Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B

Percentage Accurate: 99.9% → 99.9%

Time: 4.1s

Alternatives: 16

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x + \cos y\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(y));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + cos(y)) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.cos(y)) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x + math.cos(y)) - (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + cos(y)) - (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \cos y\right) - z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x + cos(y)) - (z * sin(y));
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + cos(y)) - (z * sin(y))
end function

Java

public static double code(double x, double y, double z) {
	return (x + Math.cos(y)) - (z * Math.sin(y));
}

Python

def code(x, y, z):
	return (x + math.cos(y)) - (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + cos(y)) - (z * sin(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos y - \left(\sin y \cdot z - x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (cos y) (- (* (sin y) z) x)))

C

double code(double x, double y, double z) {
	return cos(y) - ((sin(y) * z) - x);
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = cos(y) - ((sin(y) * z) - x)
end function

Java

public static double code(double x, double y, double z) {
	return Math.cos(y) - ((Math.sin(y) * z) - x);
}

Python

def code(x, y, z):
	return math.cos(y) - ((math.sin(y) * z) - x)

Julia

function code(x, y, z)
	return Float64(cos(y) - Float64(Float64(sin(y) * z) - x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = cos(y) - ((sin(y) * z) - x);
end

Wolfram

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] - N[(N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x + \cos y\right) - z \cdot \sin y} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + \cos y\right)} - z \cdot \sin y \]

   3. lift-cos.f64 N/A

      \[\leadsto \left(x + \color{blue}{\cos y}\right) - z \cdot \sin y \]

   4. lift-*.f64 N/A

      \[\leadsto \left(x + \cos y\right) - \color{blue}{z \cdot \sin y} \]

   5. lift-sin.f64 N/A

      \[\leadsto \left(x + \cos y\right) - z \cdot \color{blue}{\sin y} \]

   6. *-lft-identity N/A

      \[\leadsto \left(x + \cos y\right) - \color{blue}{1 \cdot \left(z \cdot \sin y\right)} \]

   7. metadata-eval N/A

      \[\leadsto \left(x + \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(z \cdot \sin y\right) \]

   8. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x + \cos y\right) + -1 \cdot \left(z \cdot \sin y\right)} \]

   9. associate-+r+ N/A

      \[\leadsto \color{blue}{x + \left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right)} \]

   10. +-commutative N/A

       \[\leadsto \color{blue}{\left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right) + x} \]

   11. fp-cancel-sign-sub-inv N/A

       \[\leadsto \color{blue}{\left(\cos y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(z \cdot \sin y\right)\right)} + x \]

   12. metadata-eval N/A

       \[\leadsto \left(\cos y - \color{blue}{1} \cdot \left(z \cdot \sin y\right)\right) + x \]

   13. *-lft-identity N/A

       \[\leadsto \left(\cos y - \color{blue}{z \cdot \sin y}\right) + x \]

   14. associate-+l- N/A

       \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

   15. lower--.f64 N/A

       \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

   16. lift-cos.f64 N/A

       \[\leadsto \color{blue}{\cos y} - \left(z \cdot \sin y - x\right) \]

   17. lower--.f64 N/A

       \[\leadsto \cos y - \color{blue}{\left(z \cdot \sin y - x\right)} \]

