math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 9

Speedup: 1.0×

• 24.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Final simplification 100.0%

   \[\leadsto e^{re} \cdot \sin im \]

Alternative 2: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999999 \lor \neg \left(e^{re} \leq 1.02\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999999999999) (not (<= (exp re) 1.02)))
   (* (exp re) im)
   (/ (sin im) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999999999999) || !(exp(re) <= 1.02)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) / (1.0 - re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.99999999999999d0) .or. (.not. (exp(re) <= 1.02d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) / (1.0d0 - re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.99999999999999) || !(Math.exp(re) <= 1.02)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) / (1.0 - re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99999999999999) or not (math.exp(re) <= 1.02):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) / (1.0 - re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99999999999999) || !(exp(re) <= 1.02))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) / Float64(1.0 - re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99999999999999) || ~((exp(re) <= 1.02)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) / (1.0 - re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99999999999999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.02]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999999001 or 1.02 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 88.4%

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

   if 0.99999999999999001 < (exp.f64 re) < 1.02

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.8%

      \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
   3. Step-by-step derivation

      1. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

      1. flip-+ 99.8%

         \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \sin im \]

      2. associate-*l/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in re around 0 99.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \sin im}}{re + -1} \]
   8. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto \frac{\color{blue}{-\sin im}}{re + -1} \]

   9. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{-\sin im}}{re + -1} \]
   10. Step-by-step derivation

       1. frac-2neg 99.8%

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

       2. div-inv 99.8%

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

       3. remove-double-neg 99.8%

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

       4. +-commutative 99.8%

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

       5. distribute-neg-in 99.8%

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

       6. metadata-eval 99.8%

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

       7. *-rgt-identity 99.8%

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

       8. sub-neg 99.8%

          \[\leadsto \sin im \cdot \frac{1}{\color{blue}{1 - re \cdot 1}} \]

       9. *-rgt-identity 99.8%

          \[\leadsto \sin im \cdot \frac{1}{1 - \color{blue}{re}} \]

   11. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\sin im \cdot \frac{1}{1 - re}} \]
   12. Step-by-step derivation

       1. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\sin im \cdot 1}{1 - re}} \]

       2. *-rgt-identity 99.8%

          \[\leadsto \frac{\color{blue}{\sin im}}{1 - re} \]

   13. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\sin im}{1 - re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999999 \lor \neg \left(e^{re} \leq 1.02\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \]

Alternative 3: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (exp re) 0.0)
   (exp re)
   (if (<= (exp re) 1.02) (* (sin im) (+ re 1.0)) (* (exp re) im))))

C

double code(double re, double im) {
	double tmp;
	if (exp(re) <= 0.0) {
		tmp = exp(re);
	} else if (exp(re) <= 1.02) {
		tmp = sin(im) * (re + 1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (exp(re) <= 0.0d0) then
        tmp = exp(re)
    else if (exp(re) <= 1.02d0) then
        tmp = sin(im) * (re + 1.0d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.exp(re) <= 0.0) {
		tmp = Math.exp(re);
	} else if (Math.exp(re) <= 1.02) {
		tmp = Math.sin(im) * (re + 1.0);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.exp(re) <= 0.0:
		tmp = math.exp(re)
	elif math.exp(re) <= 1.02:
		tmp = math.sin(im) * (re + 1.0)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.02)
		tmp = Float64(sin(im) * Float64(re + 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (exp(re) <= 0.0)
		tmp = exp(re);
	elseif (exp(re) <= 1.02)
		tmp = sin(im) * (re + 1.0);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.02], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 100.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 100.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 100.0%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 36.8%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 36.8%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 100.0%

      \[\leadsto e^{\color{blue}{re}} \]

   if 0.0 < (exp.f64 re) < 1.02

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.8%

      \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
   3. Step-by-step derivation

      1. distribute-rgt1-in 99.8%

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

   4. Simplified 99.8%

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

   if 1.02 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 74.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \leq 1.02:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

Alternative 4: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999999 \lor \neg \left(e^{re} \leq 1.02\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.99999999999999) (not (<= (exp re) 1.02)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.99999999999999) || !(exp(re) <= 1.02)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.99999999999999d0) .or. (.not. (exp(re) <= 1.02d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.99999999999999) || !(Math.exp(re) <= 1.02)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.99999999999999) or not (math.exp(re) <= 1.02):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.99999999999999) || !(exp(re) <= 1.02))
		tmp = Float64(exp(re) * im);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.99999999999999) || ~((exp(re) <= 1.02)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.99999999999999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.02]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99999999999999001 or 1.02 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 88.4%

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

   if 0.99999999999999001 < (exp.f64 re) < 1.02

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.0%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.99999999999999 \lor \neg \left(e^{re} \leq 1.02\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 5: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.0) (not (<= (exp re) 2.0))) (exp re) (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0)) {
		tmp = exp(re);
	} else {
		tmp = sin(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) <= 0.0d0) .or. (.not. (exp(re) <= 2.0d0))) then
        tmp = exp(re)
    else
        tmp = sin(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.0) || !(Math.exp(re) <= 2.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.0) or not (math.exp(re) <= 2.0):
		tmp = math.exp(re)
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.0) || !(exp(re) <= 2.0))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.0) || ~((exp(re) <= 2.0)))
		tmp = exp(re);
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 2.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.0 or 2 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 78.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 78.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 78.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 78.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 44.4%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 44.4%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 77.9%

      \[\leadsto e^{\color{blue}{re}} \]

