math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 1.7s

Alternatives: 9

Speedup: 1.0×

• 75.1% of points produce a very large (infinite) output. You may want to add a precondition.

• Could not converge on a ground truth

  1. re = 4.614906735260092e-21
  2. im = +nan.0

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(1 + re\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.004:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (+ 1.0 re) (cos im))) (t_1 (* (exp re) (cos im))))
  (if (<= t_1 (- INFINITY))
    (* (exp re) (fma (* im im) -0.5 1.0))
    (if (<= t_1 -0.004)
      t_0
      (if (<= t_1 0.0)
        (exp re)
        (if (<= t_1 0.9999999) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = (1.0 + re) * cos(im);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_1 <= -0.004) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(1.0 + re) * cos(im))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_1 <= -0.004)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.004], t$95$0, If[LessEqual[t$95$1, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 62.7%

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

   4. Applied rewrites 62.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot {im}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 62.7%

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

      6. lift-pow.f64 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 62.7%

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999990000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 52.1%

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

   4. Applied rewrites 52.1%

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

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.99999990000000005 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.004:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (cos im))))
  (if (<= t_0 (- INFINITY))
    (* (exp re) (fma (* im im) -0.5 1.0))
    (if (<= t_0 -0.004)
      (cos im)
      (if (<= t_0 0.0)
        (exp re)
        (if (<= t_0 0.9999999) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.004) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.004)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.004], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 62.7%

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

   4. Applied rewrites 62.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot {im}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 62.7%

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

      6. lift-pow.f64 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 62.7%

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999990000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   3. Step-by-step derivation

      1. lower-cos.f64 51.2%

         \[\leadsto \cos im \]

   4. Applied rewrites 51.2%

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

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0 or 0.99999990000000005 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.004:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.004)
  (* (exp re) (fma (* im im) -0.5 1.0))
  (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.004) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.004)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 62.7%

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

   4. Applied rewrites 62.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot {im}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 62.7%

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

      6. lift-pow.f64 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 62.7%

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.004)
  (fma (sqrt (* (* (* im im) im) im)) -0.5 1.0)
  (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.004) {
		tmp = fma(sqrt((((im * im) * im) * im)), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.004)
		tmp = fma(sqrt(Float64(Float64(Float64(im * im) * im) * im)), -0.5, 1.0);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[Sqrt[N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   3. Step-by-step derivation

      1. lower-cos.f64 51.2%

         \[\leadsto \cos im \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. lower-pow.f64 29.8%

         \[\leadsto 1 + -0.5 \cdot {im}^{2} \]

   7. Applied rewrites 29.8%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      4. lift-pow.f64 N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      5. pow2 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(im \cdot im\right) \cdot \frac{-1}{2} + 1 \]

      8. lift-fma.f64 29.8%

         \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

       2. fabs-sqr N/A

          \[\leadsto \mathsf{fma}\left(\left|im \cdot im\right|, \frac{-1}{2}, 1\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left|im \cdot im\right|, \frac{-1}{2}, 1\right) \]

       4. rem-sqrt-square-rev N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, \frac{-1}{2}, 1\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, \frac{-1}{2}, 1\right) \]

       6. lower-sqrt.f64 31.1%

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, -0.5, 1\right) \]

       7. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, \frac{-1}{2}, 1\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, \frac{-1}{2}, 1\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{im \cdot \left(im \cdot \left(im \cdot im\right)\right)}, \frac{-1}{2}, 1\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{im \cdot \left(im \cdot \left(im \cdot im\right)\right)}, \frac{-1}{2}, 1\right) \]

       11. cube-unmult N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{im \cdot {im}^{3}}, \frac{-1}{2}, 1\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{{im}^{3} \cdot im}, \frac{-1}{2}, 1\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{{im}^{3} \cdot im}, \frac{-1}{2}, 1\right) \]

       14. pow3 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}, \frac{-1}{2}, 1\right) \]

       15. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}, \frac{-1}{2}, 1\right) \]

       16. lower-*.f64 31.1%

           \[\leadsto \mathsf{fma}\left(\sqrt{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}, -0.5, 1\right) \]

