math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 11

Speedup: 1.0×

• 25.2% 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: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \frac{1}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999) (not (<= (exp re) 1.05)))
   (* (exp re) im)
   (* (sin im) (/ 1.0 (- 1.0 re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(im) * (1.0 / (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) <= 0.9999d0) .or. (.not. (exp(re) <= 1.05d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (1.0d0 / (1.0d0 - re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9999) || !(Math.exp(re) <= 1.05)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (1.0 / (1.0 - re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999) or not (math.exp(re) <= 1.05):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (1.0 / (1.0 - re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(1.0 / Float64(1.0 - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999) || ~((exp(re) <= 1.05)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (1.0 / (1.0 - re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.05]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(1.0 / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99990000000000001 or 1.05000000000000004 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.0%

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

   if 0.99990000000000001 < (exp.f64 re) < 1.05000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   6. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. flip3-+ 99.2%

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

      3. associate-*r/ 99.2%

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

      4. associate-/l* 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. associate-/l* 99.2%

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

      7. flip3-+ 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in re around 0 99.2%

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

       1. mul-1-neg 99.2%

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

       2. sub-neg 99.2%

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

   11. Simplified 99.2%

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

       1. div-inv 99.3%

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

       2. *-commutative 99.3%

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

   13. Applied egg-rr 99.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \frac{1}{1 - re}\\ \end{array} \]

Alternative 3: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999) (not (<= (exp re) 1.05)))
   (* (exp re) im)
   (* (sin im) (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05)) {
		tmp = exp(re) * im;
	} else {
		tmp = sin(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 ((exp(re) <= 0.9999d0) .or. (.not. (exp(re) <= 1.05d0))) then
        tmp = exp(re) * im
    else
        tmp = sin(im) * (re + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) <= 0.9999) || !(Math.exp(re) <= 1.05)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im) * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999) or not (math.exp(re) <= 1.05):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im) * (re + 1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(sin(im) * Float64(re + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) <= 0.9999) || ~((exp(re) <= 1.05)))
		tmp = exp(re) * im;
	else
		tmp = sin(im) * (re + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.05]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99990000000000001 or 1.05000000000000004 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.0%

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

   if 0.99990000000000001 < (exp.f64 re) < 1.05000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999) (not (<= (exp re) 1.05)))
   (* (exp re) im)
   (/ (sin im) (- 1.0 re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05)) {
		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) <= 0.9999d0) .or. (.not. (exp(re) <= 1.05d0))) 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) <= 0.9999) || !(Math.exp(re) <= 1.05)) {
		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) <= 0.9999) or not (math.exp(re) <= 1.05):
		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) <= 0.9999) || !(exp(re) <= 1.05))
		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) <= 0.9999) || ~((exp(re) <= 1.05)))
		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], 0.9999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.05]], $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) < 0.99990000000000001 or 1.05000000000000004 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.0%

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

   if 0.99990000000000001 < (exp.f64 re) < 1.05000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 99.2%

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

      1. distribute-rgt1-in 99.2%

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

   6. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. flip3-+ 99.2%

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

      3. associate-*r/ 99.2%

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

      4. associate-/l* 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. associate-/l* 99.2%

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

      7. flip3-+ 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in re around 0 99.2%

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

       1. mul-1-neg 99.2%

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

       2. sub-neg 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \end{array} \]

Alternative 5: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= (exp re) 0.9999) (not (<= (exp re) 1.05)))
   (* (exp re) im)
   (sin im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05)) {
		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) <= 0.9999d0) .or. (.not. (exp(re) <= 1.05d0))) 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) <= 0.9999) || !(Math.exp(re) <= 1.05)) {
		tmp = Math.exp(re) * im;
	} else {
		tmp = Math.sin(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) <= 0.9999) or not (math.exp(re) <= 1.05):
		tmp = math.exp(re) * im
	else:
		tmp = math.sin(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((exp(re) <= 0.9999) || !(exp(re) <= 1.05))
		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) <= 0.9999) || ~((exp(re) <= 1.05)))
		tmp = exp(re) * im;
	else
		tmp = sin(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[N[Exp[re], $MachinePrecision], 0.9999], N[Not[LessEqual[N[Exp[re], $MachinePrecision], 1.05]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[Sin[im], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 0.99990000000000001 or 1.05000000000000004 < (exp.f64 re)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 88.0%