   18. *-commutative N/A

       \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

   20. lift-sin.f64 99.9

       \[\leadsto \cos y - \left(\color{blue}{\sin y} \cdot z - x\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\cos y - \left(\sin y \cdot z - x\right)} \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(\sin y \cdot z - x\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\cos y - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (- (* (sin y) z) x))))
   (if (<= x -4.5e-14) t_0 (if (<= x 5.5e-15) (- (cos y) (* z (sin y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((sin(y) * z) - x);
	double tmp;
	if (x <= -4.5e-14) {
		tmp = t_0;
	} else if (x <= 5.5e-15) {
		tmp = cos(y) - (z * sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((sin(y) * z) - x)
    if (x <= (-4.5d-14)) then
        tmp = t_0
    else if (x <= 5.5d-15) then
        tmp = cos(y) - (z * sin(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - ((Math.sin(y) * z) - x);
	double tmp;
	if (x <= -4.5e-14) {
		tmp = t_0;
	} else if (x <= 5.5e-15) {
		tmp = Math.cos(y) - (z * Math.sin(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - ((math.sin(y) * z) - x)
	tmp = 0
	if x <= -4.5e-14:
		tmp = t_0
	elif x <= 5.5e-15:
		tmp = math.cos(y) - (z * math.sin(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(sin(y) * z) - x))
	tmp = 0.0
	if (x <= -4.5e-14)
		tmp = t_0;
	elseif (x <= 5.5e-15)
		tmp = Float64(cos(y) - Float64(z * sin(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - ((sin(y) * z) - x);
	tmp = 0.0;
	if (x <= -4.5e-14)
		tmp = t_0;
	elseif (x <= 5.5e-15)
		tmp = cos(y) - (z * sin(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-14], t$95$0, If[LessEqual[x, 5.5e-15], N[(N[Cos[y], $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999998e-14 or 5.5000000000000002e-15 < x

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) - z \cdot \sin y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right)} - z \cdot \sin y \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(x + \color{blue}{\cos y}\right) - z \cdot \sin y \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{z \cdot \sin y} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(x + \cos y\right) - z \cdot \color{blue}{\sin y} \]

      6. *-lft-identity N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{1 \cdot \left(z \cdot \sin y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(z \cdot \sin y\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) + -1 \cdot \left(z \cdot \sin y\right)} \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{x + \left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right) + x} \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\cos y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(z \cdot \sin y\right)\right)} + x \]

      12. metadata-eval N/A

          \[\leadsto \left(\cos y - \color{blue}{1} \cdot \left(z \cdot \sin y\right)\right) + x \]

      13. *-lft-identity N/A

          \[\leadsto \left(\cos y - \color{blue}{z \cdot \sin y}\right) + x \]

      14. associate-+l- N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \color{blue}{\cos y} - \left(z \cdot \sin y - x\right) \]

      17. lower--.f64 N/A

          \[\leadsto \cos y - \color{blue}{\left(z \cdot \sin y - x\right)} \]

      18. *-commutative N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      20. lift-sin.f64 99.9

          \[\leadsto \cos y - \left(\color{blue}{\sin y} \cdot z - x\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\cos y - \left(\sin y \cdot z - x\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]

   if -4.4999999999999998e-14 < x < 5.5000000000000002e-15

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\cos y} - z \cdot \sin y \]
   3. Step-by-step derivation

      1. lift-cos.f64 99.9

         \[\leadsto \cos y - z \cdot \sin y \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\cos y} - z \cdot \sin y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left(\sin y \cdot z - x\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 0.00176:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (- (* (sin y) z) x))))
   (if (<= z -1.35e+35) t_0 (if (<= z 0.00176) (+ (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - ((sin(y) * z) - x);
	double tmp;
	if (z <= -1.35e+35) {
		tmp = t_0;
	} else if (z <= 0.00176) {
		tmp = cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - ((sin(y) * z) - x)
    if (z <= (-1.35d+35)) then
        tmp = t_0
    else if (z <= 0.00176d0) then
        tmp = cos(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - ((Math.sin(y) * z) - x);
	double tmp;
	if (z <= -1.35e+35) {
		tmp = t_0;
	} else if (z <= 0.00176) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - ((math.sin(y) * z) - x)
	tmp = 0
	if z <= -1.35e+35:
		tmp = t_0
	elif z <= 0.00176:
		tmp = math.cos(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(Float64(sin(y) * z) - x))
	tmp = 0.0
	if (z <= -1.35e+35)
		tmp = t_0;
	elseif (z <= 0.00176)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - ((sin(y) * z) - x);
	tmp = 0.0;
	if (z <= -1.35e+35)
		tmp = t_0;
	elseif (z <= 0.00176)
		tmp = cos(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.35e+35], t$95$0, If[LessEqual[z, 0.00176], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000001e35 or 0.00176000000000000006 < z

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) - z \cdot \sin y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right)} - z \cdot \sin y \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(x + \color{blue}{\cos y}\right) - z \cdot \sin y \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{z \cdot \sin y} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(x + \cos y\right) - z \cdot \color{blue}{\sin y} \]

