math.cos on complex, imaginary part

Percentage Accurate: 65.3% → 99.6%

Time: 11.7s

Alternatives: 12

Speedup: 2.8×

• 49.8% 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: 65.3% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.01\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.01)))
     (* t_0 (* 0.5 (sin re)))
     (-
      (*
       (sin re)
       (+
        (* (pow im 3.0) -0.16666666666666666)
        (* (pow im 5.0) -0.008333333333333333)))
      (* im (sin re))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.01)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = (sin(re) * ((pow(im, 3.0) * -0.16666666666666666) + (pow(im, 5.0) * -0.008333333333333333))) - (im * sin(re));
	}
	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 <= 0.01)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = (Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) + (Math.pow(im, 5.0) * -0.008333333333333333))) - (im * Math.sin(re));
	}
	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 <= 0.01):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = (math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) + (math.pow(im, 5.0) * -0.008333333333333333))) - (im * math.sin(re))
	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 <= 0.01))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) + Float64((im ^ 5.0) * -0.008333333333333333))) - Float64(im * sin(re)));
	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 <= 0.01)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = (sin(re) * (((im ^ 3.0) * -0.16666666666666666) + ((im ^ 5.0) * -0.008333333333333333))) - (im * sin(re));
	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, 0.01]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(im * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -inf.0 or 0.0100000000000000002 < (-.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)) < 0.0100000000000000002

   1. Initial program 30.0%

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

   2. Taylor expanded in im around 0 99.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

      2. mul-1-neg 99.9%

         \[\leadsto \left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right) + \color{blue}{\left(-im \cdot \sin re\right)} \]

      3. unsub-neg 99.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right) - im \cdot \sin re} \]

      4. associate-*r* 99.9%

         \[\leadsto \left(\color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re} + -0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)\right) - im \cdot \sin re \]

      5. associate-*r* 99.9%

         \[\leadsto \left(\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re + \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \sin re}\right) - im \cdot \sin re \]

      6. distribute-rgt-out 99.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + -0.008333333333333333 \cdot {im}^{5}\right)} - im \cdot \sin re \]

      7. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} + -0.008333333333333333 \cdot {im}^{5}\right) - im \cdot \sin re \]

      8. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 + \color{blue}{{im}^{5} \cdot -0.008333333333333333}\right) - im \cdot \sin re \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \sin re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.01\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 + {im}^{5} \cdot -0.008333333333333333\right) - im \cdot \sin re\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.01\right):\\ \;\;\;\;t_1 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

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 0.01)))
     (* t_1 t_0)
     (*
      t_0
      (+
       (* im -2.0)
       (+
        (* (pow im 3.0) -0.3333333333333333)
        (* (pow im 5.0) -0.016666666666666666)))))))

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 <= 0.01)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * ((im * -2.0) + ((pow(im, 3.0) * -0.3333333333333333) + (pow(im, 5.0) * -0.016666666666666666)));
	}
	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 <= 0.01)) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_0 * ((im * -2.0) + ((Math.pow(im, 3.0) * -0.3333333333333333) + (Math.pow(im, 5.0) * -0.016666666666666666)));
	}
	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 <= 0.01):
		tmp = t_1 * t_0
	else:
		tmp = t_0 * ((im * -2.0) + ((math.pow(im, 3.0) * -0.3333333333333333) + (math.pow(im, 5.0) * -0.016666666666666666)))
	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 <= 0.01))
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 3.0) * -0.3333333333333333) + Float64((im ^ 5.0) * -0.016666666666666666))));
	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 <= 0.01)))
		tmp = t_1 * t_0;
	else
		tmp = t_0 * ((im * -2.0) + (((im ^ 3.0) * -0.3333333333333333) + ((im ^ 5.0) * -0.016666666666666666)));
	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, 0.01]], $MachinePrecision]], N[(t$95$1 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $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 0.0100000000000000002 < (-.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)) < 0.0100000000000000002

   1. Initial program 30.0%

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

   2. Taylor expanded in im around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.01\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot -2 + \left({im}^{3} \cdot -0.3333333333333333 + {im}^{5} \cdot -0.016666666666666666\right)\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-6)))
     (* 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-6)) {
		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-6)) {
		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-6):
		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-6))
		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-6)))
		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-6]], $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 1.99999999999999991e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999991e-6

