math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 21

Speedup: 1.5×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 71.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 49.2

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

   11. Applied rewrites 49.2%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 99.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 49.2

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

   11. Applied rewrites 49.2%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 49.2

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

   11. Applied rewrites 49.2%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin re \]

   5. Applied rewrites 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* (fma (* re re) -0.08333333333333333 0.5) re) (* im im))
     (if (<= t_0 1.0)
       (sin re)
       (*
        (*
         0.5
         (*
          (fma
           (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
           (* re re)
           1.0)
          re))
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (im * im);
	} else if (t_0 <= 1.0) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * (fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * re)) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(im * im));
	elseif (t_0 <= 1.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * re)) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[re], $MachinePrecision], N[(N[(0.5 * N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 49.2

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

   11. Applied rewrites 49.2%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin re \]

   5. Applied rewrites 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 84.9

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      11. lower-*.f64 66.5

          \[\leadsto \left(0.5 \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 66.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 40.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 -0.01)
     (* (* (* re re) -0.16666666666666666) re)
     (if (<= t_0 0.9995) re (* (* re 0.5) (* im im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = ((re * re) * -0.16666666666666666) * re;
	} else if (t_0 <= 0.9995) {
		tmp = re;
	} else {
		tmp = (re * 0.5) * (im * im);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
    if (t_0 <= (-0.01d0)) then
        tmp = ((re * re) * (-0.16666666666666666d0)) * re
    else if (t_0 <= 0.9995d0) then
        tmp = re
    else
        tmp = (re * 0.5d0) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
	double tmp;
	if (t_0 <= -0.01) {
		tmp = ((re * re) * -0.16666666666666666) * re;
	} else if (t_0 <= 0.9995) {
		tmp = re;
	} else {
		tmp = (re * 0.5) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))
	tmp = 0
	if t_0 <= -0.01:
		tmp = ((re * re) * -0.16666666666666666) * re
	elif t_0 <= 0.9995:
		tmp = re
	else:
		tmp = (re * 0.5) * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(Float64(Float64(re * re) * -0.16666666666666666) * re);
	elseif (t_0 <= 0.9995)
		tmp = re;
	else
		tmp = Float64(Float64(re * 0.5) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = ((re * re) * -0.16666666666666666) * re;
	elseif (t_0 <= 0.9995)
		tmp = re;
	else
		tmp = (re * 0.5) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], re, N[(N[(re * 0.5), $MachinePrecision] * N[(im * im), $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))) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 33.2

         \[\leadsto \sin re \]

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re \]

      6. lower-*.f64 12.9

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \]

   8. Applied rewrites 12.9%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{6}\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{6}\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) \cdot re \]

       4. lift-*.f64 12.5

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

   11. Applied rewrites 12.5%

       \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

   if -0.0100000000000000002 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.99950000000000006

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 98.6

         \[\leadsto \sin re \]

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

      \[\leadsto re \]

   if 0.99950000000000006 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 52.1%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 44.5

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

   11. Applied rewrites 44.5%

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

Alternative 7: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= (* t_0 (+ (exp (- 0.0 im)) (exp im))) 1.0)
     (*
      t_0
      (fma
       (fma
        (fma 0.002777777777777778 (* im im) 0.08333333333333333)
        (* im im)
        1.0)
       (* im im)
       2.0))
     (* (* 0.5 re) (+ 1.0 (exp im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((t_0 * (exp((0.0 - im)) + exp(im))) <= 1.0) {
		tmp = t_0 * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 1.0)
		tmp = Float64(t_0 * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$0 * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $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))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 94.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 94.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

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

Alternative 8: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \cdot \left(e^{0 - im} + e^{im}\right) \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= (* t_0 (+ (exp (- 0.0 im)) (exp im))) 1.0)
     (*
      t_0
      (fma
       (fma (* (* im im) 0.002777777777777778) (* im im) 1.0)
       (* im im)
       2.0))
     (* (* 0.5 re) (+ 1.0 (exp im))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if ((t_0 * (exp((0.0 - im)) + exp(im))) <= 1.0) {
		tmp = t_0 * fma(fma(((im * im) * 0.002777777777777778), (im * im), 1.0), (im * im), 2.0);
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (Float64(t_0 * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 1.0)
		tmp = Float64(t_0 * fma(fma(Float64(Float64(im * im) * 0.002777777777777778), Float64(im * im), 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$0 * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $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))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 94.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 94.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. pow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. lift-*.f64 94.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 94.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 38.9%