      6. *-lft-identity N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{1 \cdot \left(z \cdot \sin y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(z \cdot \sin y\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) + -1 \cdot \left(z \cdot \sin y\right)} \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{x + \left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right) + x} \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\cos y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(z \cdot \sin y\right)\right)} + x \]

      12. metadata-eval N/A

          \[\leadsto \left(\cos y - \color{blue}{1} \cdot \left(z \cdot \sin y\right)\right) + x \]

      13. *-lft-identity N/A

          \[\leadsto \left(\cos y - \color{blue}{z \cdot \sin y}\right) + x \]

      14. associate-+l- N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \color{blue}{\cos y} - \left(z \cdot \sin y - x\right) \]

      17. lower--.f64 N/A

          \[\leadsto \cos y - \color{blue}{\left(z \cdot \sin y - x\right)} \]

      18. *-commutative N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      20. lift-sin.f64 99.9

          \[\leadsto \cos y - \left(\color{blue}{\sin y} \cdot z - x\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\cos y - \left(\sin y \cdot z - x\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]

   if -1.35000000000000001e35 < z < 0.00176000000000000006

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 97.8

         \[\leadsto \cos y + x \]

   4. Applied rewrites 97.8%

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

Alternative 4: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \sin y\\ t_1 := \left(x + \cos y\right) - t\_0\\ t_2 := x - t\_0\\ \mathbf{if}\;t\_1 \leq -500000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))) (t_1 (- (+ x (cos y)) t_0)) (t_2 (- x t_0)))
   (if (<= t_1 -500000000.0) t_2 (if (<= t_1 4e+15) (+ (cos y) x) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	double t_1 = (x + cos(y)) - t_0;
	double t_2 = x - t_0;
	double tmp;
	if (t_1 <= -500000000.0) {
		tmp = t_2;
	} else if (t_1 <= 4e+15) {
		tmp = cos(y) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z * sin(y)
    t_1 = (x + cos(y)) - t_0
    t_2 = x - t_0
    if (t_1 <= (-500000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 4d+15) then
        tmp = cos(y) + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * Math.sin(y);
	double t_1 = (x + Math.cos(y)) - t_0;
	double t_2 = x - t_0;
	double tmp;
	if (t_1 <= -500000000.0) {
		tmp = t_2;
	} else if (t_1 <= 4e+15) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * math.sin(y)
	t_1 = (x + math.cos(y)) - t_0
	t_2 = x - t_0
	tmp = 0
	if t_1 <= -500000000.0:
		tmp = t_2
	elif t_1 <= 4e+15:
		tmp = math.cos(y) + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	t_1 = Float64(Float64(x + cos(y)) - t_0)
	t_2 = Float64(x - t_0)
	tmp = 0.0
	if (t_1 <= -500000000.0)
		tmp = t_2;
	elseif (t_1 <= 4e+15)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * sin(y);
	t_1 = (x + cos(y)) - t_0;
	t_2 = x - t_0;
	tmp = 0.0;
	if (t_1 <= -500000000.0)
		tmp = t_2;
	elseif (t_1 <= 4e+15)
		tmp = cos(y) + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -500000000.0], t$95$2, If[LessEqual[t$95$1, 4e+15], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -5e8 or 4e15 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]
   3. Step-by-step derivation

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{x} - z \cdot \sin y \]

   if -5e8 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 4e15

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 95.3

         \[\leadsto \cos y + x \]

