math.cos on complex, imaginary part

Percentage Accurate: 65.1% → 99.9%

Time: 5.7s

Alternatives: 14

Speedup: 1.2×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (sin(re) * 0.5d0) * ((-2.0d0) * sinh(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-sin.f64 65.1

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lower-neg.f64 N/A

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

   12. sinh-undef N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sinh.f64 99.9

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

3. Applied rewrites 99.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sinh.f64 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sinh.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-300)
     (* (* (sinh im) (+ re re)) -0.5)
     (if (<= t_0 0.02)
       (* (* (sin re) 0.5) (* (fma -0.3333333333333333 (* im im) -2.0) im))
       (* (* (fma (* re re) -0.08333333333333333 0.5) re) (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = (sinh(im) * (re + re)) * -0.5;
	} else if (t_0 <= 0.02) {
		tmp = (sin(re) * 0.5) * (fma(-0.3333333333333333, (im * im), -2.0) * im);
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 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(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(Float64(sinh(im) * Float64(re + re)) * -0.5);
	elseif (t_0 <= 0.02)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 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[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * 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 (neg.f64 im)) (exp.f64 im))) < -2.00000000000000005e-300

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

   if -2.00000000000000005e-300 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. sinh-undef-rev N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{im} \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \]

      5. sub-flip N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\sin re \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot {im}^{2} + -2\right) \cdot im\right) \]

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.9

         \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

   6. Applied rewrites 83.9%

      \[\leadsto \left(\sin re \cdot 0.5\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 36.4%

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

Alternative 3: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -2e-300)
     (* (* (sinh im) (+ re re)) -0.5)
     (if (<= t_0 0.02)
       (* (- im) (sin re))
       (* (* (fma (* re re) -0.08333333333333333 0.5) re) (- 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -2e-300) {
		tmp = (sinh(im) * (re + re)) * -0.5;
	} else if (t_0 <= 0.02) {
		tmp = -im * sin(re);
	} else {
		tmp = (fma((re * re), -0.08333333333333333, 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(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -2e-300)
		tmp = Float64(Float64(sinh(im) * Float64(re + re)) * -0.5);
	elseif (t_0 <= 0.02)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 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[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-300], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * 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 (neg.f64 im)) (exp.f64 im))) < -2.00000000000000005e-300

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

   if -2.00000000000000005e-300 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0200000000000000004

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   if 0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 36.4%

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

Alternative 4: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(-2 \cdot \sinh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 2e-7)
   (* (* -2.0 (sinh im)) (* (fma (* re re) -0.08333333333333333 0.5) re))
   (* (* 0.5 re) (- 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 2e-7) {
		tmp = (-2.0 * sinh(im)) * (fma((re * re), -0.08333333333333333, 0.5) * re);
	} else {
		tmp = (0.5 * re) * (1.0 - exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 2e-7)
		tmp = Float64(Float64(-2.0 * sinh(im)) * Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(-2.0 * N[Sinh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $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 #s(literal 1/2 binary64) (sin.f64 re)) < 1.9999999999999999e-7

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sinh-undef-rev N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sinh.f64 62.6

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

   6. Applied rewrites 62.6%

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

   if 1.9999999999999999e-7 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.1%

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

Alternative 5: 61.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * 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 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 36.4%

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

Alternative 6: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma -0.3333333333333333 (* im im) -2.0) im)
    (* (* (* re re) -0.08333333333333333) re))
   (* (* (sinh im) (+ re re)) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(-0.3333333333333333, (im * im), -2.0) * im) * (((re * re) * -0.08333333333333333) * re);
	} else {
		tmp = (sinh(im) * (re + re)) * -0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im) * Float64(Float64(Float64(re * re) * -0.08333333333333333) * re));
	else
		tmp = Float64(Float64(sinh(im) * Float64(re + re)) * -0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sinh-undef-rev N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sinh.f64 62.6

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

   6. Applied rewrites 62.6%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      3. sub-flip N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} + -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, {im}^{2}, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      7. lower-*.f64 53.7

         \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   10. Taylor expanded in re around inf

       \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left({re}^{2} \cdot \frac{-1}{12}\right) \cdot re\right) \]

       3. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12}\right) \cdot re\right) \]

       4. lift-*.f64 24.3

          \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \]

   12. Applied rewrites 24.3%

       \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\left(\left(re \cdot re\right) \cdot -0.08333333333333333\right) \cdot re\right) \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

Alternative 7: 57.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) 0.0)
   (* (* (sinh im) (+ re re)) -0.5)
   (*
    (* (* (* im im) -0.3333333333333333) im)
    (* (fma (* re re) -0.08333333333333333 0.5) re))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (sinh(im) * (re + re)) * -0.5;
	} else {
		tmp = (((im * im) * -0.3333333333333333) * im) * (fma((re * re), -0.08333333333333333, 0.5) * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(Float64(sinh(im) * Float64(re + re)) * -0.5);
	else
		tmp = Float64(Float64(Float64(Float64(im * im) * -0.3333333333333333) * im) * Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. sinh-undef-rev N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sinh.f64 62.6

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

   6. Applied rewrites 62.6%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot \color{blue}{im}\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      3. sub-flip N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2} + -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, {im}^{2}, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{3}, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      7. lower-*.f64 53.7

         \[\leadsto \left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   9. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   10. Taylor expanded in im around inf

       \[\leadsto \left(\left(\frac{-1}{3} \cdot {im}^{2}\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{3}\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({im}^{2} \cdot \frac{-1}{3}\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