   1. Initial program 29.5%

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

   2. Taylor expanded in im around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. +-rgt-identity 99.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} \cdot \sin re + -1 \cdot \left(im \cdot \sin re\right) \]

      4. associate-*r* 99.9%

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} + 0\right) \cdot \sin re + \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} \]

      5. distribute-rgt-in 99.9%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + 0\right) + -1 \cdot im\right)} \]

      6. +-commutative 99.9%

         \[\leadsto \sin re \cdot \color{blue}{\left(-1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right)} \]

      7. mul-1-neg 99.9%

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(-im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right) \]

      8. +-rgt-identity 99.9%

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

      9. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in re around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+122} \lor \neg \left(im \leq -0.0031\right) \land \left(im \leq 0.235 \lor \neg \left(im \leq 5.6 \cdot 10^{+97}\right)\right):\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.8e+122)
         (and (not (<= im -0.0031)) (or (<= im 0.235) (not (<= im 5.6e+97)))))
   (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
   (* (- (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.8e+122) || (!(im <= -0.0031) && ((im <= 0.235) || !(im <= 5.6e+97)))) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (exp(-im) - exp(im)) * (0.5 * 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 <= (-2.8d+122)) .or. (.not. (im <= (-0.0031d0))) .and. (im <= 0.235d0) .or. (.not. (im <= 5.6d+97))) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = (exp(-im) - exp(im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.8e+122) || (!(im <= -0.0031) && ((im <= 0.235) || !(im <= 5.6e+97)))) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.exp(-im) - Math.exp(im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.8e+122) or (not (im <= -0.0031) and ((im <= 0.235) or not (im <= 5.6e+97))):
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.exp(-im) - math.exp(im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.8e+122) || (!(im <= -0.0031) && ((im <= 0.235) || !(im <= 5.6e+97))))
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(exp(Float64(-im)) - exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.8e+122) || (~((im <= -0.0031)) && ((im <= 0.235) || ~((im <= 5.6e+97)))))
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (exp(-im) - exp(im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.8e+122], And[N[Not[LessEqual[im, -0.0031]], $MachinePrecision], Or[LessEqual[im, 0.235], N[Not[LessEqual[im, 5.6e+97]], $MachinePrecision]]]], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.8e122 or -0.00309999999999999989 < im < 0.23499999999999999 or 5.5999999999999998e97 < im

   1. Initial program 58.8%

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

   2. Taylor expanded in im around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*r* 99.1%

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

      3. +-rgt-identity 99.1%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} \cdot \sin re + -1 \cdot \left(im \cdot \sin re\right) \]

      4. associate-*r* 99.1%

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} + 0\right) \cdot \sin re + \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} \]

      5. distribute-rgt-in 99.1%

         \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + 0\right) + -1 \cdot im\right)} \]

      6. +-commutative 99.1%

         \[\leadsto \sin re \cdot \color{blue}{\left(-1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right)} \]

      7. mul-1-neg 99.1%

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(-im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right) \]

      8. +-rgt-identity 99.1%

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

      9. *-commutative 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in re around inf 99.1%

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

   if -2.8e122 < im < -0.00309999999999999989 or 0.23499999999999999 < im < 5.5999999999999998e97