   4. Applied rewrites 95.3%

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

Alternative 5: 84.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \sin y \cdot z\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (sin y) z))))
   (if (<= z -4.4e+81) t_0 (if (<= z 7.2e+18) (+ (cos y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (sin(y) * z);
	double tmp;
	if (z <= -4.4e+81) {
		tmp = t_0;
	} else if (z <= 7.2e+18) {
		tmp = cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (sin(y) * z)
    if (z <= (-4.4d+81)) then
        tmp = t_0
    else if (z <= 7.2d+18) then
        tmp = cos(y) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (Math.sin(y) * z);
	double tmp;
	if (z <= -4.4e+81) {
		tmp = t_0;
	} else if (z <= 7.2e+18) {
		tmp = Math.cos(y) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (math.sin(y) * z)
	tmp = 0
	if z <= -4.4e+81:
		tmp = t_0
	elif z <= 7.2e+18:
		tmp = math.cos(y) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(sin(y) * z))
	tmp = 0.0
	if (z <= -4.4e+81)
		tmp = t_0;
	elseif (z <= 7.2e+18)
		tmp = Float64(cos(y) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (sin(y) * z);
	tmp = 0.0;
	if (z <= -4.4e+81)
		tmp = t_0;
	elseif (z <= 7.2e+18)
		tmp = cos(y) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(N[Sin[y], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.4e+81], t$95$0, If[LessEqual[z, 7.2e+18], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.39999999999999974e81 or 7.2e18 < z

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) - z \cdot \sin y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right)} - z \cdot \sin y \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(x + \color{blue}{\cos y}\right) - z \cdot \sin y \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{z \cdot \sin y} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(x + \cos y\right) - z \cdot \color{blue}{\sin y} \]

      6. *-lft-identity N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{1 \cdot \left(z \cdot \sin y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(z \cdot \sin y\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) + -1 \cdot \left(z \cdot \sin y\right)} \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{x + \left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right) + x} \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\cos y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(z \cdot \sin y\right)\right)} + x \]

      12. metadata-eval N/A

          \[\leadsto \left(\cos y - \color{blue}{1} \cdot \left(z \cdot \sin y\right)\right) + x \]

      13. *-lft-identity N/A

          \[\leadsto \left(\cos y - \color{blue}{z \cdot \sin y}\right) + x \]

      14. associate-+l- N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \color{blue}{\cos y} - \left(z \cdot \sin y - x\right) \]

      17. lower--.f64 N/A

          \[\leadsto \cos y - \color{blue}{\left(z \cdot \sin y - x\right)} \]

      18. *-commutative N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      20. lift-sin.f64 99.9

          \[\leadsto \cos y - \left(\color{blue}{\sin y} \cdot z - x\right) \]

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\cos y - \left(\sin y \cdot z - x\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{z \cdot \sin y} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \sin y \cdot \color{blue}{z} \]

      2. lift-sin.f64 N/A

         \[\leadsto 1 - \sin y \cdot z \]

      3. lift-*.f64 69.6

         \[\leadsto 1 - \sin y \cdot \color{blue}{z} \]

   9. Applied rewrites 69.6%

      \[\leadsto 1 - \color{blue}{\sin y \cdot z} \]

   if -4.39999999999999974e81 < z < 7.2e18

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 95.1

         \[\leadsto \cos y + x \]

   4. Applied rewrites 95.1%

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

Alternative 6: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+86}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+163}:\\ \;\;\;\;\cos y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \sin y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.4e+86)
   (- x (fma z y -1.0))
   (if (<= z 1.05e+163) (+ (cos y) x) (* (- z) (sin y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.4e+86) {
		tmp = x - fma(z, y, -1.0);
	} else if (z <= 1.05e+163) {
		tmp = cos(y) + x;
	} else {
		tmp = -z * sin(y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.4e+86)
		tmp = Float64(x - fma(z, y, -1.0));
	elseif (z <= 1.05e+163)
		tmp = Float64(cos(y) + x);
	else
		tmp = Float64(Float64(-z) * sin(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.4e+86], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e+163], N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision], N[((-z) * N[Sin[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4000000000000001e86

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \]

      6. lower--.f64 N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{-1} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x - 1 \cdot \left(y \cdot z\right)\right) - -1 \]

      9. *-lft-identity N/A

         \[\leadsto \left(x - y \cdot z\right) - -1 \]

      10. lower--.f64 N/A

          \[\leadsto \left(x - y \cdot z\right) - -1 \]

      11. *-commutative N/A

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

      12. lower-*.f64 54.8

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

   4. Applied rewrites 54.8%

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

      1. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - \color{blue}{-1} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      3. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      4. associate--l- N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      6. lower-fma.f64 54.8

         \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{y}, -1\right) \]