       4. lift-*.f64 41.9

          \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   12. Applied rewrites 41.9%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.3333333333333333\right) \cdot im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* re re) im) 0.16666666666666666 (- im)) re)
   (* (* (sinh im) (+ re re)) -0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((re * re) * im), 0.16666666666666666, -im) * re;
	} else {
		tmp = (sinh(im) * (re + re)) * -0.5;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Sinh[im], $MachinePrecision] * N[(re + re), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 35.7

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

   7. Applied rewrites 35.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sin.f64 65.1

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-neg.f64 N/A

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

      12. sinh-undef N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. rec-exp N/A

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

      3. sinh-undef-rev N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 62.4

          \[\leadsto \left(\sinh im \cdot \left(re + re\right)\right) \cdot -0.5 \]

   8. Applied rewrites 62.4%

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

Alternative 9: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))) -1e-11)
   (* (* 0.5 re) (- 1.0 (exp im)))
   (* (fma (* (* re re) im) 0.16666666666666666 (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) - exp(im))) <= -1e-11) {
		tmp = (0.5 * re) * (1.0 - exp(im));
	} else {
		tmp = fma(((re * re) * im), 0.16666666666666666, -im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im))) <= -1e-11)
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 - exp(im)));
	else
		tmp = Float64(fma(Float64(Float64(re * re) * im), 0.16666666666666666, Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-11], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -9.99999999999999939e-12

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 33.1%

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

   if -9.99999999999999939e-12 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 35.7

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

   7. Applied rewrites 35.7%

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

Alternative 10: 48.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* re re) im) 0.16666666666666666 (- im)) re)
   (* (* (* (fma -0.3333333333333333 (* im im) -2.0) im) re) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((re * re) * im), 0.16666666666666666, -im) * re;
	} else {
		tmp = ((fma(-0.3333333333333333, (im * im), -2.0) * im) * re) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(re * re) * im), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(Float64(fma(-0.3333333333333333, Float64(im * im), -2.0) * im) * re) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 35.7

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

   7. Applied rewrites 35.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 62.4

         \[\leadsto \left(\left(-2 \cdot \sinh im\right) \cdot re\right) \cdot 0.5 \]

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in im around 0

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

      1. sinh-undef-rev N/A

         \[\leadsto \left(\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \cdot re\right) \cdot \frac{1}{2} \]

      2. sub-negate-rev N/A

         \[\leadsto \left(\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right) \cdot re\right) \cdot \frac{1}{2} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \cdot re\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right) \cdot re\right) \cdot \frac{1}{2} \]

      5. sub-flip N/A

         \[\leadsto \left(\left(\left(\frac{-1}{3} \cdot {im}^{2} + \left(\mathsf{neg}\left(2\right)\right)\right) \cdot im\right) \cdot re\right) \cdot \frac{1}{2} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{3} \cdot {im}^{2} + -2\right) \cdot im\right) \cdot re\right) \cdot \frac{1}{2} \]

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 52.5

         \[\leadsto \left(\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot re\right) \cdot 0.5 \]

   7. Applied rewrites 52.5%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \cdot re\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot re, -0.16666666666666666, -re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* re re) im) 0.16666666666666666 (- im)) re)
   (* (fma (* (* im im) re) -0.16666666666666666 (- re)) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((re * re) * im), 0.16666666666666666, -im) * re;
	} else {
		tmp = fma(((im * im) * re), -0.16666666666666666, -re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(re * re) * im), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(fma(Float64(Float64(im * im) * re), -0.16666666666666666, Float64(-re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666 + (-re)), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 35.7

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

   7. Applied rewrites 35.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lower-neg.f64 N/A

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

      7. sinh-undef N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sinh.f64 62.4

         \[\leadsto \left(\left(-2 \cdot \sinh im\right) \cdot re\right) \cdot 0.5 \]

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.2

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

   7. Applied rewrites 49.2%

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

Alternative 12: 33.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 0.05)
   (* (fma (* (* re re) im) 0.16666666666666666 (- im)) re)
   (* (- im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 0.05) {
		tmp = fma(((re * re) * im), 0.16666666666666666, -im) * re;
	} else {
		tmp = -im * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 0.05)
		tmp = Float64(fma(Float64(Float64(re * re) * im), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(-im) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-im) * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 0.050000000000000003

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lift-neg.f64 35.7

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

   7. Applied rewrites 35.7%

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

   if 0.050000000000000003 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot re \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot re \]

      4. lift-neg.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 13: 33.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(-im\right) \cdot \left(\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (- im) (* (* (* re re) -0.16666666666666666) re))
   (* (- im) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = -im * (((re * re) * -0.16666666666666666) * re);
	} else {
		tmp = -im * 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)) <= (-0.01d0)) then
        tmp = -im * (((re * re) * (-0.16666666666666666d0)) * re)
    else
        tmp = -im * re
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = -im * (((re * re) * -0.16666666666666666) * re)
	else:
		tmp = -im * re
	return tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[((-im) * N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], N[((-im) * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 35.7

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

   7. Applied rewrites 35.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 23.3

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

   10. Applied rewrites 23.3%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 65.1%

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

   2. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot re \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot re \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot re \]

      4. lift-neg.f64 31.9

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

   7. Applied rewrites 31.9%

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

Alternative 14: 31.9% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \left(-im\right) \cdot re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- im) re))

C

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

Java

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

Python

def code(re, im):
	return -im * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

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

2. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sin.f64 51.7

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

4. Applied rewrites 51.7%

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

5. Taylor expanded in re around 0

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

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot im\right) \cdot re \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot im\right) \cdot re \]

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot re \]

   4. lift-neg.f64 31.9

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

7. Applied rewrites 31.9%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))