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

   if 0.99990000000000001 < (exp.f64 re) < 1.05000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 98.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0.9999 \lor \neg \left(e^{re} \leq 1.05\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im\\ \end{array} \]

Alternative 6: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -86:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\frac{1}{im} - re \cdot \frac{-1}{im}}{\frac{1}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -86.0)
   0.0
   (if (<= re 0.075)
     (sin im)
     (* im (/ (- (/ 1.0 im) (* re (/ -1.0 im))) (/ 1.0 im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -86.0) {
		tmp = 0.0;
	} else if (re <= 0.075) {
		tmp = sin(im);
	} else {
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-86.0d0)) then
        tmp = 0.0d0
    else if (re <= 0.075d0) then
        tmp = sin(im)
    else
        tmp = im * (((1.0d0 / im) - (re * ((-1.0d0) / im))) / (1.0d0 / im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -86.0) {
		tmp = 0.0;
	} else if (re <= 0.075) {
		tmp = Math.sin(im);
	} else {
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -86.0:
		tmp = 0.0
	elif re <= 0.075:
		tmp = math.sin(im)
	else:
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -86.0)
		tmp = 0.0;
	elseif (re <= 0.075)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(Float64(1.0 / im) - Float64(re * Float64(-1.0 / im))) / Float64(1.0 / im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -86.0)
		tmp = 0.0;
	elseif (re <= 0.075)
		tmp = sin(im);
	else
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -86.0], 0.0, If[LessEqual[re, 0.075], N[Sin[im], $MachinePrecision], N[(im * N[(N[(N[(1.0 / im), $MachinePrecision] - N[(re * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -86

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin im \cdot e^{re}\right)} - 1} \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -86 < re < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 96.9%

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

   if 0.0749999999999999972 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 5.0%

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

      1. distribute-rgt1-in 5.0%

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

   6. Simplified 5.0%

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

   7. Taylor expanded in im around 0 19.5%

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

      1. +-commutative 19.5%

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

   9. Simplified 19.5%

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

       1. *-commutative 19.5%

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

       2. /-rgt-identity 19.5%

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

       3. frac-2neg 19.5%

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

       4. associate-*l/ 19.5%

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

       5. neg-sub0 19.5%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(re + 1\right)\right)} \cdot im}{-1} \]

       6. +-commutative 19.5%

          \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + re\right)}\right) \cdot im}{-1} \]

       7. associate--r+ 19.5%

          \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - re\right)} \cdot im}{-1} \]

       8. metadata-eval 19.5%

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

       9. metadata-eval 19.5%

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

   11. Applied egg-rr 19.5%

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

       1. associate-/l* 19.5%

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

   13. Simplified 19.5%

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

       1. div-sub 19.5%

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

       2. frac-sub 32.6%

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

       3. associate-*l/ 32.6%

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

       4. neg-mul-1 32.6%

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

       5. distribute-neg-frac 32.6%

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

       6. metadata-eval 32.6%

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

       7. associate-/r/ 43.2%

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

       8. neg-mul-1 43.2%

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

       9. distribute-neg-frac 43.2%

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

       10. metadata-eval 43.2%

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

   15. Applied egg-rr 43.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -86:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.075:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\frac{1}{im} - re \cdot \frac{-1}{im}}{\frac{1}{im}}\\ \end{array} \]

Alternative 7: 59.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\frac{1}{im} - re \cdot \frac{-1}{im}}{\frac{1}{im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.0) 0.0 (* im (/ (- (/ 1.0 im) (* re (/ -1.0 im))) (/ 1.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else {
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.0d0)) then
        tmp = 0.0d0
    else
        tmp = im * (((1.0d0 / im) - (re * ((-1.0d0) / im))) / (1.0d0 / im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} else {
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	else:
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.0)
		tmp = 0.0;
	else
		tmp = Float64(im * Float64(Float64(Float64(1.0 / im) - Float64(re * Float64(-1.0 / im))) / Float64(1.0 / im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.0)
		tmp = 0.0;
	else
		tmp = im * (((1.0 / im) - (re * (-1.0 / im))) / (1.0 / im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.0], 0.0, N[(im * N[(N[(N[(1.0 / im), $MachinePrecision] - N[(re * N[(-1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin im \cdot e^{re}\right)} - 1} \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 60.2%