   6. Applied rewrites 54.8%

      \[\leadsto \color{blue}{x - \mathsf{fma}\left(z, y, -1\right)} \]

   if -6.4000000000000001e86 < z < 1.05e163

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 89.3

         \[\leadsto \cos y + x \]

   4. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\cos y + x} \]

   if 1.05e163 < z

   1. Initial program 99.8%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \sin y\right) \]

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

         \[\leadsto \left(-z\right) \cdot \sin \color{blue}{y} \]

      5. lift-sin.f64 69.6

         \[\leadsto \left(-z\right) \cdot \sin y \]

   4. Applied rewrites 69.6%

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

Alternative 7: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos y + x\\ \mathbf{if}\;y \leq -4.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (cos y) x)))
   (if (<= y -4.2)
     t_0
     (if (<= y 26000000000000.0)
       (fma (- (* (- (* 0.16666666666666666 (* z y)) 0.5) y) z) y (- x -1.0))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = cos(y) + x;
	double tmp;
	if (y <= -4.2) {
		tmp = t_0;
	} else if (y <= 26000000000000.0) {
		tmp = fma(((((0.16666666666666666 * (z * y)) - 0.5) * y) - z), y, (x - -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(cos(y) + x)
	tmp = 0.0
	if (y <= -4.2)
		tmp = t_0;
	elseif (y <= 26000000000000.0)
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(z * y)) - 0.5) * y) - z), y, Float64(x - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cos[y], $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -4.2], t$95$0, If[LessEqual[y, 26000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000018 or 2.6e13 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 62.8

         \[\leadsto \cos y + x \]

   4. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\cos y + x} \]

   if -4.20000000000000018 < y < 2.6e13

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, y, 1 + x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1 \cdot 1\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1 \cdot 1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1\right) \]

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + \cos y\right) - z \cdot \sin y\\ t_1 := x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{if}\;t\_0 \leq -200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;\cos y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x (cos y)) (* z (sin y)))) (t_1 (- x (fma z y -1.0))))
   (if (<= t_0 -200000000.0) t_1 (if (<= t_0 0.9998) (cos y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + cos(y)) - (z * sin(y));
	double t_1 = x - fma(z, y, -1.0);
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.9998) {
		tmp = cos(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + cos(y)) - Float64(z * sin(y)))
	t_1 = Float64(x - fma(z, y, -1.0))
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 0.9998)
		tmp = cos(y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$1, If[LessEqual[t$95$0, 0.9998], N[Cos[y], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < -2e8 or 0.99980000000000002 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y)))

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \]

      6. lower--.f64 N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{-1} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x - 1 \cdot \left(y \cdot z\right)\right) - -1 \]

      9. *-lft-identity N/A

         \[\leadsto \left(x - y \cdot z\right) - -1 \]

      10. lower--.f64 N/A

          \[\leadsto \left(x - y \cdot z\right) - -1 \]

      11. *-commutative N/A

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

      12. lower-*.f64 72.5

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

   4. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - \color{blue}{-1} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      3. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      4. associate--l- N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      6. lower-fma.f64 72.5

         \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{y}, -1\right) \]

   6. Applied rewrites 72.5%

      \[\leadsto \color{blue}{x - \mathsf{fma}\left(z, y, -1\right)} \]

   if -2e8 < (-.f64 (+.f64 x (cos.f64 y)) (*.f64 z (sin.f64 y))) < 0.99980000000000002

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \cos y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto \cos y + \color{blue}{x} \]

      3. lift-cos.f64 95.0

         \[\leadsto \cos y + x \]

   4. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\cos y + x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \cos y \]
   6. Step-by-step derivation

      1. lift-cos.f64 90.8

         \[\leadsto \cos y \]