   1. Initial program 99.8%

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

   2. Taylor expanded in re around 0 86.2%

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

      1. associate-*r* 86.2%

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

      2. *-commutative 86.2%

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

   4. Simplified 86.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.8 \cdot 10^{+122} \lor \neg \left(im \leq -0.0031\right) \land \left(im \leq 0.235 \lor \neg \left(im \leq 5.6 \cdot 10^{+97}\right)\right):\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 5: 79.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.00027:\\ \;\;\;\;re \cdot \left(\sqrt{{im}^{6} \cdot 0.027777777777777776} - im\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\frac{-0.16666666666666666}{\frac{im}{im \cdot {im}^{3}}} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -0.00027)
   (* re (- (sqrt (* (pow im 6.0) 0.027777777777777776)) im))
   (if (<= im 2.4e+23)
     (* (- im) (sin re))
     (* re (- (/ -0.16666666666666666 (/ im (* im (pow im 3.0)))) im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -0.00027) {
		tmp = re * (sqrt((pow(im, 6.0) * 0.027777777777777776)) - im);
	} else if (im <= 2.4e+23) {
		tmp = -im * sin(re);
	} else {
		tmp = re * ((-0.16666666666666666 / (im / (im * pow(im, 3.0)))) - 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 <= (-0.00027d0)) then
        tmp = re * (sqrt(((im ** 6.0d0) * 0.027777777777777776d0)) - im)
    else if (im <= 2.4d+23) then
        tmp = -im * sin(re)
    else
        tmp = re * (((-0.16666666666666666d0) / (im / (im * (im ** 3.0d0)))) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -0.00027) {
		tmp = re * (Math.sqrt((Math.pow(im, 6.0) * 0.027777777777777776)) - im);
	} else if (im <= 2.4e+23) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = re * ((-0.16666666666666666 / (im / (im * Math.pow(im, 3.0)))) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -0.00027:
		tmp = re * (math.sqrt((math.pow(im, 6.0) * 0.027777777777777776)) - im)
	elif im <= 2.4e+23:
		tmp = -im * math.sin(re)
	else:
		tmp = re * ((-0.16666666666666666 / (im / (im * math.pow(im, 3.0)))) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -0.00027)
		tmp = Float64(re * Float64(sqrt(Float64((im ^ 6.0) * 0.027777777777777776)) - im));
	elseif (im <= 2.4e+23)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 / Float64(im / Float64(im * (im ^ 3.0)))) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -0.00027)
		tmp = re * (sqrt(((im ^ 6.0) * 0.027777777777777776)) - im);
	elseif (im <= 2.4e+23)
		tmp = -im * sin(re);
	else
		tmp = re * ((-0.16666666666666666 / (im / (im * (im ^ 3.0)))) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -0.00027], N[(re * N[(N[Sqrt[N[(N[Power[im, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.4e+23], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(-0.16666666666666666 / N[(im / N[(im * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.70000000000000003e-4

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. *-commutative 73.4%

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

   4. Simplified 73.4%

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

   5. Taylor expanded in im around 0 54.8%

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

      1. +-commutative 54.8%

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

      2. mul-1-neg 54.8%

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

      3. unsub-neg 54.8%

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

      4. associate-*r* 54.8%

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

      5. distribute-rgt-out-- 54.8%

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

   7. Simplified 54.8%

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

      1. add-sqr-sqrt 54.8%

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

      2. sqrt-unprod 61.8%

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

      3. swap-sqr 61.8%

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

      4. *-commutative 61.8%

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

      5. pow-sqr 61.8%

         \[\leadsto re \cdot \left(\sqrt{\color{blue}{{im}^{\left(2 \cdot 3\right)}} \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)} - im\right) \]

      6. metadata-eval 61.8%

         \[\leadsto re \cdot \left(\sqrt{{im}^{\color{blue}{6}} \cdot \left(-0.16666666666666666 \cdot -0.16666666666666666\right)} - im\right) \]

      7. metadata-eval 61.8%

         \[\leadsto re \cdot \left(\sqrt{{im}^{6} \cdot \color{blue}{0.027777777777777776}} - im\right) \]

   9. Applied egg-rr 61.8%

      \[\leadsto re \cdot \left(\color{blue}{\sqrt{{im}^{6} \cdot 0.027777777777777776}} - im\right) \]

   if -2.70000000000000003e-4 < im < 2.4e23

   1. Initial program 31.7%

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

   2. Taylor expanded in im around 0 96.7%

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

      1. associate-*r* 96.7%

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

      2. mul-1-neg 96.7%

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

   4. Simplified 96.7%

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

   if 2.4e23 < im

   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 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around 0 57.5%

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

      1. +-commutative 57.5%

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

      2. mul-1-neg 57.5%

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

      3. unsub-neg 57.5%

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

      4. associate-*r* 57.5%

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

      5. distribute-rgt-out-- 57.5%

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

   7. Simplified 57.5%

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

   9. Applied egg-rr 60.7%

      \[\leadsto re \cdot \left(\color{blue}{\frac{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \left({im}^{2} + 0\right)}{im}} - im\right) \]
   10. Step-by-step derivation