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

Alternative 9: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.005:\\ \;\;\;\;\left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.005)
   (* (* 0.5 (* (fma -0.16666666666666666 (* re re) 1.0) re)) (fma im im 2.0))
   (*
    (* re 0.5)
    (fma
     (fma (* (* im im) 0.002777777777777778) (* im im) 1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.005) {
		tmp = (0.5 * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * fma(im, im, 2.0);
	} else {
		tmp = (re * 0.5) * fma(fma(((im * im) * 0.002777777777777778), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.005)
		tmp = Float64(Float64(0.5 * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(re * 0.5) * fma(fma(Float64(Float64(im * im) * 0.002777777777777778), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(0.5 * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lower-*.f64 59.7

         \[\leadsto \left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 59.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 89.8

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 89.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. pow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. lift-*.f64 89.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 89.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

       2. lower-*.f64 45.0

          \[\leadsto \left(re \cdot \color{blue}{0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   11. Applied rewrites 45.0%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.005:\\ \;\;\;\;\left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.005)
   (* (* 0.5 (* (fma -0.16666666666666666 (* re re) 1.0) re)) (fma im im 2.0))
   (* (* 0.5 re) (fma (* (fma (* im im) 0.08333333333333333 1.0) im) im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.005) {
		tmp = (0.5 * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * fma(im, im, 2.0);
	} else {
		tmp = (0.5 * re) * fma((fma((im * im), 0.08333333333333333, 1.0) * im), im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.005)
		tmp = Float64(Float64(0.5 * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(0.5 * re) * fma(Float64(fma(Float64(im * im), 0.08333333333333333, 1.0) * im), im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(0.5 * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lower-*.f64 59.7

         \[\leadsto \left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 59.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 84.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 41.4%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 41.4

         \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 41.4%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, \color{blue}{im}, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) -0.01)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (* im im))
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (im * im);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(im * im));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $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))) < -0.0100000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 37.2

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 37.2%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 34.0

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

   11. Applied rewrites 34.0%

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

   if -0.0100000000000000002 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.5%

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

Alternative 12: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.005)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.005) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.005)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 59.1

         \[\leadsto \sin re \]

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re \]

      6. lower-*.f64 46.5

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \]

   8. Applied rewrites 46.5%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 31.4%

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

Alternative 13: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.005)
   (* (fma -0.16666666666666666 (* re re) 1.0) re)
   (* (* re 0.5) (* im im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.005) {
		tmp = fma(-0.16666666666666666, (re * re), 1.0) * re;
	} else {
		tmp = (re * 0.5) * (im * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.005)
		tmp = Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re);
	else
		tmp = Float64(Float64(re * 0.5) * Float64(im * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.005], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(im * im), $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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 59.1

         \[\leadsto \sin re \]

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re \]

      6. lower-*.f64 46.5

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \]

   8. Applied rewrites 46.5%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]

   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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 36.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 36.9%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 31.5

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

   11. Applied rewrites 31.5%

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

Alternative 14: 49.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= t_0 -0.005)
     (* (* (fma (* re re) -0.08333333333333333 0.5) re) (* im im))
     (if (<= t_0 5e-7)
       (* (* 0.5 re) (fma im im 2.0))
       (*
        (fma
         (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
         (* re re)
         1.0)
        re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (im * im);
	} else if (t_0 <= 5e-7) {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	} else {
		tmp = fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(im * im));
	elseif (t_0 <= 5e-7)
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-7], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 29.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 24.9

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

   11. Applied rewrites 24.9%

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

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 4.99999999999999977e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.3%

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

   if 4.99999999999999977e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 50.0

         \[\leadsto \sin re \]

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right) \cdot re \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) \cdot {re}^{2} + 1\right) \cdot re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, {re}^{2}, 1\right) \cdot re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, {re}^{2}, 1\right) \cdot re \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, {re}^{2}, 1\right) \cdot re \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot \left(re \cdot re\right) - \frac{1}{6}, re \cdot re, 1\right) \cdot re \]

      11. lower-*.f64 21.6

          \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re \]

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \cdot 0.5 \end{array} \]

FPCore

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

C

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

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 = ((2.0d0 * cosh(im)) * sin(re)) * 0.5d0
end function

Java

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

Python

def code(re, im):
	return ((2.0 * math.cosh(im)) * math.sin(re)) * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. 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. lift-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. sub0-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \cdot 0.5} \]
5. Add Preprocessing

Alternative 16: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(1 + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 (sin re)) (+ 1.0 (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (1.0 + 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)) * (1.0d0 + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (1.0 + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (1.0 + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(1.0 + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (1.0 + exp(im));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 74.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]
6. Add Preprocessing