   7. Applied rewrites 90.8%

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

Alternative 9: 70.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+20}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+20)
   (- x -1.0)
   (if (<= y 26000000000000.0)
     (fma (- (* (- (* 0.16666666666666666 (* z y)) 0.5) y) z) y (- x -1.0))
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+20) {
		tmp = x - -1.0;
	} else if (y <= 26000000000000.0) {
		tmp = fma(((((0.16666666666666666 * (z * y)) - 0.5) * y) - z), y, (x - -1.0));
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+20)
		tmp = Float64(x - -1.0);
	elseif (y <= 26000000000000.0)
		tmp = fma(Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(z * y)) - 0.5) * y) - z), y, Float64(x - -1.0));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+20], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 26000000000000.0], N[(N[(N[(N[(N[(0.16666666666666666 * N[(z * y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e20 or 2.6e13 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 40.2

         \[\leadsto x - \color{blue}{-1} \]

   4. Applied rewrites 40.2%

      \[\leadsto \color{blue}{x - -1} \]

   if -7.8e20 < y < 2.6e13

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, y, 1 + x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1 \cdot 1\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1 \cdot 1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1\right) \]

   4. Applied rewrites 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3300000:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 26000000000000:\\ \;\;\;\;1 - \left(\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\right) \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3300000.0)
   (- x -1.0)
   (if (<= y 26000000000000.0)
     (- 1.0 (- (* (* (fma (* y y) -0.16666666666666666 1.0) y) z) x))
     (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3300000.0) {
		tmp = x - -1.0;
	} else if (y <= 26000000000000.0) {
		tmp = 1.0 - (((fma((y * y), -0.16666666666666666, 1.0) * y) * z) - x);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3300000.0)
		tmp = Float64(x - -1.0);
	elseif (y <= 26000000000000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) * z) - x));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3300000.0], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 26000000000000.0], N[(1.0 - N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e6 or 2.6e13 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 40.1

         \[\leadsto x - \color{blue}{-1} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{x - -1} \]

   if -3.3e6 < y < 2.6e13

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) - z \cdot \sin y} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right)} - z \cdot \sin y \]

      3. lift-cos.f64 N/A

         \[\leadsto \left(x + \color{blue}{\cos y}\right) - z \cdot \sin y \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{z \cdot \sin y} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(x + \cos y\right) - z \cdot \color{blue}{\sin y} \]

      6. *-lft-identity N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{1 \cdot \left(z \cdot \sin y\right)} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \cos y\right) - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(z \cdot \sin y\right) \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x + \cos y\right) + -1 \cdot \left(z \cdot \sin y\right)} \]

      9. associate-+r+ N/A

         \[\leadsto \color{blue}{x + \left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\cos y + -1 \cdot \left(z \cdot \sin y\right)\right) + x} \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\left(\cos y - \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(z \cdot \sin y\right)\right)} + x \]

      12. metadata-eval N/A

          \[\leadsto \left(\cos y - \color{blue}{1} \cdot \left(z \cdot \sin y\right)\right) + x \]

      13. *-lft-identity N/A

          \[\leadsto \left(\cos y - \color{blue}{z \cdot \sin y}\right) + x \]

      14. associate-+l- N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{\cos y - \left(z \cdot \sin y - x\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \color{blue}{\cos y} - \left(z \cdot \sin y - x\right) \]

      17. lower--.f64 N/A

          \[\leadsto \cos y - \color{blue}{\left(z \cdot \sin y - x\right)} \]

      18. *-commutative N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      19. lower-*.f64 N/A

          \[\leadsto \cos y - \left(\color{blue}{\sin y \cdot z} - x\right) \]

      20. lift-sin.f64 100.0

          \[\leadsto \cos y - \left(\color{blue}{\sin y} \cdot z - x\right) \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cos y - \left(\sin y \cdot z - x\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 98.9%

      \[\leadsto \color{blue}{1} - \left(\sin y \cdot z - x\right) \]

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto 1 - \left(\left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \cdot z - x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 1 - \left(\left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{y}\right) \cdot z - x\right) \]

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 97.9

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

   9. Applied rewrites 97.9%

      \[\leadsto 1 - \left(\color{blue}{\left(\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y\right)} \cdot z - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+34}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 23.5:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+34)
   (- x -1.0)
   (if (<= y 23.5) (fma (- (* -0.5 y) z) y (- x -1.0)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+34) {
		tmp = x - -1.0;
	} else if (y <= 23.5) {
		tmp = fma(((-0.5 * y) - z), y, (x - -1.0));
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+34)
		tmp = Float64(x - -1.0);
	elseif (y <= 23.5)
		tmp = fma(Float64(Float64(-0.5 * y) - z), y, Float64(x - -1.0));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+34], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 23.5], N[(N[(N[(-0.5 * y), $MachinePrecision] - z), $MachinePrecision] * y + N[(x - -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000004e34 or 23.5 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 40.2

         \[\leadsto x - \color{blue}{-1} \]