       1. associate-/l* 57.5%

          \[\leadsto re \cdot \left(\color{blue}{\frac{-0.16666666666666666 \cdot {im}^{2}}{\frac{im}{{im}^{2} + 0}}} - im\right) \]

       2. +-rgt-identity 57.5%

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

       3. associate-/l* 60.7%

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

       4. associate-*r* 60.7%

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

       5. unpow2 60.7%

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

       6. associate-*r* 60.7%

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

       7. unpow2 60.7%

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

       8. cube-mult 60.7%

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

       9. associate-/l* 60.7%

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

   11. Simplified 60.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00027:\\ \;\;\;\;re \cdot \left(\sqrt{{im}^{6} \cdot 0.027777777777777776} - im\right)\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{+23}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\frac{-0.16666666666666666}{\frac{im}{im \cdot {im}^{3}}} - im\right)\\ \end{array} \]

Alternative 6: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

2. Taylor expanded in im around 0 83.8%

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

   1. +-commutative 83.8%

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

   2. associate-*r* 83.8%

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

   3. +-rgt-identity 83.8%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} + 0\right)} \cdot \sin re + -1 \cdot \left(im \cdot \sin re\right) \]

   4. associate-*r* 83.8%

      \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} + 0\right) \cdot \sin re + \color{blue}{\left(-1 \cdot im\right) \cdot \sin re} \]

   5. distribute-rgt-in 83.8%

      \[\leadsto \color{blue}{\sin re \cdot \left(\left(-0.16666666666666666 \cdot {im}^{3} + 0\right) + -1 \cdot im\right)} \]

   6. +-commutative 83.8%

      \[\leadsto \sin re \cdot \color{blue}{\left(-1 \cdot im + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right)} \]

   7. mul-1-neg 83.8%

      \[\leadsto \sin re \cdot \left(\color{blue}{\left(-im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} + 0\right)\right) \]

   8. +-rgt-identity 83.8%

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

   9. *-commutative 83.8%

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

4. Simplified 83.8%

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

5. Taylor expanded in re around inf 83.8%

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

6. Final simplification 83.8%

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

Alternative 7: 78.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.0031 \lor \neg \left(im \leq 4.6 \cdot 10^{+22}\right):\\ \;\;\;\;re \cdot \left(\frac{-0.16666666666666666}{\frac{im}{im \cdot {im}^{3}}} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.0031) (not (<= im 4.6e+22)))
   (* re (- (/ -0.16666666666666666 (/ im (* im (pow im 3.0)))) im))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.0031) || !(im <= 4.6e+22)) {
		tmp = re * ((-0.16666666666666666 / (im / (im * pow(im, 3.0)))) - im);
	} 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 <= (-0.0031d0)) .or. (.not. (im <= 4.6d+22))) then
        tmp = re * (((-0.16666666666666666d0) / (im / (im * (im ** 3.0d0)))) - im)
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.0031) || !(im <= 4.6e+22)) {
		tmp = re * ((-0.16666666666666666 / (im / (im * Math.pow(im, 3.0)))) - im);
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.0031) or not (im <= 4.6e+22):
		tmp = re * ((-0.16666666666666666 / (im / (im * math.pow(im, 3.0)))) - im)
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.0031) || !(im <= 4.6e+22))
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 / Float64(im / Float64(im * (im ^ 3.0)))) - im));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.0031) || ~((im <= 4.6e+22)))
		tmp = re * ((-0.16666666666666666 / (im / (im * (im ^ 3.0)))) - im);
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.0031], N[Not[LessEqual[im, 4.6e+22]], $MachinePrecision]], N[(re * N[(N[(-0.16666666666666666 / N[(im / N[(im * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.00309999999999999989 or 4.6000000000000004e22 < im

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

   4. Simplified 74.2%

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

   5. Taylor expanded in im around 0 56.1%

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

      1. +-commutative 56.1%

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

      2. mul-1-neg 56.1%

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

      3. unsub-neg 56.1%

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

      4. associate-*r* 56.1%

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

      5. distribute-rgt-out-- 56.1%

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

   7. Simplified 56.1%

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

   9. Applied egg-rr 58.3%

      \[\leadsto re \cdot \left(\color{blue}{\frac{\left(-0.16666666666666666 \cdot {im}^{2}\right) \cdot \left({im}^{2} + 0\right)}{im}} - im\right) \]
   10. Step-by-step derivation

