := -512.0

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

6. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-512} \]

7. Final simplification 2.9%

   \[\leadsto -512 \]

Alternative 12: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -0.004629629629629629 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -0.004629629629629629)

C

double code(double re, double im) {
	return -0.004629629629629629;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -0.004629629629629629d0
end function

Java

public static double code(double re, double im) {
	return -0.004629629629629629;
}

Python

def code(re, im):
	return -0.004629629629629629

Julia

function code(re, im)
	return -0.004629629629629629
end

MATLAB

function tmp = code(re, im)
	tmp = -0.004629629629629629;
end

Wolfram

code[re_, im_] := -0.004629629629629629

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

6. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-0.004629629629629629} \]

7. Final simplification 2.9%

   \[\leadsto -0.004629629629629629 \]

Alternative 13: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -5.080526342529086 \cdot 10^{-5} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -5.080526342529086e-5)

C

double code(double re, double im) {
	return -5.080526342529086e-5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -5.080526342529086d-5
end function

Java

public static double code(double re, double im) {
	return -5.080526342529086e-5;
}

Python

def code(re, im):
	return -5.080526342529086e-5

Julia

function code(re, im)
	return -5.080526342529086e-5
end

MATLAB

function tmp = code(re, im)
	tmp = -5.080526342529086e-5;
end

Wolfram

code[re_, im_] := -5.080526342529086e-5

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

6. Applied egg-rr 2.9%

   \[\leadsto \color{blue}{-5.080526342529086 \cdot 10^{-5}} \]

7. Final simplification 2.9%

   \[\leadsto -5.080526342529086 \cdot 10^{-5} \]

Alternative 14: 2.8% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ -9.92290301275212 \cdot 10^{-8} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -9.92290301275212e-8)

C

double code(double re, double im) {
	return -9.92290301275212e-8;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -9.92290301275212d-8
end function

Java

public static double code(double re, double im) {
	return -9.92290301275212e-8;
}

Python

def code(re, im):
	return -9.92290301275212e-8

Julia

function code(re, im)
	return -9.92290301275212e-8
end

MATLAB

function tmp = code(re, im)
	tmp = -9.92290301275212e-8;
end

Wolfram

code[re_, im_] := -9.92290301275212e-8

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

6. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{-9.92290301275212 \cdot 10^{-8}} \]

7. Final simplification 3.0%

   \[\leadsto -9.92290301275212 \cdot 10^{-8} \]

Alternative 15: 15.2% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 64.3%

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

2. Taylor expanded in im around 0 53.8%

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

   1. associate-*r* 53.8%

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

   2. neg-mul-1 53.8%

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

4. Simplified 53.8%

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

6. Applied egg-rr 16.2%

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

7. Final simplification 16.2%

   \[\leadsto 0 \]

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 2023283 
(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))))