(sin (* z1 (- z0 1)))

Percentage Accurate: 53.8% → 55.9%

Time: 2.7s

Alternatives: 12

Speedup: 0.3×

Specification

Math

\[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* z1 (- z0 1.0))))

C

double code(double z1, double z0) {
	return sin((z1 * (z0 - 1.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = sin((z1 * (z0 - 1.0d0)))
end function

Java

public static double code(double z1, double z0) {
	return Math.sin((z1 * (z0 - 1.0)));
}

Python

def code(z1, z0):
	return math.sin((z1 * (z0 - 1.0)))

Julia

function code(z1, z0)
	return sin(Float64(z1 * Float64(z0 - 1.0)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((z1 * (z0 - 1.0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(z1 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.8% accurate, 1.0× speedup

Math

\[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* z1 (- z0 1.0))))

C

double code(double z1, double z0) {
	return sin((z1 * (z0 - 1.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = sin((z1 * (z0 - 1.0d0)))
end function

Java

public static double code(double z1, double z0) {
	return Math.sin((z1 * (z0 - 1.0)));
}

Python

def code(z1, z0):
	return math.sin((z1 * (z0 - 1.0)))

Julia

function code(z1, z0)
	return sin(Float64(z1 * Float64(z0 - 1.0)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((z1 * (z0 - 1.0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(z1 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 55.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin \left(z1 \cdot \left(z0 - 1\right)\right) \leq 0.999:\\ \;\;\;\;\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{\frac{\left(z0 \cdot z1\right) \cdot 2}{\sqrt[3]{\pi \cdot \pi}}}{\sqrt[3]{\pi}}\right) \cdot \left(0.5 \cdot \pi\right)\right) \cdot \sin z1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(z1 \cdot \frac{\left(0.25 - z0\right) - 0.25}{\left(z0 - 0.5\right) + 0.5}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (if (<= (sin (* z1 (- z0 1.0))) 0.999)
  (-
   (* (sin (* z1 z0)) (cos z1))
   (*
    (sin
     (*
      (+ 1.0 (/ (/ (* (* z0 z1) 2.0) (cbrt (* PI PI))) (cbrt PI)))
      (* 0.5 PI)))
    (sin z1)))
  (sin (* z1 (/ (- (- 0.25 z0) 0.25) (+ (- z0 0.5) 0.5))))))

C

double code(double z1, double z0) {
	double tmp;
	if (sin((z1 * (z0 - 1.0))) <= 0.999) {
		tmp = (sin((z1 * z0)) * cos(z1)) - (sin(((1.0 + ((((z0 * z1) * 2.0) / cbrt((((double) M_PI) * ((double) M_PI)))) / cbrt(((double) M_PI)))) * (0.5 * ((double) M_PI)))) * sin(z1));
	} else {
		tmp = sin((z1 * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.sin((z1 * (z0 - 1.0))) <= 0.999) {
		tmp = (Math.sin((z1 * z0)) * Math.cos(z1)) - (Math.sin(((1.0 + ((((z0 * z1) * 2.0) / Math.cbrt((Math.PI * Math.PI))) / Math.cbrt(Math.PI))) * (0.5 * Math.PI))) * Math.sin(z1));
	} else {
		tmp = Math.sin((z1 * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return tmp;
}

Julia

function code(z1, z0)
	tmp = 0.0
	if (sin(Float64(z1 * Float64(z0 - 1.0))) <= 0.999)
		tmp = Float64(Float64(sin(Float64(z1 * z0)) * cos(z1)) - Float64(sin(Float64(Float64(1.0 + Float64(Float64(Float64(Float64(z0 * z1) * 2.0) / cbrt(Float64(pi * pi))) / cbrt(pi))) * Float64(0.5 * pi))) * sin(z1)));
	else
		tmp = sin(Float64(z1 * Float64(Float64(Float64(0.25 - z0) - 0.25) / Float64(Float64(z0 - 0.5) + 0.5))));
	end
	return tmp
end

Wolfram

code[z1_, z0_] := If[LessEqual[N[Sin[N[(z1 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.999], N[(N[(N[Sin[N[(z1 * z0), $MachinePrecision]], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(N[(1.0 + N[(N[(N[(N[(z0 * z1), $MachinePrecision] * 2.0), $MachinePrecision] / N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(z1 * N[(N[(N[(0.25 - z0), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(z0 - 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   3. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \color{blue}{\cos \left(z1 \cdot z0\right)} \cdot \sin z1 \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \color{blue}{\sin \left(z1 \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin z1 \]

      3. lower-sin.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \color{blue}{\sin \left(z1 \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin z1 \]

      4. +-commutative N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + z1 \cdot z0\right)} \cdot \sin z1 \]

      5. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + z1 \cdot z0\right)} \cdot \sin z1 \]

      6. lift-PI.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\frac{\color{blue}{\pi}}{2} + z1 \cdot z0\right) \cdot \sin z1 \]

      7. mult-flip N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\color{blue}{\pi \cdot \frac{1}{2}} + z1 \cdot z0\right) \cdot \sin z1 \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\pi \cdot \color{blue}{\frac{1}{2}} + z1 \cdot z0\right) \cdot \sin z1 \]

      9. lower-*.f64 54.8%

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\color{blue}{\pi \cdot 0.5} + z1 \cdot z0\right) \cdot \sin z1 \]

      10. lift-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\pi \cdot \frac{1}{2} + \color{blue}{z1 \cdot z0}\right) \cdot \sin z1 \]

      11. *-commutative N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot z1}\right) \cdot \sin z1 \]

      12. lift-*.f64 54.8%

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot z1}\right) \cdot \sin z1 \]

   5. Applied rewrites 54.8%

      \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \color{blue}{\sin \left(\pi \cdot 0.5 + z0 \cdot z1\right)} \cdot \sin z1 \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\pi \cdot \frac{1}{2} + z0 \cdot z1\right)} \cdot \sin z1 \]

