math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 10

Speedup: 1.3×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(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 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(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 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\sin re \cdot \cosh im \]

FPCore

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

C

double code(double re, double im) {
	return sin(re) * cosh(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 = sin(re) * cosh(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \frac{1}{2}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \sin re \cdot \left(\left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

   7. mult-flip N/A

      \[\leadsto \sin re \cdot \color{blue}{\frac{e^{0 - im} + e^{im}}{2}} \]

   8. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{0 - im} + e^{im}}}{2} \]

   9. +-commutative N/A

      \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im} + e^{0 - im}}}{2} \]

   10. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{\color{blue}{e^{im}} + e^{0 - im}}{2} \]

   11. lift-exp.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + \color{blue}{e^{0 - im}}}{2} \]

   12. lift--.f64 N/A

       \[\leadsto \sin re \cdot \frac{e^{im} + e^{\color{blue}{0 - im}}}{2} \]

   13. sub0-neg N/A

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

   14. cosh-def N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|re\right|\right)\\ t_1 := \left(0.5 \cdot t\_0\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(e^{im} + e^{-im}\right) \cdot \left(\mathsf{fma}\left(\left|re\right| \cdot \left|re\right|, -0.08333333333333333, 0.5\right) \cdot \left|re\right|\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left|re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs re)))
       (t_1 (* (* 0.5 t_0) (+ (exp (- 0.0 im)) (exp im)))))
  (*
   (copysign 1.0 re)
   (if (<= t_1 (- INFINITY))
     (*
      (+ (exp im) (exp (- im)))
      (*
       (fma (* (fabs re) (fabs re)) -0.08333333333333333 0.5)
       (fabs re)))
     (if (<= t_1 1.0) t_0 (* (cosh im) (fabs re)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(re));
	double t_1 = (0.5 * t_0) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (exp(im) + exp(-im)) * (fma((fabs(re) * fabs(re)), -0.08333333333333333, 0.5) * fabs(re));
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = cosh(im) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = sin(abs(re))
	t_1 = Float64(Float64(0.5 * t_0) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(fma(Float64(abs(re) * abs(re)), -0.08333333333333333, 0.5) * abs(re)));
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(cosh(im) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], t$95$0, N[(N[Cosh[im], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 34.3%

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      6. lower-*.f64 34.3%

         \[\leadsto 2 \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      11. lower-fma.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 34.3%

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]

   10. Taylor expanded in im around inf

       \[\leadsto \color{blue}{\left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{im} + \color{blue}{e^{\mathsf{neg}\left(im\right)}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       3. lower-exp.f64 N/A

          \[\leadsto \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       4. lower-neg.f64 62.7%

          \[\leadsto \left(e^{im} + e^{-im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   12. Applied rewrites 62.7%

       \[\leadsto \color{blue}{\left(e^{im} + e^{-im}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 63.3%

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 63.2%

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-exp.f64 N/A

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

      13. exp-neg N/A

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

      14. distribute-lft-in N/A

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

      15. lift-exp.f64 N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

          \[\leadsto \frac{1}{\frac{2}{re \cdot \left(2 \cdot \cosh im\right)}} \]

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-rgt-in N/A

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

      22. *-lft-identity N/A

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

      23. *-lft-identity N/A

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

      24. lower-+.f64 63.2%

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

   8. Applied rewrites 63.2%

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

   9. Taylor expanded in im around 0

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

       1. lower-sin.f64 50.5%

          \[\leadsto \sin re \]

   11. Applied rewrites 50.5%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. metadata-eval N/A

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

      10. mult-flip-rev N/A

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

      11. div-add N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. cosh-def N/A

          \[\leadsto \cosh im \cdot re \]

      16. lift-cosh.f64 N/A

          \[\leadsto \cosh im \cdot re \]