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

      1. distribute-rgt1-in 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in im around 0 33.1%

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

      1. +-commutative 33.1%

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

   9. Simplified 33.1%

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

       1. *-commutative 33.1%

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

       2. /-rgt-identity 33.1%

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

       3. frac-2neg 33.1%

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

       4. associate-*l/ 33.1%

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

       5. neg-sub0 33.1%

          \[\leadsto \frac{\color{blue}{\left(0 - \left(re + 1\right)\right)} \cdot im}{-1} \]

       6. +-commutative 33.1%

          \[\leadsto \frac{\left(0 - \color{blue}{\left(1 + re\right)}\right) \cdot im}{-1} \]

       7. associate--r+ 33.1%

          \[\leadsto \frac{\color{blue}{\left(\left(0 - 1\right) - re\right)} \cdot im}{-1} \]

       8. metadata-eval 33.1%

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

       9. metadata-eval 33.1%

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

   11. Applied egg-rr 33.1%

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

       1. associate-/l* 33.1%

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

   13. Simplified 33.1%

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

       1. div-sub 33.1%

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

       2. frac-sub 25.5%

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

       3. associate-*l/ 25.5%

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

       4. neg-mul-1 25.5%

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

       5. distribute-neg-frac 25.5%

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

       6. metadata-eval 25.5%

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

       7. associate-/r/ 42.8%

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

       8. neg-mul-1 42.8%

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

       9. distribute-neg-frac 42.8%

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

       10. metadata-eval 42.8%

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

   15. Applied egg-rr 42.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{\frac{1}{im} - re \cdot \frac{-1}{im}}{\frac{1}{im}}\\ \end{array} \]

Alternative 8: 53.1% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -76:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -76.0) 0.0 (if (<= re 1.0) im (* re im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -76.0) {
		tmp = 0.0;
	} else if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = 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 (re <= (-76.0d0)) then
        tmp = 0.0d0
    else if (re <= 1.0d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -76.0) {
		tmp = 0.0;
	} else if (re <= 1.0) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -76.0:
		tmp = 0.0
	elif re <= 1.0:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -76.0)
		tmp = 0.0;
	elseif (re <= 1.0)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -76.0)
		tmp = 0.0;
	elseif (re <= 1.0)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -76.0], 0.0, If[LessEqual[re, 1.0], im, N[(re * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -76

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin im \cdot e^{re}\right)} - 1} \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -76 < re < 1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 43.7%

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

   5. Taylor expanded in re around 0 42.0%

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

   if 1 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 4.6%

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

      1. distribute-rgt1-in 4.6%

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

   6. Simplified 4.6%

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

   7. Taylor expanded in im around 0 19.3%

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

      1. +-commutative 19.3%

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

   9. Simplified 19.3%

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

   10. Taylor expanded in re around inf 19.3%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -76:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 9: 53.3% accurate, 28.7× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (if (<= re -1.0) 0.0 (* im (+ re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.0) {
		tmp = 0.0;
	} 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 <= (-1.0d0)) then
        tmp = 0.0d0
    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 <= -1.0) {
		tmp = 0.0;
	} else {
		tmp = im * (re + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.0:
		tmp = 0.0
	else:
		tmp = im * (re + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin im \cdot e^{re}\right)} - 1} \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 60.2%

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

      1. distribute-rgt1-in 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in im around 0 33.1%

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

      1. +-commutative 33.1%

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

   9. Simplified 33.1%

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

4. Final simplification 48.0%

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

Alternative 10: 27.6% accurate, 40.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.85e+14) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+14) {
		tmp = im;
	} else {
		tmp = 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 (im <= 1.85d+14) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.85e+14) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.85e+14:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.85e+14)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.85e+14)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.85e+14], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.85e14

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.4%

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

   5. Taylor expanded in re around 0 27.0%

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

   if 1.85e14 < im

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 55.3%

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

      1. distribute-rgt1-in 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in im around 0 14.1%

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

      1. +-commutative 14.1%

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

   9. Simplified 14.1%

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

   10. Taylor expanded in re around inf 14.8%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.85 \cdot 10^{+14}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

Alternative 11: 26.0% accurate, 203.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 67.6%

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

5. Taylor expanded in re around 0 21.3%

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

6. Final simplification 21.3%

   \[\leadsto im \]

Reproduce

herbie shell --seed 2023326 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))