      2. sum-to-mult N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \frac{1}{2}\right)\right)} \cdot \sin z1 \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}\right) \cdot \sin z1 \]

      4. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin z1 \]

      5. mult-flip N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \color{blue}{\frac{\pi}{2}}\right) \cdot \sin z1 \]

      6. lift-PI.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \cdot \sin z1 \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sin z1 \]

      8. lower-unsound-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\color{blue}{\left(1 + \frac{z0 \cdot z1}{\pi \cdot \frac{1}{2}}\right)} \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      9. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\pi \cdot \frac{1}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \color{blue}{\frac{1}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      11. mult-flip N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\frac{\pi}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      12. lift-PI.f64 N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

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

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{z0 \cdot z1}{\frac{\mathsf{PI}\left(\right)}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      14. lift-PI.f64 N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{\color{blue}{\pi}}{2}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      15. mult-flip N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\pi \cdot \frac{1}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      16. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\pi \cdot \color{blue}{\frac{1}{2}}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      17. *-commutative N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\frac{1}{2} \cdot \pi}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      18. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\frac{1}{2} \cdot \pi}}\right) \cdot \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin z1 \]

      19. lift-PI.f64 N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{1}{2} \cdot \pi}\right) \cdot \frac{\color{blue}{\pi}}{2}\right) \cdot \sin z1 \]

      20. mult-flip N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{1}{2} \cdot \pi}\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)}\right) \cdot \sin z1 \]

      21. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{1}{2} \cdot \pi}\right) \cdot \left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin z1 \]

      22. *-commutative N/A

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\frac{1}{2} \cdot \pi}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \pi\right)}\right) \cdot \sin z1 \]

      23. lower-*.f64 54.8%

          \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{0.5 \cdot \pi}\right) \cdot \color{blue}{\left(0.5 \cdot \pi\right)}\right) \cdot \sin z1 \]

   7. Applied rewrites 54.8%

      \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \color{blue}{\left(\left(1 + \frac{z0 \cdot z1}{0.5 \cdot \pi}\right) \cdot \left(0.5 \cdot \pi\right)\right)} \cdot \sin z1 \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{z0 \cdot z1}{\frac{1}{2} \cdot \pi}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{z0 \cdot z1}{\color{blue}{\frac{1}{2} \cdot \pi}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      3. associate-/r* N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{\frac{z0 \cdot z1}{\frac{1}{2}}}{\pi}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      4. lift-PI.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{\frac{z0 \cdot z1}{\frac{1}{2}}}{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      5. add-cube-cbrt N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \frac{\frac{z0 \cdot z1}{\frac{1}{2}}}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      6. associate-/r* N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{\frac{\frac{z0 \cdot z1}{\frac{1}{2}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

      7. lower-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{\frac{\frac{z0 \cdot z1}{\frac{1}{2}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}{\sqrt[3]{\mathsf{PI}\left(\right)}}}\right) \cdot \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \sin z1 \]

   9. Applied rewrites 54.8%

      \[\leadsto \sin \left(z1 \cdot z0\right) \cdot \cos z1 - \sin \left(\left(1 + \color{blue}{\frac{\frac{\left(z0 \cdot z1\right) \cdot 2}{\sqrt[3]{\pi \cdot \pi}}}{\sqrt[3]{\pi}}}\right) \cdot \left(0.5 \cdot \pi\right)\right) \cdot \sin z1 \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. *-inverses N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}\right)\right) \]

      3. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{\color{blue}{1} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + e^{\color{blue}{0}}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      5. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + \color{blue}{1}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      6. div-add N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\left(\frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right) \]

      8. flip-- N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

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

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

   3. Applied rewrites 12.4%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - 0.5\right) \cdot \left(z0 - 0.5\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}}\right) \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(\frac{1}{4} + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lower-*.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 + -1 \cdot \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

   6. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + -1 \cdot \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(z0\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      6. sub-flip-reverse N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      7. lower--.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.5 \cdot 0.5 - \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      9. metadata-eval 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 - z0\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{4}\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

   8. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 - z0\right) - 0.25}}{\left(z0 - 0.5\right) + 0.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 55.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z1\right| \cdot z0\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.8:\\ \;\;\;\;\sin t\_0 \cdot \cos \left(\left|z1\right|\right) - \cos t\_0 \cdot \sin \left(\left|z1\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (* (fabs z1) z0)))
  (*
   (copysign 1.0 z1)
   (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.8)
     (- (* (sin t_0) (cos (fabs z1))) (* (cos t_0) (sin (fabs z1))))
     (sin (* (fabs z1) -1.0))))))

C

double code(double z1, double z0) {
	double t_0 = fabs(z1) * z0;
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.8) {
		tmp = (sin(t_0) * cos(fabs(z1))) - (cos(t_0) * sin(fabs(z1)));
	} else {
		tmp = sin((fabs(z1) * -1.0));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.abs(z1) * z0;
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.8) {
		tmp = (Math.sin(t_0) * Math.cos(Math.abs(z1))) - (Math.cos(t_0) * Math.sin(Math.abs(z1)));
	} else {
		tmp = Math.sin((Math.abs(z1) * -1.0));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	t_0 = math.fabs(z1) * z0
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.8:
		tmp = (math.sin(t_0) * math.cos(math.fabs(z1))) - (math.cos(t_0) * math.sin(math.fabs(z1)))
	else:
		tmp = math.sin((math.fabs(z1) * -1.0))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(abs(z1) * z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.8)
		tmp = Float64(Float64(sin(t_0) * cos(abs(z1))) - Float64(cos(t_0) * sin(abs(z1))));
	else
		tmp = sin(Float64(abs(z1) * -1.0));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = abs(z1) * z0;
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.8)
		tmp = (sin(t_0) * cos(abs(z1))) - (cos(t_0) * sin(abs(z1)));
	else
		tmp = sin((abs(z1) * -1.0));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[N[Abs[z1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[N[Abs[z1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.80000000000000004