      17. lower-*.f64 63.3%

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

   6. Applied rewrites 63.3%

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

Alternative 3: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|re\right|\right)\\ t_1 := \left(0.5 \cdot t\_0\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(2 + {im}^{2}\right) \cdot \left(\mathsf{fma}\left(\left|re\right| \cdot \left|re\right|, -0.08333333333333333, 0.5\right) \cdot \left|re\right|\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left|re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs re)))
       (t_1 (* (* 0.5 t_0) (+ (exp (- 0.0 im)) (exp im)))))
  (*
   (copysign 1.0 re)
   (if (<= t_1 (- INFINITY))
     (*
      (+ 2.0 (pow im 2.0))
      (*
       (fma (* (fabs re) (fabs re)) -0.08333333333333333 0.5)
       (fabs re)))
     (if (<= t_1 1.0) t_0 (* (cosh im) (fabs re)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(re));
	double t_1 = (0.5 * t_0) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (2.0 + pow(im, 2.0)) * (fma((fabs(re) * fabs(re)), -0.08333333333333333, 0.5) * fabs(re));
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = cosh(im) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = sin(abs(re))
	t_1 = Float64(Float64(0.5 * t_0) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 + (im ^ 2.0)) * Float64(fma(Float64(abs(re) * abs(re)), -0.08333333333333333, 0.5) * abs(re)));
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(cosh(im) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(N[(2.0 + N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], t$95$0, N[(N[Cosh[im], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 34.3%

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      6. lower-*.f64 34.3%

         \[\leadsto 2 \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      11. lower-fma.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 34.3%

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]
   11. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(2 + \color{blue}{{im}^{2}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       2. lower-pow.f64 49.7%

          \[\leadsto \left(2 + {im}^{\color{blue}{2}}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   12. Applied rewrites 49.7%

       \[\leadsto \color{blue}{\left(2 + {im}^{2}\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 63.3%

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 63.2%

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-exp.f64 N/A

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

      13. exp-neg N/A

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

      14. distribute-lft-in N/A

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

      15. lift-exp.f64 N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

          \[\leadsto \frac{1}{\frac{2}{re \cdot \left(2 \cdot \cosh im\right)}} \]

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-rgt-in N/A

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

      22. *-lft-identity N/A

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

      23. *-lft-identity N/A

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

      24. lower-+.f64 63.2%

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

   8. Applied rewrites 63.2%

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

   9. Taylor expanded in im around 0

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

       1. lower-sin.f64 50.5%

          \[\leadsto \sin re \]

   11. Applied rewrites 50.5%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. metadata-eval N/A

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

      10. mult-flip-rev N/A

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

      11. div-add N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. cosh-def N/A

          \[\leadsto \cosh im \cdot re \]

      16. lift-cosh.f64 N/A

          \[\leadsto \cosh im \cdot re \]

      17. lower-*.f64 63.3%

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

   6. Applied rewrites 63.3%

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

Alternative 4: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|re\right|\right)\\ t_1 := \left(0.5 \cdot t\_0\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ t_2 := \left|re\right| \cdot \left|re\right|\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\sqrt{t\_2 \cdot t\_2}, -0.08333333333333333, 0.5\right) \cdot \left|re\right|\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left|re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sin (fabs re)))
       (t_1 (* (* 0.5 t_0) (+ (exp (- 0.0 im)) (exp im))))
       (t_2 (* (fabs re) (fabs re))))
  (*
   (copysign 1.0 re)
   (if (<= t_1 (- INFINITY))
     (*
      2.0
      (* (fma (sqrt (* t_2 t_2)) -0.08333333333333333 0.5) (fabs re)))
     (if (<= t_1 1.0) t_0 (* (cosh im) (fabs re)))))))

C

double code(double re, double im) {
	double t_0 = sin(fabs(re));
	double t_1 = (0.5 * t_0) * (exp((0.0 - im)) + exp(im));
	double t_2 = fabs(re) * fabs(re);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * (fma(sqrt((t_2 * t_2)), -0.08333333333333333, 0.5) * fabs(re));
	} else if (t_1 <= 1.0) {
		tmp = t_0;
	} else {
		tmp = cosh(im) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = sin(abs(re))
	t_1 = Float64(Float64(0.5 * t_0) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	t_2 = Float64(abs(re) * abs(re))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(fma(sqrt(Float64(t_2 * t_2)), -0.08333333333333333, 0.5) * abs(re)));
	elseif (t_1 <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(cosh(im) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * t$95$0), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, (-Infinity)], N[(2.0 * N[(N[(N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], t$95$0, N[(N[Cosh[im], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 34.3%

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      6. lower-*.f64 34.3%

         \[\leadsto 2 \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      11. lower-fma.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 34.3%

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{re \cdot re} \cdot \sqrt{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       2. sqrt-unprod N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       4. lower-*.f64 35.1%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   11. Applied rewrites 35.1%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-neg N/A

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

      9. lift-exp.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 63.3%

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

   6. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip-rev N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 63.2%

