math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 7

Speedup: 1.0×

• 74.8% of points produce a very large (infinite) output. You may want to add a precondition.

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.00035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 56000:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
  (if (<= re -0.00035)
    t_0
    (if (<= re 56000.0) (* (+ 1.0 re) (cos im)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.00035) {
		tmp = t_0;
	} else if (re <= 56000.0) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.00035)
		tmp = t_0;
	elseif (re <= 56000.0)
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00035], t$95$0, If[LessEqual[re, 56000.0], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5e-4 or 56000 < re

   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 63.8%

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

   4. Applied rewrites 63.8%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   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 63.8%

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

      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 63.8%

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

   6. Applied rewrites 63.8%

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

   if -3.5e-4 < re < 56000

   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 51.6%

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

   4. Applied rewrites 51.6%

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

Alternative 2: 86.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -0.00035:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 7200000000:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (exp re) (fma (* im im) -0.5 1.0))))
  (if (<= re -0.00035) t_0 (if (<= re 7200000000.0) (cos im) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(re) * fma((im * im), -0.5, 1.0);
	double tmp;
	if (re <= -0.00035) {
		tmp = t_0;
	} else if (re <= 7200000000.0) {
		tmp = cos(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
	tmp = 0.0
	if (re <= -0.00035)
		tmp = t_0;
	elseif (re <= 7200000000.0)
		tmp = cos(im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.00035], t$95$0, If[LessEqual[re, 7200000000.0], N[Cos[im], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5e-4 or 7.2e9 < re

   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 63.8%

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

   4. Applied rewrites 63.8%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   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 63.8%

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

      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 63.8%

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

   6. Applied rewrites 63.8%

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

   if -3.5e-4 < re < 7.2e9

   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 50.8%

         \[\leadsto \cos im \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\cos im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (exp re) (fma (* im im) -0.5 1.0)))

C

double code(double re, double im) {
	return exp(re) * fma((im * im), -0.5, 1.0);
}

Julia

function code(re, im)
	return Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0))
end

Wolfram

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

Derivation

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 63.8%

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

4. Applied rewrites 63.8%

   \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
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 63.8%

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

   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 63.8%

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

6. Applied rewrites 63.8%

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

Alternative 4: 38.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;re \leq -210:\\ \;\;\;\;-0.5 \cdot {im}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= re -210.0)
  (* -0.5 (pow im 2.0))
  (* (- re -1.0) (fma -0.5 (* im im) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -210.0) {
		tmp = -0.5 * pow(im, 2.0);
	} else {
		tmp = (re - -1.0) * fma(-0.5, (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -210.0)
		tmp = Float64(-0.5 * (im ^ 2.0));
	else
		tmp = Float64(Float64(re - -1.0) * fma(-0.5, Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -210.0], N[(-0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(re - -1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -210

   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 50.8%

         \[\leadsto \cos im \]

   4. Applied rewrites 50.8%

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

   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 30.2%

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

   7. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 30.2%

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

   9. Applied rewrites 30.2%

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

   10. Taylor expanded in im around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 12.1%

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

   12. Applied rewrites 12.1%

       \[\leadsto -0.5 \cdot {im}^{\color{blue}{2}} \]

   if -210 < re

   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 51.6%

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 31.9%

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

   7. Applied rewrites 31.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval 31.9%

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

      6. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) + 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) + 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(pow2, \left(im \cdot im\right)\right) + 1\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(im \cdot im\right)\right) + 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{-1}{2}, im \cdot im, 1\right)\right)\right) \]

   9. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\left(re - -1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 31.9% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (- re -1.0) (fma -0.5 (* im im) 1.0)))

C

double code(double re, double im) {
	return (re - -1.0) * fma(-0.5, (im * im), 1.0);
}

Julia

function code(re, im)
	return Float64(Float64(re - -1.0) * fma(-0.5, Float64(im * im), 1.0))
end

Wolfram

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

Derivation

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 51.6%

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 31.9%

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

7. Applied rewrites 31.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f64 N/A

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

   5. metadata-eval 31.9%

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

   6. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) + 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) + 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(pow2, \left(im \cdot im\right)\right) + 1\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(re - -1\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(im \cdot im\right)\right) + 1\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(re - -1\right) \cdot \mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{-1}{2}, im \cdot im, 1\right)\right)\right) \]

9. Applied rewrites 31.9%

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

Alternative 6: 30.2% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (fma -0.5 (* im im) 1.0))

C

double code(double re, double im) {
	return fma(-0.5, (im * im), 1.0);
}

Julia

function code(re, im)
	return fma(-0.5, Float64(im * im), 1.0)
end

Wolfram

code[re_, im_] := N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

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 50.8%

      \[\leadsto \cos im \]

4. Applied rewrites 50.8%

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

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 30.2%

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

7. Applied rewrites 30.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 30.2%

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

9. Applied rewrites 30.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, im \cdot im, 1\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025227 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))