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   3. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]

   if 0.80000000000000004 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 41.2%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 55.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-e}{z0}\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.995:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{t\_0}\right) \cdot t\_0\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left|z1\right|\right) \cdot \sin \pi + \cos \pi \cdot \sin \left(\pi - \left|z1\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ (- E) z0)))
  (*
   (copysign 1.0 z1)
   (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.995)
     (sin (* (fabs z1) (/ (* z0 (* (+ 1.0 (/ E t_0)) t_0)) E)))
     (+
      (* (cos (fabs z1)) (sin PI))
      (* (cos PI) (sin (- PI (fabs z1)))))))))

C

double code(double z1, double z0) {
	double t_0 = -((double) M_E) / z0;
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.995) {
		tmp = sin((fabs(z1) * ((z0 * ((1.0 + (((double) M_E) / t_0)) * t_0)) / ((double) M_E))));
	} else {
		tmp = (cos(fabs(z1)) * sin(((double) M_PI))) + (cos(((double) M_PI)) * sin((((double) M_PI) - fabs(z1))));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = -Math.E / z0;
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.995) {
		tmp = Math.sin((Math.abs(z1) * ((z0 * ((1.0 + (Math.E / t_0)) * t_0)) / Math.E)));
	} else {
		tmp = (Math.cos(Math.abs(z1)) * Math.sin(Math.PI)) + (Math.cos(Math.PI) * Math.sin((Math.PI - Math.abs(z1))));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	t_0 = -math.e / z0
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.995:
		tmp = math.sin((math.fabs(z1) * ((z0 * ((1.0 + (math.e / t_0)) * t_0)) / math.e)))
	else:
		tmp = (math.cos(math.fabs(z1)) * math.sin(math.pi)) + (math.cos(math.pi) * math.sin((math.pi - math.fabs(z1))))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(-exp(1)) / z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.995)
		tmp = sin(Float64(abs(z1) * Float64(Float64(z0 * Float64(Float64(1.0 + Float64(exp(1) / t_0)) * t_0)) / exp(1))));
	else
		tmp = Float64(Float64(cos(abs(z1)) * sin(pi)) + Float64(cos(pi) * sin(Float64(pi - abs(z1)))));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = -2.71828182845904523536 / z0;
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.995)
		tmp = sin((abs(z1) * ((z0 * ((1.0 + (2.71828182845904523536 / t_0)) * t_0)) / 2.71828182845904523536)));
	else
		tmp = (cos(abs(z1)) * sin(pi)) + (cos(pi) * sin((pi - abs(z1))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[((-E) / z0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.995], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(z0 * N[(N[(1.0 + N[(E / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Cos[N[Abs[z1], $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[Pi], $MachinePrecision] * N[Sin[N[(Pi - N[Abs[z1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.995

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right) \]

      3. sub-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right) \]

      4. add-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)\right)}\right)\right)\right) \]

      5. sub-negate N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right) - 1\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{z0} - 1\right)\right) \]

      7. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{e^{0}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - e^{\color{blue}{1 - 1}}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{1}}{e^{1}}}\right)\right) \]

      10. sub-to-fraction N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1} - e^{1}}}{e^{1}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1}} - e^{1}}{e^{1}}\right) \]

      14. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\mathsf{E}\left(\right)} - e^{1}}{e^{1}}\right) \]

      15. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{e} - e^{1}}{e^{1}}\right) \]

      16. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{\mathsf{E}\left(\right)}}{e^{1}}\right) \]

      17. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{e}}{e^{1}}\right) \]

      18. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

      19. lower-E.f64 53.8%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{e}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e - e}{e}}\right) \]

   4. Taylor expanded in z0 around inf

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\left(\mathsf{E}\left(\right) + -1 \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}}{e}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\mathsf{E}\left(\right) + \color{blue}{-1 \cdot \frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      3. lower-E.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + \color{blue}{-1} \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}{e}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \color{blue}{\frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{\mathsf{E}\left(\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      6. lower-E.f64 53.3%

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}{e}\right) \]

   6. Applied rewrites 53.3%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + \color{blue}{-1 \cdot \frac{e}{z0}}\right)}{e}\right) \]

      2. +-commutative N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(-1 \cdot \frac{e}{z0} + \color{blue}{e}\right)}{e}\right) \]

      3. sum-to-mult N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{e}{z0}\right)}\right)}{e}\right) \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{e}{z0}\right)}\right)}{e}\right) \]

      5. lower-unsound-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      6. lower-unsound-/.f64 53.3%

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\mathsf{neg}\left(\frac{e}{z0}\right)}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\mathsf{neg}\left(\frac{e}{z0}\right)}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{\mathsf{neg}\left(e\right)}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{\mathsf{neg}\left(e\right)}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      12. lower-neg.f64 53.3%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(-1 \cdot \color{blue}{\frac{e}{z0}}\right)\right)}{e}\right) \]

      14. mul-1-neg N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(\mathsf{neg}\left(\frac{e}{z0}\right)\right)\right)}{e}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(\mathsf{neg}\left(\frac{e}{z0}\right)\right)\right)}{e}\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{\mathsf{neg}\left(e\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{\mathsf{neg}\left(e\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      18. lower-neg.f64 53.3%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{-e}{z0}\right)}{e}\right) \]

   8. Applied rewrites 53.3%

      \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \color{blue}{\frac{-e}{z0}}\right)}{e}\right) \]

   if 0.995 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(z1 \cdot \left(z0 - 1\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(z1 \cdot \left(z0 - 1\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(z0 - 1\right) \cdot z1\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(z0 - 1\right)} \cdot z1\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin \left(\color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)} \cdot z1\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right) \cdot z1\right)\right)} \]

      7. sin-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sin \left(\left(1 - z0\right) \cdot z1\right)\right)} \]

      8. sin-+PI-rev N/A

         \[\leadsto \color{blue}{\sin \left(\left(1 - z0\right) \cdot z1 + \mathsf{PI}\left(\right)\right)} \]