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-exp.f64 N/A

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

      13. exp-neg N/A

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

      14. distribute-lft-in N/A

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

      15. lift-exp.f64 N/A

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

      16. cosh-undef N/A

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

      17. lift-cosh.f64 N/A

          \[\leadsto \frac{1}{\frac{2}{re \cdot \left(2 \cdot \cosh im\right)}} \]

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. distribute-rgt-in N/A

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

      22. *-lft-identity N/A

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

      23. *-lft-identity N/A

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

      24. lower-+.f64 63.2%

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

   8. Applied rewrites 63.2%

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

   9. Taylor expanded in im around 0

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

       1. lower-sin.f64 50.5%

          \[\leadsto \sin re \]

   11. Applied rewrites 50.5%

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

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. metadata-eval N/A

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

      10. mult-flip-rev N/A

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

      11. div-add N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. cosh-def N/A

          \[\leadsto \cosh im \cdot re \]

      16. lift-cosh.f64 N/A

          \[\leadsto \cosh im \cdot re \]

      17. lower-*.f64 63.3%

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

   6. Applied rewrites 63.3%

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

Alternative 5: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|re\right| \cdot \left|re\right|\\ \mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\sqrt{t\_0 \cdot t\_0}, -0.08333333333333333, 0.5\right) \cdot \left|re\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left|re\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* (fabs re) (fabs re))))
  (*
   (copysign 1.0 re)
   (if (<=
        (* (* 0.5 (sin (fabs re))) (+ (exp (- 0.0 im)) (exp im)))
        -0.005)
     (*
      2.0
      (* (fma (sqrt (* t_0 t_0)) -0.08333333333333333 0.5) (fabs re)))
     (* (cosh im) (fabs re))))))

C

double code(double re, double im) {
	double t_0 = fabs(re) * fabs(re);
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= -0.005) {
		tmp = 2.0 * (fma(sqrt((t_0 * t_0)), -0.08333333333333333, 0.5) * fabs(re));
	} else {
		tmp = cosh(im) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	t_0 = Float64(abs(re) * abs(re))
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.005)
		tmp = Float64(2.0 * Float64(fma(sqrt(Float64(t_0 * t_0)), -0.08333333333333333, 0.5) * abs(re)));
	else
		tmp = Float64(cosh(im) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Abs[re], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(2.0 * N[(N[(N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 34.3%

         \[\leadsto \color{blue}{2 \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      6. lower-*.f64 34.3%

         \[\leadsto 2 \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      8. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      9. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      11. lower-fma.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      13. unpow2 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      14. lower-*.f64 34.3%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 34.3%

      \[\leadsto \color{blue}{2 \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{re \cdot re} \cdot \sqrt{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       2. sqrt-unprod N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       4. lower-*.f64 35.1%

          \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   11. Applied rewrites 35.1%

       \[\leadsto 2 \cdot \left(\mathsf{fma}\left(\sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. metadata-eval N/A

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

      10. mult-flip-rev N/A

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

      11. div-add N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. cosh-def N/A

          \[\leadsto \cosh im \cdot re \]

      16. lift-cosh.f64 N/A

          \[\leadsto \cosh im \cdot re \]

      17. lower-*.f64 63.3%

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

   6. Applied rewrites 63.3%

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

Alternative 6: 72.1% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;\left(\left|re\right| \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot \left|re\right|, \left|re\right|, 0.5\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot \left|re\right|\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (* (* 0.5 (sin (fabs re))) (+ (exp (- 0.0 im)) (exp im)))
      -0.005)
   (*
    (*
     (fabs re)
     (fma (* -0.08333333333333333 (fabs re)) (fabs re) 0.5))
    2.0)
   (* (cosh im) (fabs re)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= -0.005) {
		tmp = (fabs(re) * fma((-0.08333333333333333 * fabs(re)), fabs(re), 0.5)) * 2.0;
	} else {
		tmp = cosh(im) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.005)
		tmp = Float64(Float64(abs(re) * fma(Float64(-0.08333333333333333 * abs(re)), abs(re), 0.5)) * 2.0);
	else
		tmp = Float64(cosh(im) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(-0.08333333333333333 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)\right) \cdot 2 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)\right) \cdot 2 \]

      5. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot \left(re \cdot re\right) + \frac{1}{2}\right)\right) \cdot 2 \]

      6. associate-*r* N/A

         \[\leadsto \left(re \cdot \left(\left(\frac{-1}{12} \cdot re\right) \cdot re + \frac{1}{2}\right)\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\frac{-1}{12} \cdot re, \color{blue}{re}, \frac{1}{2}\right)\right) \cdot 2 \]