   4. Applied rewrites 40.2%

      \[\leadsto \color{blue}{x - -1} \]

   if -1.40000000000000004e34 < y < 23.5

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(\frac{-1}{2} \cdot y - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

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

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(\frac{-1}{2} \cdot y - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{-1}{2} \cdot y - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{-1}{2} \cdot y - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, 1 + x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, 1 + x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x + 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x + 1 \cdot 1\right) \]

      11. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1 \cdot 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot y - z, y, x - -1\right) \]

      14. lower--.f64 95.6

          \[\leadsto \mathsf{fma}\left(-0.5 \cdot y - z, y, x - -1\right) \]

   4. Applied rewrites 95.6%

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

Alternative 12: 69.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+44}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+32}:\\ \;\;\;\;x - \mathsf{fma}\left(z, y, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+44)
   (- x -1.0)
   (if (<= y 5.6e+32) (- x (fma z y -1.0)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+44) {
		tmp = x - -1.0;
	} else if (y <= 5.6e+32) {
		tmp = x - fma(z, y, -1.0);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+44)
		tmp = Float64(x - -1.0);
	elseif (y <= 5.6e+32)
		tmp = Float64(x - fma(z, y, -1.0));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+44], N[(x - -1.0), $MachinePrecision], If[LessEqual[y, 5.6e+32], N[(x - N[(z * y + -1.0), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000026e44 or 5.6e32 < y

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 40.1

         \[\leadsto x - \color{blue}{-1} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{x - -1} \]

   if -4.80000000000000026e44 < y < 5.6e32

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \]

      6. lower--.f64 N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{-1} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x - 1 \cdot \left(y \cdot z\right)\right) - -1 \]

      9. *-lft-identity N/A

         \[\leadsto \left(x - y \cdot z\right) - -1 \]

      10. lower--.f64 N/A

          \[\leadsto \left(x - y \cdot z\right) - -1 \]

      11. *-commutative N/A

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

      12. lower-*.f64 92.6

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

   4. Applied rewrites 92.6%

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

      1. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - \color{blue}{-1} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      3. lift--.f64 N/A

         \[\leadsto \left(x - z \cdot y\right) - -1 \]

      4. associate--l- N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      5. lower--.f64 N/A

         \[\leadsto x - \color{blue}{\left(z \cdot y + -1\right)} \]

      6. lower-fma.f64 92.6

         \[\leadsto x - \mathsf{fma}\left(z, \color{blue}{y}, -1\right) \]

   6. Applied rewrites 92.6%

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

Alternative 13: 66.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-60}:\\ \;\;\;\;x - -1\\ \mathbf{elif}\;x \leq 18:\\ \;\;\;\;1 - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.6e-60) (- x -1.0) (if (<= x 18.0) (- 1.0 (* z y)) (- x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e-60) {
		tmp = x - -1.0;
	} else if (x <= 18.0) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.6d-60)) then
        tmp = x - (-1.0d0)
    else if (x <= 18.0d0) then
        tmp = 1.0d0 - (z * y)
    else
        tmp = x - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e-60) {
		tmp = x - -1.0;
	} else if (x <= 18.0) {
		tmp = 1.0 - (z * y);
	} else {
		tmp = x - -1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.6e-60:
		tmp = x - -1.0
	elif x <= 18.0:
		tmp = 1.0 - (z * y)
	else:
		tmp = x - -1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.6e-60)
		tmp = Float64(x - -1.0);
	elseif (x <= 18.0)
		tmp = Float64(1.0 - Float64(z * y));
	else
		tmp = Float64(x - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.6e-60)
		tmp = x - -1.0;
	elseif (x <= 18.0)
		tmp = 1.0 - (z * y);
	else
		tmp = x - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.6e-60], N[(x - -1.0), $MachinePrecision], If[LessEqual[x, 18.0], N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x - -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6000000000000005e-60 or 18 < x