      9. sin-sum N/A

         \[\leadsto \color{blue}{\sin \left(\left(1 - z0\right) \cdot z1\right) \cdot \cos \mathsf{PI}\left(\right) + \cos \left(\left(1 - z0\right) \cdot z1\right) \cdot \sin \mathsf{PI}\left(\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \color{blue}{\sin \left(\left(1 - z0\right) \cdot z1\right) \cdot \cos \mathsf{PI}\left(\right) + \cos \left(\left(1 - z0\right) \cdot z1\right) \cdot \sin \mathsf{PI}\left(\right)} \]

   3. Applied rewrites 18.9%

      \[\leadsto \color{blue}{\sin \left(z1 \cdot \left(1 - z0\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(z1 \cdot \left(1 - z0\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(z1 \cdot \left(1 - z0\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      3. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(1 - z0\right)}\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      4. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(z0 - 1\right)\right)\right)}\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(z1 \cdot \left(z0 - 1\right)\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      6. sub-negate-rev N/A

         \[\leadsto \sin \left(\mathsf{neg}\left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      7. lift--.f64 N/A

         \[\leadsto \sin \left(\mathsf{neg}\left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - z0\right)}\right)\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z1 \cdot \left(1 - z0\right)\right)\right)}\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      9. lift-*.f64 N/A

         \[\leadsto \sin \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{z1 \cdot \left(1 - z0\right)}\right)\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      10. sin-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(z1 \cdot \left(1 - z0\right)\right)\right)\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      11. sin-neg-rev N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sin \left(z1 \cdot \left(1 - z0\right)\right)\right)\right)}\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      12. sin-+PI N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin \left(z1 \cdot \left(1 - z0\right) + \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      13. lift-PI.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\sin \left(z1 \cdot \left(1 - z0\right) + \color{blue}{\pi}\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      14. sin-sum-rev N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\sin \left(z1 \cdot \left(1 - z0\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi\right)}\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      15. lift-sin.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{\sin \left(z1 \cdot \left(1 - z0\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      16. lift-cos.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\sin \left(z1 \cdot \left(1 - z0\right)\right) \cdot \color{blue}{\cos \pi} + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\color{blue}{\sin \left(z1 \cdot \left(1 - z0\right)\right) \cdot \cos \pi} + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

      18. lift-cos.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(\sin \left(z1 \cdot \left(1 - z0\right)\right) \cdot \cos \pi + \color{blue}{\cos \left(z1 \cdot \left(1 - z0\right)\right)} \cdot \sin \pi\right)\right)\right) \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

   5. Applied rewrites 8.8%

      \[\leadsto \color{blue}{\sin \left(-\left(\left(1 - z0\right) \cdot z1 - \pi\right)\right)} \cdot \cos \pi + \cos \left(z1 \cdot \left(1 - z0\right)\right) \cdot \sin \pi \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\pi - z1\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \mathsf{PI}\left(\right) + \color{blue}{\cos \mathsf{PI}\left(\right) \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \mathsf{PI}\left(\right) + \color{blue}{\cos \mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \mathsf{PI}\left(\right) + \cos \color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \mathsf{PI}\left(\right) + \cos \mathsf{PI}\left(\right) \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \pi + \cos \mathsf{PI}\left(\right) \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \pi + \cos \mathsf{PI}\left(\right) \cdot \color{blue}{\sin \left(\mathsf{PI}\left(\right) - z1\right)} \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \pi + \cos \mathsf{PI}\left(\right) \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) - z1\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} - z1\right) \]

      9. lower-sin.f64 N/A

         \[\leadsto \cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      10. lower--.f64 N/A

          \[\leadsto \cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\mathsf{PI}\left(\right) - z1\right) \]

      11. lower-PI.f64 7.8%

          \[\leadsto \cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\pi - z1\right) \]

   8. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\cos z1 \cdot \sin \pi + \cos \pi \cdot \sin \left(\pi - z1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-e}{z0}\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.999:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{t\_0}\right) \cdot t\_0\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{\left(0.25 - z0\right) - 0.25}{\left(z0 - 0.5\right) + 0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (/ (- E) z0)))
  (*
   (copysign 1.0 z1)
   (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.999)
     (sin (* (fabs z1) (/ (* z0 (* (+ 1.0 (/ E t_0)) t_0)) E)))
     (sin
      (* (fabs z1) (/ (- (- 0.25 z0) 0.25) (+ (- z0 0.5) 0.5))))))))

C

double code(double z1, double z0) {
	double t_0 = -((double) M_E) / z0;
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = sin((fabs(z1) * ((z0 * ((1.0 + (((double) M_E) / t_0)) * t_0)) / ((double) M_E))));
	} else {
		tmp = sin((fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = -Math.E / z0;
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = Math.sin((Math.abs(z1) * ((z0 * ((1.0 + (Math.E / t_0)) * t_0)) / Math.E)));
	} else {
		tmp = Math.sin((Math.abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	t_0 = -math.e / z0
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.999:
		tmp = math.sin((math.fabs(z1) * ((z0 * ((1.0 + (math.e / t_0)) * t_0)) / math.e)))
	else:
		tmp = math.sin((math.fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	t_0 = Float64(Float64(-exp(1)) / z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.999)
		tmp = sin(Float64(abs(z1) * Float64(Float64(z0 * Float64(Float64(1.0 + Float64(exp(1) / t_0)) * t_0)) / exp(1))));
	else
		tmp = sin(Float64(abs(z1) * Float64(Float64(Float64(0.25 - z0) - 0.25) / Float64(Float64(z0 - 0.5) + 0.5))));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = -2.71828182845904523536 / z0;
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.999)
		tmp = sin((abs(z1) * ((z0 * ((1.0 + (2.71828182845904523536 / t_0)) * t_0)) / 2.71828182845904523536)));
	else
		tmp = sin((abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[((-E) / z0), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.999], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(z0 * N[(N[(1.0 + N[(E / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(0.25 - z0), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(z0 - 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right) \]

      3. sub-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right) \]