   if 0.0 < (exp.f64 re) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 98.2%

      \[\leadsto \color{blue}{\sin im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0 \lor \neg \left(e^{re} \leq 2\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 6: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.94 \lor \neg \left(re \leq 5\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -0.94) (not (<= re 5.0))) (exp re) (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -0.94) || !(re <= 5.0)) {
		tmp = exp(re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-0.94d0)) .or. (.not. (re <= 5.0d0))) then
        tmp = exp(re)
    else
        tmp = im * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -0.94) || !(re <= 5.0)) {
		tmp = Math.exp(re);
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -0.94) or not (re <= 5.0):
		tmp = math.exp(re)
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -0.94) || !(re <= 5.0))
		tmp = exp(re);
	else
		tmp = Float64(im * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -0.94) || ~((re <= 5.0)))
		tmp = exp(re);
	else
		tmp = im * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -0.94], N[Not[LessEqual[re, 5.0]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.93999999999999995 or 5 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 78.5%

         \[\leadsto \color{blue}{{\left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 78.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(e^{re} \cdot \sin im\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot \left(e^{re} \cdot \sin im\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 78.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(e^{re} \cdot \sin im\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 78.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(e^{re} \cdot \sin im\right)\right)} \cdot 0.3333333333333333} \]

      6. log-prod 44.4%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{re}\right) + \log \sin im\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 44.4%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{re} + \log \sin im\right)\right) \cdot 0.3333333333333333} \]

   3. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(re + \log \sin im\right)\right) \cdot 0.3333333333333333}} \]

   4. Taylor expanded in re around inf 77.9%

      \[\leadsto e^{\color{blue}{re}} \]

   if -0.93999999999999995 < re < 5

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.3%

      \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
   3. Step-by-step derivation

      1. distribute-rgt1-in 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in im around 0 55.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.94 \lor \neg \left(re \leq 5\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 7: 32.8% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \frac{-im}{re + -1} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (/ (- im) (+ re -1.0)))

C

double code(double re, double im) {
	return -im / (re + -1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -im / (re + (-1.0d0))
end function

Java

public static double code(double re, double im) {
	return -im / (re + -1.0);
}

Python

def code(re, im):
	return -im / (re + -1.0)

Julia

function code(re, im)
	return Float64(Float64(-im) / Float64(re + -1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = -im / (re + -1.0);
end

Wolfram

code[re_, im_] := N[((-im) / N[(re + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in re around 0 52.2%

   \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
3. Step-by-step derivation

   1. distribute-rgt1-in 52.2%

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

4. Simplified 52.2%

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

   1. flip-+ 61.7%

      \[\leadsto \color{blue}{\frac{re \cdot re - 1 \cdot 1}{re - 1}} \cdot \sin im \]

   2. associate-*l/ 61.7%

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

   3. metadata-eval 61.7%

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

   4. fma-neg 61.7%

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

   5. metadata-eval 61.7%

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

   6. sub-neg 61.7%

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

   7. metadata-eval 61.7%

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

6. Applied egg-rr 61.7%

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

7. Taylor expanded in re around 0 58.9%

   \[\leadsto \frac{\color{blue}{-1 \cdot \sin im}}{re + -1} \]
8. Step-by-step derivation

   1. mul-1-neg 58.9%

      \[\leadsto \frac{\color{blue}{-\sin im}}{re + -1} \]

9. Simplified 58.9%

   \[\leadsto \frac{\color{blue}{-\sin im}}{re + -1} \]

10. Taylor expanded in im around 0 36.3%

    \[\leadsto \color{blue}{-1 \cdot \frac{im}{re - 1}} \]
11. Step-by-step derivation

    1. sub-neg 36.3%

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

    2. metadata-eval 36.3%

       \[\leadsto -1 \cdot \frac{im}{re + \color{blue}{-1}} \]

    3. associate-*r/ 36.3%

       \[\leadsto \color{blue}{\frac{-1 \cdot im}{re + -1}} \]

    4. mul-1-neg 36.3%

       \[\leadsto \frac{\color{blue}{-im}}{re + -1} \]

    5. +-commutative 36.3%

       \[\leadsto \frac{-im}{\color{blue}{-1 + re}} \]

12. Simplified 36.3%

    \[\leadsto \color{blue}{\frac{-im}{-1 + re}} \]

13. Final simplification 36.3%

    \[\leadsto \frac{-im}{re + -1} \]

Alternative 8: 29.7% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ im \cdot \left(re + 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* im (+ re 1.0)))

C

double code(double re, double im) {
	return im * (re + 1.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = im * (re + 1.0d0)
end function

Java

public static double code(double re, double im) {
	return im * (re + 1.0);
}

Python

def code(re, im):
	return im * (re + 1.0)

Julia

function code(re, im)
	return Float64(im * Float64(re + 1.0))
end

MATLAB

function tmp = code(re, im)
	tmp = im * (re + 1.0);
end

Wolfram

code[re_, im_] := N[(im * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in re around 0 52.2%

   \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
3. Step-by-step derivation

   1. distribute-rgt1-in 52.2%

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

4. Simplified 52.2%

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

5. Taylor expanded in im around 0 30.5%

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

6. Final simplification 30.5%

   \[\leadsto im \cdot \left(re + 1\right) \]

Alternative 9: 6.6% accurate, 67.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in re around 0 52.2%

   \[\leadsto \color{blue}{\sin im + re \cdot \sin im} \]
3. Step-by-step derivation

   1. distribute-rgt1-in 52.2%

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

4. Simplified 52.2%

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

5. Taylor expanded in re around inf 4.3%

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

   1. *-commutative 4.3%

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

7. Simplified 4.3%

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

8. Taylor expanded in im around 0 5.0%

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

9. Final simplification 5.0%

   \[\leadsto re \cdot im \]

Reproduce

herbie shell --seed 2023339 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))