       1. associate-/l* 56.1%

          \[\leadsto re \cdot \left(\color{blue}{\frac{-0.16666666666666666 \cdot {im}^{2}}{\frac{im}{{im}^{2} + 0}}} - im\right) \]

       2. +-rgt-identity 56.1%

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

       3. associate-/l* 58.3%

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

       4. associate-*r* 58.3%

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

       5. unpow2 58.3%

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

       6. associate-*r* 58.3%

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

       7. unpow2 58.3%

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

       8. cube-mult 58.3%

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

       9. associate-/l* 58.3%

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

   11. Simplified 58.3%

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

   if -0.00309999999999999989 < im < 4.6000000000000004e22

   1. Initial program 31.7%

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

   2. Taylor expanded in im around 0 96.7%

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

      1. associate-*r* 96.7%

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

      2. mul-1-neg 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.0031 \lor \neg \left(im \leq 4.6 \cdot 10^{+22}\right):\\ \;\;\;\;re \cdot \left(\frac{-0.16666666666666666}{\frac{im}{im \cdot {im}^{3}}} - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 8: 77.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.0024 \lor \neg \left(im \leq 7.5 \cdot 10^{+22}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.0024) (not (<= im 7.5e+22)))
   (* re (- (* (pow im 3.0) -0.16666666666666666) im))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.0024) || !(im <= 7.5e+22)) {
		tmp = re * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} 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 <= (-0.0024d0)) .or. (.not. (im <= 7.5d+22))) then
        tmp = re * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.0024) || !(im <= 7.5e+22)) {
		tmp = re * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.0024) or not (im <= 7.5e+22):
		tmp = re * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.0024) || !(im <= 7.5e+22))
		tmp = Float64(re * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.0024) || ~((im <= 7.5e+22)))
		tmp = re * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.0024], N[Not[LessEqual[im, 7.5e+22]], $MachinePrecision]], N[(re * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.00239999999999999979 or 7.5000000000000002e22 < im

   1. Initial program 99.9%

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

   2. Taylor expanded in re around 0 74.2%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

   4. Simplified 74.2%

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

   5. Taylor expanded in im around 0 56.1%

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

      1. +-commutative 56.1%

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

      2. mul-1-neg 56.1%

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

      3. unsub-neg 56.1%

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

      4. associate-*r* 56.1%

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

      5. distribute-rgt-out-- 56.1%

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

   7. Simplified 56.1%

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

   if -0.00239999999999999979 < im < 7.5000000000000002e22

   1. Initial program 31.7%

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

   2. Taylor expanded in im around 0 96.7%

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

      1. associate-*r* 96.7%

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

      2. mul-1-neg 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.0024 \lor \neg \left(im \leq 7.5 \cdot 10^{+22}\right):\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 9: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+51} \lor \neg \left(im \leq 3.75 \cdot 10^{+22}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.32e+51) (not (<= im 3.75e+22)))
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.32e+51) || !(im <= 3.75e+22)) {
		tmp = -0.16666666666666666 * (re * pow(im, 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 <= (-1.32d+51)) .or. (.not. (im <= 3.75d+22))) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 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 <= -1.32e+51) || !(im <= 3.75e+22)) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.32e+51) or not (im <= 3.75e+22):
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.32e+51) || !(im <= 3.75e+22))
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.32e+51) || ~((im <= 3.75e+22)))
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.32e+51], N[Not[LessEqual[im, 3.75e+22]], $MachinePrecision]], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.32e51 or 3.7500000000000001e22 < im

   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 73.5%

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

      1. associate-*r* 73.5%

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

      2. *-commutative 73.5%

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

   4. Simplified 73.5%

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

   5. Taylor expanded in im around 0 60.5%

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

      1. +-commutative 60.5%

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

      2. mul-1-neg 60.5%

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

      3. unsub-neg 60.5%

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

      4. associate-*r* 60.5%

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

      5. distribute-rgt-out-- 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in im around inf 60.5%