      4. add-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)\right)}\right)\right)\right) \]

      5. sub-negate N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right) - 1\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{z0} - 1\right)\right) \]

      7. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{e^{0}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - e^{\color{blue}{1 - 1}}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{1}}{e^{1}}}\right)\right) \]

      10. sub-to-fraction N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1} - e^{1}}}{e^{1}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1}} - e^{1}}{e^{1}}\right) \]

      14. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\mathsf{E}\left(\right)} - e^{1}}{e^{1}}\right) \]

      15. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{e} - e^{1}}{e^{1}}\right) \]

      16. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{\mathsf{E}\left(\right)}}{e^{1}}\right) \]

      17. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{e}}{e^{1}}\right) \]

      18. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

      19. lower-E.f64 53.8%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{e}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e - e}{e}}\right) \]

   4. Taylor expanded in z0 around inf

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\left(\mathsf{E}\left(\right) + -1 \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}}{e}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\mathsf{E}\left(\right) + \color{blue}{-1 \cdot \frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      3. lower-E.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + \color{blue}{-1} \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}{e}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \color{blue}{\frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{\mathsf{E}\left(\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      6. lower-E.f64 53.3%

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}{e}\right) \]

   6. Applied rewrites 53.3%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + \color{blue}{-1 \cdot \frac{e}{z0}}\right)}{e}\right) \]

      2. +-commutative N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(-1 \cdot \frac{e}{z0} + \color{blue}{e}\right)}{e}\right) \]

      3. sum-to-mult N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{e}{z0}\right)}\right)}{e}\right) \]

      4. lower-unsound-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \color{blue}{\left(-1 \cdot \frac{e}{z0}\right)}\right)}{e}\right) \]

      5. lower-unsound-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(\color{blue}{-1} \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      6. lower-unsound-/.f64 53.3%

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{-1 \cdot \frac{e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\mathsf{neg}\left(\frac{e}{z0}\right)}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\mathsf{neg}\left(\frac{e}{z0}\right)}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{\mathsf{neg}\left(e\right)}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{\mathsf{neg}\left(e\right)}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      12. lower-neg.f64 53.3%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(-1 \cdot \frac{e}{z0}\right)\right)}{e}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(-1 \cdot \color{blue}{\frac{e}{z0}}\right)\right)}{e}\right) \]

      14. mul-1-neg N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(\mathsf{neg}\left(\frac{e}{z0}\right)\right)\right)}{e}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \left(\mathsf{neg}\left(\frac{e}{z0}\right)\right)\right)}{e}\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{\mathsf{neg}\left(e\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{\mathsf{neg}\left(e\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      18. lower-neg.f64 53.3%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \frac{-e}{z0}\right)}{e}\right) \]

   8. Applied rewrites 53.3%

      \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\left(1 + \frac{e}{\frac{-e}{z0}}\right) \cdot \color{blue}{\frac{-e}{z0}}\right)}{e}\right) \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. *-inverses N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}\right)\right) \]

      3. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{\color{blue}{1} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + e^{\color{blue}{0}}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      5. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + \color{blue}{1}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      6. div-add N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\left(\frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right) \]

      8. flip-- N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

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

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

   3. Applied rewrites 12.4%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - 0.5\right) \cdot \left(z0 - 0.5\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}}\right) \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(\frac{1}{4} + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lower-*.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 + -1 \cdot \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

   6. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + -1 \cdot \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(z0\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      6. sub-flip-reverse N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      7. lower--.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.5 \cdot 0.5 - \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      9. metadata-eval 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 - z0\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{4}\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

   8. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 - z0\right) - 0.25}}{\left(z0 - 0.5\right) + 0.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 54.8% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.999:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{\left(0.25 - z0\right) - 0.25}{\left(z0 - 0.5\right) + 0.5}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.999)
   (sin (* (fabs z1) (/ (* z0 (+ E (* -1.0 (/ E z0)))) E)))
   (sin (* (fabs z1) (/ (- (- 0.25 z0) 0.25) (+ (- z0 0.5) 0.5)))))))

C

double code(double z1, double z0) {
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = sin((fabs(z1) * ((z0 * (((double) M_E) + (-1.0 * (((double) M_E) / z0)))) / ((double) M_E))));
	} else {
		tmp = sin((fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = Math.sin((Math.abs(z1) * ((z0 * (Math.E + (-1.0 * (Math.E / z0)))) / Math.E)));
	} else {
		tmp = Math.sin((Math.abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.999:
		tmp = math.sin((math.fabs(z1) * ((z0 * (math.e + (-1.0 * (math.e / z0)))) / math.e)))
	else:
		tmp = math.sin((math.fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.999)
		tmp = sin(Float64(abs(z1) * Float64(Float64(z0 * Float64(exp(1) + Float64(-1.0 * Float64(exp(1) / z0)))) / exp(1))));
	else
		tmp = sin(Float64(abs(z1) * Float64(Float64(Float64(0.25 - z0) - 0.25) / Float64(Float64(z0 - 0.5) + 0.5))));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.999)
		tmp = sin((abs(z1) * ((z0 * (2.71828182845904523536 + (-1.0 * (2.71828182845904523536 / z0)))) / 2.71828182845904523536)));
	else
		tmp = sin((abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.999], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(z0 * N[(E + N[(-1.0 * N[(E / z0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(0.25 - z0), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(z0 - 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right) \]

      3. sub-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right) \]

      4. add-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)\right)}\right)\right)\right) \]

      5. sub-negate N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right) - 1\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{z0} - 1\right)\right) \]

      7. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{e^{0}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - e^{\color{blue}{1 - 1}}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{1}}{e^{1}}}\right)\right) \]

      10. sub-to-fraction N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1} - e^{1}}}{e^{1}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1}} - e^{1}}{e^{1}}\right) \]

      14. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\mathsf{E}\left(\right)} - e^{1}}{e^{1}}\right) \]

      15. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{e} - e^{1}}{e^{1}}\right) \]