Alternative 17: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.004:\\ \;\;\;\;\left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.004)
   (*
    (* 0.5 (* (fma -0.16666666666666666 (* re re) 1.0) re))
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))
   (*
    (* re 0.5)
    (fma
     (fma (* (* im im) 0.002777777777777778) (* im im) 1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.004) {
		tmp = (0.5 * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	} else {
		tmp = (re * 0.5) * fma(fma(((im * im) * 0.002777777777777778), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.004)
		tmp = Float64(Float64(0.5 * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(Float64(re * 0.5) * fma(fma(Float64(Float64(im * im) * 0.002777777777777778), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.004], N[(N[(0.5 * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(re * 0.5), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.0040000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 91.9

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 91.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\left(re \cdot \left(1 + \frac{-1}{6} \cdot {re}^{2}\right)\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. lower-*.f64 70.0

         \[\leadsto \left(0.5 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 70.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   if 0.0040000000000000001 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + \color{blue}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), \color{blue}{{im}^{2}}, 2\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1, {\color{blue}{im}}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right), {\color{blue}{im}}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

      14. lower-*.f64 92.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 92.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({im}^{2} \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. pow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. lift-*.f64 92.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 92.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \frac{1}{360}, im \cdot im, 1\right), im \cdot im, 2\right) \]

       2. lower-*.f64 25.3

          \[\leadsto \left(re \cdot \color{blue}{0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]

   11. Applied rewrites 25.3%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 55.9% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (* im im))
   (* (* 0.5 re) (fma (* (fma (* im im) 0.08333333333333333 1.0) im) im 2.0))))

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333 + 1.0), $MachinePrecision] * im), $MachinePrecision] * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 29.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 24.9

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

   11. Applied rewrites 24.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 66.3

         \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, im, 2\right) \]

   10. Applied rewrites 66.3%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right) \cdot im, \color{blue}{im}, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 55.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* (fma (* re re) -0.08333333333333333 0.5) re) (* im im))
   (* (* 0.5 re) (fma (* (* im im) 0.08333333333333333) (* im im) 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (im * im);
	} else {
		tmp = (0.5 * re) * fma(((im * im) * 0.08333333333333333), (im * im), 2.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in im around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 29.8

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right) \]

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 24.9

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

   11. Applied rewrites 24.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot \color{blue}{im}, 2\right) \]

   5. Applied rewrites 87.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, \frac{1}{12}, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.08333333333333333, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(\frac{1}{12} \cdot {im}^{2}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 66.1

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

   11. Applied rewrites 66.1%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.08333333333333333, \color{blue}{im} \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.005:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.005)
   (* (* (* re re) -0.16666666666666666) re)
   re))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.005) {
		tmp = ((re * re) * -0.16666666666666666) * re;
	} else {
		tmp = re;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.005d0)) then
        tmp = ((re * re) * (-0.16666666666666666d0)) * re
    else
        tmp = re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.005) {
		tmp = ((re * re) * -0.16666666666666666) * re;
	} else {
		tmp = re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.005:
		tmp = ((re * re) * -0.16666666666666666) * re
	else:
		tmp = re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.005)
		tmp = Float64(Float64(Float64(re * re) * -0.16666666666666666) * re);
	else
		tmp = re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.005)
		tmp = ((re * re) * -0.16666666666666666) * re;
	else
		tmp = re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], re]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 49.4

         \[\leadsto \sin re \]

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {re}^{2}\right) \cdot re \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {re}^{2} + 1\right) \cdot re \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, {re}^{2}, 1\right) \cdot re \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, re \cdot re, 1\right) \cdot re \]

      6. lower-*.f64 18.2

         \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re \]

   8. Applied rewrites 18.2%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{6}\right) \cdot re \]

       2. lower-*.f64 N/A

          \[\leadsto \left({re}^{2} \cdot \frac{-1}{6}\right) \cdot re \]

       3. pow2 N/A

          \[\leadsto \left(\left(re \cdot re\right) \cdot \frac{-1}{6}\right) \cdot re \]

       4. lift-*.f64 18.1

          \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

   11. Applied rewrites 18.1%

       \[\leadsto \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lift-sin.f64 50.1

         \[\leadsto \sin re \]

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto re \]
   7. Step-by-step derivation

   8. Applied rewrites 34.5%

      \[\leadsto re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 26.6% accurate, 317.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

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 = re
end function

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lift-sin.f64 49.9

      \[\leadsto \sin re \]

5. Applied rewrites 49.9%

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

6. Taylor expanded in re around 0

   \[\leadsto re \]
7. Step-by-step derivation

8. Applied rewrites 26.6%

   \[\leadsto re \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))