   11. Applied rewrites 31.1%

       \[\leadsto \mathsf{fma}\left(\sqrt{\left(\left(im \cdot im\right) \cdot im\right) \cdot im}, -0.5, 1\right) \]

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.004:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.004)
  (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
  (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.004) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.004)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 62.7%

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

   4. Applied rewrites 62.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

         \[\leadsto e^{re} \cdot \left(\frac{-1}{2} \cdot {im}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 62.7%

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

      6. lift-pow.f64 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left({im}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 62.7%

         \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto e^{re} \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 31.6%

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   9. Applied rewrites 31.6%

      \[\leadsto \color{blue}{\left(1 + re\right)} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.004:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) -0.004)
  (fma (* im im) -0.5 1.0)
  (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.004) {
		tmp = fma((im * im), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.004)
		tmp = fma(Float64(im * im), -0.5, 1.0);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.004], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0040000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   3. Step-by-step derivation

      1. lower-cos.f64 51.2%

         \[\leadsto \cos im \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. lower-pow.f64 29.8%

         \[\leadsto 1 + -0.5 \cdot {im}^{2} \]

   7. Applied rewrites 29.8%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      4. lift-pow.f64 N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      5. pow2 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(im \cdot im\right) \cdot \frac{-1}{2} + 1 \]

      8. lift-fma.f64 29.8%

         \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   if -0.0040000000000000001 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   10. Step-by-step derivation

       1. lower-exp.f64 70.8%

          \[\leadsto e^{re} \]

   11. Applied rewrites 70.8%

       \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 33.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + re\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (exp re) (cos im)) 0.0)
  (fma (* im im) -0.5 1.0)
  (+ 1.0 re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = fma((im * im), -0.5, 1.0);
	} else {
		tmp = 1.0 + re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = fma(Float64(im * im), -0.5, 1.0);
	else
		tmp = Float64(1.0 + re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(1.0 + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   3. Step-by-step derivation

      1. lower-cos.f64 51.2%

         \[\leadsto \cos im \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. lower-pow.f64 29.8%

         \[\leadsto 1 + -0.5 \cdot {im}^{2} \]

   7. Applied rewrites 29.8%

      \[\leadsto 1 + \color{blue}{-0.5 \cdot {im}^{2}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{{im}^{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      4. lift-pow.f64 N/A

         \[\leadsto 1 + {im}^{2} \cdot \frac{-1}{2} \]

      5. pow2 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      6. lift-*.f64 N/A

         \[\leadsto 1 + \left(im \cdot im\right) \cdot \frac{-1}{2} \]

      7. +-commutative N/A

         \[\leadsto \left(im \cdot im\right) \cdot \frac{-1}{2} + 1 \]

      8. lift-fma.f64 29.8%

         \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto \mathsf{fma}\left(im \cdot im, -0.5, 1\right) \]

   if -0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 52.1%

         \[\leadsto \cos im + re \cdot \cos im \]

   4. Applied rewrites 52.1%

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

   6. Applied rewrites 63.5%

      \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

      2. remove-sound-/ N/A

         \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   8. Applied rewrites 52.1%

      \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

   9. Taylor expanded in im around 0

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

       1. lower-+.f64 29.3%

          \[\leadsto 1 + re \]

   11. Applied rewrites 29.3%

       \[\leadsto 1 + \color{blue}{re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.3% accurate, 13.5× speedup

Math

\[1 + re \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 52.1%

      \[\leadsto \cos im + re \cdot \cos im \]

4. Applied rewrites 52.1%

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

6. Applied rewrites 63.5%

   \[\leadsto \frac{\left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) - \left(0.5 + 0.5 \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

   2. remove-sound-/ N/A

      \[\leadsto \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) - \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(im + im\right)\right) \cdot \left(re \cdot re\right)}{\color{blue}{\cos im - \cos im \cdot re}} \]

8. Applied rewrites 52.1%

   \[\leadsto \mathsf{fma}\left(\cos im, \color{blue}{re}, \cos im\right) \]

9. Taylor expanded in im around 0

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

    1. lower-+.f64 29.3%

       \[\leadsto 1 + re \]

11. Applied rewrites 29.3%

    \[\leadsto 1 + \color{blue}{re} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))