      16. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{\mathsf{E}\left(\right)}}{e^{1}}\right) \]

      17. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{e}}{e^{1}}\right) \]

      18. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

      19. lower-E.f64 53.8%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{e}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e - e}{e}}\right) \]

   4. Taylor expanded in z0 around inf

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\left(\mathsf{E}\left(\right) + -1 \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}}{e}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(\mathsf{E}\left(\right) + \color{blue}{-1 \cdot \frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      3. lower-E.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + \color{blue}{-1} \cdot \frac{\mathsf{E}\left(\right)}{z0}\right)}{e}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \color{blue}{\frac{\mathsf{E}\left(\right)}{z0}}\right)}{e}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{\mathsf{E}\left(\right)}{\color{blue}{z0}}\right)}{e}\right) \]

      6. lower-E.f64 53.3%

         \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}{e}\right) \]

   6. Applied rewrites 53.3%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot \left(e + -1 \cdot \frac{e}{z0}\right)}}{e}\right) \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. *-inverses N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}\right)\right) \]

      3. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{\color{blue}{1} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + e^{\color{blue}{0}}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      5. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + \color{blue}{1}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      6. div-add N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\left(\frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right) \]

      8. flip-- N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

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

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

   3. Applied rewrites 12.4%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - 0.5\right) \cdot \left(z0 - 0.5\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}}\right) \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(\frac{1}{4} + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lower-*.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 + -1 \cdot \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

   6. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + -1 \cdot \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(z0\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      6. sub-flip-reverse N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      7. lower--.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.5 \cdot 0.5 - \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      9. metadata-eval 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 - z0\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{4}\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

   8. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 - z0\right) - 0.25}}{\left(z0 - 0.5\right) + 0.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.3% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.999:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{z0 \cdot e - e}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot -1\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.999)
   (sin (* (fabs z1) (/ (- (* z0 E) E) E)))
   (sin (* (fabs z1) -1.0)))))

C

double code(double z1, double z0) {
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = sin((fabs(z1) * (((z0 * ((double) M_E)) - ((double) M_E)) / ((double) M_E))));
	} else {
		tmp = sin((fabs(z1) * -1.0));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = Math.sin((Math.abs(z1) * (((z0 * Math.E) - Math.E) / Math.E)));
	} else {
		tmp = Math.sin((Math.abs(z1) * -1.0));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.999:
		tmp = math.sin((math.fabs(z1) * (((z0 * math.e) - math.e) / math.e)))
	else:
		tmp = math.sin((math.fabs(z1) * -1.0))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.999)
		tmp = sin(Float64(abs(z1) * Float64(Float64(Float64(z0 * exp(1)) - exp(1)) / exp(1))));
	else
		tmp = sin(Float64(abs(z1) * -1.0));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.999)
		tmp = sin((abs(z1) * (((z0 * 2.71828182845904523536) - 2.71828182845904523536) / 2.71828182845904523536)));
	else
		tmp = sin((abs(z1) * -1.0));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.999], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(z0 * E), $MachinePrecision] - E), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right) \]

      3. sub-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right) \]

      4. add-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)\right)}\right)\right)\right) \]

      5. sub-negate N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right) - 1\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{z0} - 1\right)\right) \]

      7. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{e^{0}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - e^{\color{blue}{1 - 1}}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{1}}{e^{1}}}\right)\right) \]

      10. sub-to-fraction N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1} - e^{1}}}{e^{1}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1}} - e^{1}}{e^{1}}\right) \]

      14. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\mathsf{E}\left(\right)} - e^{1}}{e^{1}}\right) \]

      15. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{e} - e^{1}}{e^{1}}\right) \]

      16. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{\mathsf{E}\left(\right)}}{e^{1}}\right) \]

      17. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{e}}{e^{1}}\right) \]

      18. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

      19. lower-E.f64 53.8%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{e}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e - e}{e}}\right) \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 41.2%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.3% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right) \leq 0.999:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{z0 \cdot e - e}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot \frac{\left(0.25 - z0\right) - 0.25}{\left(z0 - 0.5\right) + 0.5}\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (sin (* (fabs z1) (- z0 1.0))) 0.999)
   (sin (* (fabs z1) (/ (- (* z0 E) E) E)))
   (sin (* (fabs z1) (/ (- (- 0.25 z0) 0.25) (+ (- z0 0.5) 0.5)))))))

C

double code(double z1, double z0) {
	double tmp;
	if (sin((fabs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = sin((fabs(z1) * (((z0 * ((double) M_E)) - ((double) M_E)) / ((double) M_E))));
	} else {
		tmp = sin((fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.sin((Math.abs(z1) * (z0 - 1.0))) <= 0.999) {
		tmp = Math.sin((Math.abs(z1) * (((z0 * Math.E) - Math.E) / Math.E)));
	} else {
		tmp = Math.sin((Math.abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.sin((math.fabs(z1) * (z0 - 1.0))) <= 0.999:
		tmp = math.sin((math.fabs(z1) * (((z0 * math.e) - math.e) / math.e)))
	else:
		tmp = math.sin((math.fabs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (sin(Float64(abs(z1) * Float64(z0 - 1.0))) <= 0.999)
		tmp = sin(Float64(abs(z1) * Float64(Float64(Float64(z0 * exp(1)) - exp(1)) / exp(1))));
	else
		tmp = sin(Float64(abs(z1) * Float64(Float64(Float64(0.25 - z0) - 0.25) / Float64(Float64(z0 - 0.5) + 0.5))));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (sin((abs(z1) * (z0 - 1.0))) <= 0.999)
		tmp = sin((abs(z1) * (((z0 * 2.71828182845904523536) - 2.71828182845904523536) / 2.71828182845904523536)));
	else
		tmp = sin((abs(z1) * (((0.25 - z0) - 0.25) / ((z0 - 0.5) + 0.5))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.999], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(z0 * E), $MachinePrecision] - E), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(N[(N[(0.25 - z0), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(z0 - 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. sub-negate-rev N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - z0\right)\right)\right)}\right) \]