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

   if -1.32e51 < im < 3.7500000000000001e22

   1. Initial program 37.1%

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

   2. Taylor expanded in im around 0 89.6%

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

      1. associate-*r* 89.6%

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

      2. mul-1-neg 89.6%

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

   4. Simplified 89.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+51} \lor \neg \left(im \leq 3.75 \cdot 10^{+22}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 10: 76.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+50}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+23}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5e+50)
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (if (<= im 1.85e+23)
     (* (- im) (sin re))
     (* re (* (pow im 3.0) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5e+50) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} else if (im <= 1.85e+23) {
		tmp = -im * sin(re);
	} else {
		tmp = re * (pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-5d+50)) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else if (im <= 1.85d+23) then
        tmp = -im * sin(re)
    else
        tmp = re * ((im ** 3.0d0) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5e+50) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else if (im <= 1.85e+23) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = re * (Math.pow(im, 3.0) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5e+50:
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	elif im <= 1.85e+23:
		tmp = -im * math.sin(re)
	else:
		tmp = re * (math.pow(im, 3.0) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5e+50)
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	elseif (im <= 1.85e+23)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = Float64(re * Float64((im ^ 3.0) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5e+50)
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	elseif (im <= 1.85e+23)
		tmp = -im * sin(re);
	else
		tmp = re * ((im ^ 3.0) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5e+50], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.85e+23], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(re * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5e50

   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 71.9%

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

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

   4. Simplified 71.9%

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

   5. Taylor expanded in im around 0 63.5%

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

      1. +-commutative 63.5%

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

      2. mul-1-neg 63.5%

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

      3. unsub-neg 63.5%

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

      4. associate-*r* 63.5%

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

      5. distribute-rgt-out-- 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in im around inf 63.5%

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

   if -5e50 < im < 1.85000000000000006e23

   1. Initial program 37.1%

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

   2. Taylor expanded in im around 0 89.6%

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

      1. associate-*r* 89.6%

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

      2. mul-1-neg 89.6%

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

   4. Simplified 89.6%

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

   if 1.85000000000000006e23 < im

   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 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in im around 0 57.5%

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

      1. +-commutative 57.5%

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

      2. mul-1-neg 57.5%

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

      3. unsub-neg 57.5%

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

      4. associate-*r* 57.5%

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

      5. distribute-rgt-out-- 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in im around inf 57.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+50}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{+23}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 11: 57.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{+61} \lor \neg \left(im \leq 6 \cdot 10^{+22}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.6e+61) (not (<= im 6e+22)))
   (* (- im) re)
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.6e+61) || !(im <= 6e+22)) {
		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 <= (-3.6d+61)) .or. (.not. (im <= 6d+22))) 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 <= -3.6e+61) || !(im <= 6e+22)) {
		tmp = -im * re;
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.6e+61) or not (im <= 6e+22):
		tmp = -im * re
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.6e+61) || !(im <= 6e+22))
		tmp = Float64(Float64(-im) * re);
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.6e+61) || ~((im <= 6e+22)))
		tmp = -im * re;
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.6e+61], N[Not[LessEqual[im, 6e+22]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.6000000000000001e61 or 6e22 < 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 4.9%

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

      1. associate-*r* 4.9%

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

      2. mul-1-neg 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in re around 0 19.9%

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

      1. associate-*r* 19.9%

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

      2. mul-1-neg 19.9%

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

   7. Simplified 19.9%

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

   if -3.6000000000000001e61 < im < 6e22

   1. Initial program 38.4%

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

   2. Taylor expanded in im around 0 87.8%

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

      1. associate-*r* 87.8%

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

      2. mul-1-neg 87.8%

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

   4. Simplified 87.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{+61} \lor \neg \left(im \leq 6 \cdot 10^{+22}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 12: 33.5% accurate, 77.0× speedup

Math

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

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(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 65.8%

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

2. Taylor expanded in im around 0 50.9%

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

   1. associate-*r* 50.9%

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

   2. mul-1-neg 50.9%

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

4. Simplified 50.9%

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

5. Taylor expanded in re around 0 36.6%

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

   1. associate-*r* 36.6%

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

   2. mul-1-neg 36.6%

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

7. Simplified 36.6%

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

8. Final simplification 36.6%

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

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 2023305 
(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))))