      8. lower-*.f64 34.3%

         \[\leadsto \left(re \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot re, re, 0.5\right)\right) \cdot 2 \]

   9. Applied rewrites 34.3%

      \[\leadsto \left(re \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot re, \color{blue}{re}, 0.5\right)\right) \cdot 2 \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. metadata-eval N/A

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

      10. mult-flip-rev N/A

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

      11. div-add N/A

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

      12. lift-exp.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. cosh-def N/A

          \[\leadsto \cosh im \cdot re \]

      16. lift-cosh.f64 N/A

          \[\leadsto \cosh im \cdot re \]

      17. lower-*.f64 63.3%

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

   6. Applied rewrites 63.3%

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

Alternative 7: 57.0% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, re\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin \left(\left|re\right|\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.0004:\\ \;\;\;\;\left(\left|re\right| \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot \left|re\right|, \left|re\right|, 0.5\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot \left|re\right|\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 re)
 (if (<=
      (* (* 0.5 (sin (fabs re))) (+ (exp (- 0.0 im)) (exp im)))
      0.0004)
   (*
    (*
     (fabs re)
     (fma (* -0.08333333333333333 (fabs re)) (fabs re) 0.5))
    2.0)
   (* (fma (* im im) 0.5 1.0) (fabs re)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(fabs(re))) * (exp((0.0 - im)) + exp(im))) <= 0.0004) {
		tmp = (fabs(re) * fma((-0.08333333333333333 * fabs(re)), fabs(re), 0.5)) * 2.0;
	} else {
		tmp = fma((im * im), 0.5, 1.0) * fabs(re);
	}
	return copysign(1.0, re) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(abs(re))) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.0004)
		tmp = Float64(Float64(abs(re) * fma(Float64(-0.08333333333333333 * abs(re)), abs(re), 0.5)) * 2.0);
	else
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * abs(re));
	end
	return Float64(copysign(1.0, re) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[re]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Sin[N[Abs[re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0004], N[(N[(N[Abs[re], $MachinePrecision] * N[(N[(-0.08333333333333333 * N[Abs[re], $MachinePrecision]), $MachinePrecision] * N[Abs[re], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[Abs[re], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 4.0000000000000002e-4

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot 2 \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot 2 \]

      4. lower-pow.f64 34.3%

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot 2 \]

   7. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot 2 \]

      2. +-commutative N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)\right) \cdot 2 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)\right) \cdot 2 \]

      5. unpow2 N/A

         \[\leadsto \left(re \cdot \left(\frac{-1}{12} \cdot \left(re \cdot re\right) + \frac{1}{2}\right)\right) \cdot 2 \]

      6. associate-*r* N/A

         \[\leadsto \left(re \cdot \left(\left(\frac{-1}{12} \cdot re\right) \cdot re + \frac{1}{2}\right)\right) \cdot 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(re \cdot \mathsf{fma}\left(\frac{-1}{12} \cdot re, \color{blue}{re}, \frac{1}{2}\right)\right) \cdot 2 \]

      8. lower-*.f64 34.3%

         \[\leadsto \left(re \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot re, re, 0.5\right)\right) \cdot 2 \]

   9. Applied rewrites 34.3%

      \[\leadsto \left(re \cdot \mathsf{fma}\left(-0.08333333333333333 \cdot re, \color{blue}{re}, 0.5\right)\right) \cdot 2 \]

   if 4.0000000000000002e-4 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 63.3%

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 48.3%

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

   7. Applied rewrites 48.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 48.3%

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 48.3%

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

   9. Applied rewrites 48.3%

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

Alternative 8: 48.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 63.3%

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 48.3%

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

7. Applied rewrites 48.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 48.3%

      \[\leadsto \mathsf{fma}\left({im}^{2} \cdot re, 0.5, re\right) \]

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 48.3%

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

9. Applied rewrites 48.3%

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

Alternative 9: 48.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 63.3%

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in im around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 48.3%

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

7. Applied rewrites 48.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 48.3%

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 48.3%

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

9. Applied rewrites 48.3%

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

Alternative 10: 26.6% accurate, 9.7× speedup

Math

\[\left(0.5 \cdot re\right) \cdot 2 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 re) 2.0))

C

double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

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 = (0.5d0 * re) * 2.0d0
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Python

def code(re, im):
	return (0.5 * re) * 2.0

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * 2.0;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 50.5%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

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

   1. lower-*.f64 26.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot 2 \]

7. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))