      3. sub-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right) \]

      4. add-flip N/A

         \[\leadsto \sin \left(z1 \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right)\right)}\right)\right)\right) \]

      5. sub-negate N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z0\right)\right)\right)\right) - 1\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \sin \left(z1 \cdot \left(\color{blue}{z0} - 1\right)\right) \]

      7. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{e^{0}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - e^{\color{blue}{1 - 1}}\right)\right) \]

      9. exp-diff N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{1}}{e^{1}}}\right)\right) \]

      10. sub-to-fraction N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e^{1} - e^{1}}{e^{1}}}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1} - e^{1}}}{e^{1}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{z0 \cdot e^{1}} - e^{1}}{e^{1}}\right) \]

      14. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{\mathsf{E}\left(\right)} - e^{1}}{e^{1}}\right) \]

      15. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot \color{blue}{e} - e^{1}}{e^{1}}\right) \]

      16. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{\mathsf{E}\left(\right)}}{e^{1}}\right) \]

      17. lower-E.f64 N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - \color{blue}{e}}{e^{1}}\right) \]

      18. exp-1-e N/A

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{\mathsf{E}\left(\right)}}\right) \]

      19. lower-E.f64 53.8%

          \[\leadsto \sin \left(z1 \cdot \frac{z0 \cdot e - e}{\color{blue}{e}}\right) \]

   3. Applied rewrites 53.8%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{z0 \cdot e - e}{e}}\right) \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(z0 - 1\right)}\right) \]

      2. *-inverses N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\frac{e^{0} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}\right)\right) \]

      3. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{\color{blue}{1} + e^{\mathsf{neg}\left(0\right)}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + e^{\color{blue}{0}}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      5. 1-exp N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \frac{1 + \color{blue}{1}}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)\right) \]

      6. div-add N/A

         \[\leadsto \sin \left(z1 \cdot \left(z0 - \color{blue}{\left(\frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\left(\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right)}\right) \]

      8. flip-- N/A

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

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

         \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) \cdot \left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}} \cdot \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}{\left(z0 - \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}\right) + \frac{1}{e^{0} + e^{\mathsf{neg}\left(0\right)}}}}\right) \]

   3. Applied rewrites 12.4%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{\frac{\left(z0 - 0.5\right) \cdot \left(z0 - 0.5\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}}\right) \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(\frac{1}{4} + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lower-*.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 + -1 \cdot \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

   6. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 + -1 \cdot z0\right)} - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} + \color{blue}{-1 \cdot z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      2. metadata-eval N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \color{blue}{-1} \cdot z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + -1 \cdot \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(z0\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      6. sub-flip-reverse N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \color{blue}{z0}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      7. lower--.f64 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.5 \cdot 0.5 - \color{blue}{z0}\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{2} \cdot \frac{1}{2} - z0\right) - \frac{1}{2} \cdot \frac{1}{2}}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      9. metadata-eval 4.6%

         \[\leadsto \sin \left(z1 \cdot \frac{\left(0.25 - z0\right) - 0.5 \cdot 0.5}{\left(z0 - 0.5\right) + 0.5}\right) \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(z1 \cdot \frac{\left(\frac{1}{4} - z0\right) - \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{4}\right)}{\left(z0 - \frac{1}{2}\right) + \frac{1}{2}}\right) \]

   8. Applied rewrites 4.6%

      \[\leadsto \sin \left(z1 \cdot \frac{\color{blue}{\left(0.25 - z0\right) - 0.25}}{\left(z0 - 0.5\right) + 0.5}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|z1\right| \cdot \left(z0 - 1\right)\right)\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.999:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (let* ((t_0 (sin (* (fabs z1) (- z0 1.0)))))
  (*
   (copysign 1.0 z1)
   (if (<= t_0 0.999) t_0 (sin (* (fabs z1) -1.0))))))

C

double code(double z1, double z0) {
	double t_0 = sin((fabs(z1) * (z0 - 1.0)));
	double tmp;
	if (t_0 <= 0.999) {
		tmp = t_0;
	} else {
		tmp = sin((fabs(z1) * -1.0));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double t_0 = Math.sin((Math.abs(z1) * (z0 - 1.0)));
	double tmp;
	if (t_0 <= 0.999) {
		tmp = t_0;
	} else {
		tmp = Math.sin((Math.abs(z1) * -1.0));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	t_0 = math.sin((math.fabs(z1) * (z0 - 1.0)))
	tmp = 0
	if t_0 <= 0.999:
		tmp = t_0
	else:
		tmp = math.sin((math.fabs(z1) * -1.0))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	t_0 = sin(Float64(abs(z1) * Float64(z0 - 1.0)))
	tmp = 0.0
	if (t_0 <= 0.999)
		tmp = t_0;
	else
		tmp = sin(Float64(abs(z1) * -1.0));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	t_0 = sin((abs(z1) * (z0 - 1.0)));
	tmp = 0.0;
	if (t_0 <= 0.999)
		tmp = t_0;
	else
		tmp = sin((abs(z1) * -1.0));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := Block[{t$95$0 = N[Sin[N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.999], t$95$0, N[Sin[N[(N[Abs[z1], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64)))) < 0.999

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   if 0.999 < (sin.f64 (*.f64 z1 (-.f64 z0 #s(literal 1 binary64))))

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 41.2%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.1% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;\left|z1\right| \cdot \left(z0 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|z1\right| \cdot -1\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (fabs z1) 3.5e-44)
   (* (fabs z1) (- z0 1.0))
   (sin (* (fabs z1) -1.0)))))

C

double code(double z1, double z0) {
	double tmp;
	if (fabs(z1) <= 3.5e-44) {
		tmp = fabs(z1) * (z0 - 1.0);
	} else {
		tmp = sin((fabs(z1) * -1.0));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.abs(z1) <= 3.5e-44) {
		tmp = Math.abs(z1) * (z0 - 1.0);
	} else {
		tmp = Math.sin((Math.abs(z1) * -1.0));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.fabs(z1) <= 3.5e-44:
		tmp = math.fabs(z1) * (z0 - 1.0)
	else:
		tmp = math.sin((math.fabs(z1) * -1.0))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (abs(z1) <= 3.5e-44)
		tmp = Float64(abs(z1) * Float64(z0 - 1.0));
	else
		tmp = sin(Float64(abs(z1) * -1.0));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (abs(z1) <= 3.5e-44)
		tmp = abs(z1) * (z0 - 1.0);
	else
		tmp = sin((abs(z1) * -1.0));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z1], $MachinePrecision], 3.5e-44], N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision], N[Sin[N[(N[Abs[z1], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z1 < 3.4999999999999998e-44

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   3. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 \cdot \color{blue}{\left(z0 - 1\right)} \]

      2. lower--.f64 38.9%

         \[\leadsto z1 \cdot \left(z0 - \color{blue}{1}\right) \]

   6. Applied rewrites 38.9%

      \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]

   if 3.4999999999999998e-44 < z1

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 41.2%

      \[\leadsto \sin \left(z1 \cdot \color{blue}{-1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 40.8% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 0.0019:\\ \;\;\;\;\left|z1\right| \cdot \left(z0 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(z0 \cdot \left|z1\right|\right)\\ \end{array} \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (fabs z1) 0.0019)
   (* (fabs z1) (- z0 1.0))
   (sin (* z0 (fabs z1))))))

C

double code(double z1, double z0) {
	double tmp;
	if (fabs(z1) <= 0.0019) {
		tmp = fabs(z1) * (z0 - 1.0);
	} else {
		tmp = sin((z0 * fabs(z1)));
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if (Math.abs(z1) <= 0.0019) {
		tmp = Math.abs(z1) * (z0 - 1.0);
	} else {
		tmp = Math.sin((z0 * Math.abs(z1)));
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z1, z0):
	tmp = 0
	if math.fabs(z1) <= 0.0019:
		tmp = math.fabs(z1) * (z0 - 1.0)
	else:
		tmp = math.sin((z0 * math.fabs(z1)))
	return math.copysign(1.0, z1) * tmp

Julia

function code(z1, z0)
	tmp = 0.0
	if (abs(z1) <= 0.0019)
		tmp = Float64(abs(z1) * Float64(z0 - 1.0));
	else
		tmp = sin(Float64(z0 * abs(z1)));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if (abs(z1) <= 0.0019)
		tmp = abs(z1) * (z0 - 1.0);
	else
		tmp = sin((z0 * abs(z1)));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z1], $MachinePrecision], 0.0019], N[(N[Abs[z1], $MachinePrecision] * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision], N[Sin[N[(z0 * N[Abs[z1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z1 < 0.0019

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   3. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z1 \cdot \color{blue}{\left(z0 - 1\right)} \]

      2. lower--.f64 38.9%

         \[\leadsto z1 \cdot \left(z0 - \color{blue}{1}\right) \]

   6. Applied rewrites 38.9%

      \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]

   if 0.0019 < z1

   1. Initial program 53.8%

      \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

   2. Taylor expanded in z0 around inf

      \[\leadsto \sin \color{blue}{\left(z0 \cdot z1\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 18.6%

         \[\leadsto \sin \left(z0 \cdot \color{blue}{z1}\right) \]

   4. Applied rewrites 18.6%

      \[\leadsto \sin \color{blue}{\left(z0 \cdot z1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.9% accurate, 12.1× speedup

Math

\[z1 \cdot \left(z0 - 1\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* z1 (- z0 1.0)))

C

double code(double z1, double z0) {
	return z1 * (z0 - 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = z1 * (z0 - 1.0d0)
end function

Java

public static double code(double z1, double z0) {
	return z1 * (z0 - 1.0);
}

Python

def code(z1, z0):
	return z1 * (z0 - 1.0)

Julia

function code(z1, z0)
	return Float64(z1 * Float64(z0 - 1.0))
end

MATLAB

function tmp = code(z1, z0)
	tmp = z1 * (z0 - 1.0);
end

Wolfram

code[z1_, z0_] := N[(z1 * N[(z0 - 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

3. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z1 \cdot \color{blue}{\left(z0 - 1\right)} \]

   2. lower--.f64 38.9%

      \[\leadsto z1 \cdot \left(z0 - \color{blue}{1}\right) \]

6. Applied rewrites 38.9%

   \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]
7. Add Preprocessing

Alternative 12: 27.0% accurate, 18.2× speedup

Math

\[z1 \cdot -1 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* z1 -1.0))

C

double code(double z1, double z0) {
	return z1 * -1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = z1 * (-1.0d0)
end function

Java

public static double code(double z1, double z0) {
	return z1 * -1.0;
}

Python

def code(z1, z0):
	return z1 * -1.0

Julia

function code(z1, z0)
	return Float64(z1 * -1.0)
end

MATLAB

function tmp = code(z1, z0)
	tmp = z1 * -1.0;
end

Wolfram

code[z1_, z0_] := N[(z1 * -1.0), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\sin \left(z1 \cdot \left(z0 - 1\right)\right) \]

3. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\sin \left(z1 \cdot z0\right) \cdot \cos z1 - \cos \left(z1 \cdot z0\right) \cdot \sin z1} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z1 \cdot \color{blue}{\left(z0 - 1\right)} \]

   2. lower--.f64 38.9%

      \[\leadsto z1 \cdot \left(z0 - \color{blue}{1}\right) \]

6. Applied rewrites 38.9%

   \[\leadsto \color{blue}{z1 \cdot \left(z0 - 1\right)} \]

7. Taylor expanded in z0 around 0

   \[\leadsto z1 \cdot -1 \]
8. Step-by-step derivation

9. Applied rewrites 27.0%

   \[\leadsto z1 \cdot -1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(sin (* z1 (- z0 1)))"
  :precision binary64
  (sin (* z1 (- z0 1.0))))