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto x + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto x - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto x - -1 \]

      6. lower--.f64 79.4

         \[\leadsto x - \color{blue}{-1} \]

   4. Applied rewrites 79.4%

      \[\leadsto \color{blue}{x - -1} \]

   if -5.6000000000000005e-60 < x < 18

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + -1 \cdot \left(y \cdot z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + \color{blue}{1} \]

      2. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) + 1 \cdot \color{blue}{1} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \cdot 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - -1 \]

      6. lower--.f64 N/A

         \[\leadsto \left(x + -1 \cdot \left(y \cdot z\right)\right) - \color{blue}{-1} \]

      7. metadata-eval N/A

         \[\leadsto \left(x + \left(\mathsf{neg}\left(1\right)\right) \cdot \left(y \cdot z\right)\right) - -1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x - 1 \cdot \left(y \cdot z\right)\right) - -1 \]

      9. *-lft-identity N/A

         \[\leadsto \left(x - y \cdot z\right) - -1 \]

      10. lower--.f64 N/A

          \[\leadsto \left(x - y \cdot z\right) - -1 \]

      11. *-commutative N/A

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

      12. lower-*.f64 51.7

          \[\leadsto \left(x - z \cdot y\right) - -1 \]

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\left(x - z \cdot y\right) - -1} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{y \cdot z} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - y \cdot \color{blue}{z} \]

      2. *-commutative N/A

         \[\leadsto 1 - z \cdot y \]

      3. lift-*.f64 51.1

         \[\leadsto 1 - z \cdot y \]

   7. Applied rewrites 51.1%

      \[\leadsto 1 - \color{blue}{z \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.5% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ x - -1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x -1.0))

C

double code(double x, double y, double z) {
	return x - -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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x - (-1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return x - -1.0;
}

Python

def code(x, y, z):
	return x - -1.0

Julia

function code(x, y, z)
	return Float64(x - -1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x - -1.0;
end

Wolfram

code[x_, y_, z_] := N[(x - -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + x} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x + \color{blue}{1} \]

   2. metadata-eval N/A

      \[\leadsto x + 1 \cdot \color{blue}{1} \]

   3. fp-cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

      \[\leadsto x - -1 \cdot 1 \]

   5. metadata-eval N/A

      \[\leadsto x - -1 \]

   6. lower--.f64 61.5

      \[\leadsto x - \color{blue}{-1} \]

4. Applied rewrites 61.5%

   \[\leadsto \color{blue}{x - -1} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 220000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -480.0) x (if (<= x 220000.0) 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -480.0) {
		tmp = x;
	} else if (x <= 220000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-480.0d0)) then
        tmp = x
    else if (x <= 220000.0d0) then
        tmp = 1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -480.0) {
		tmp = x;
	} else if (x <= 220000.0) {
		tmp = 1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -480.0:
		tmp = x
	elif x <= 220000.0:
		tmp = 1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -480.0)
		tmp = x;
	elseif (x <= 220000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -480.0)
		tmp = x;
	elseif (x <= 220000.0)
		tmp = 1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -480.0], x, If[LessEqual[x, 220000.0], 1.0, x]]

Derivation

1. Split input into 2 regimes

2. if x < -480 or 2.2e5 < x

   1. Initial program 100.0%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 82.9%

      \[\leadsto \color{blue}{x} \]

   if -480 < x < 2.2e5

   1. Initial program 99.9%

      \[\left(x + \cos y\right) - z \cdot \sin y \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(x + y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right)\right) + \color{blue}{1} \]

      2. +-commutative N/A

         \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + x\right) + 1 \]

      3. associate-+l+ N/A

         \[\leadsto y \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) + \color{blue}{\left(x + 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(\color{blue}{x} + 1\right) \]

      5. +-commutative N/A

         \[\leadsto \left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z\right) \cdot y + \left(1 + \color{blue}{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, \color{blue}{y}, 1 + x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) - z, y, 1 + x\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(y \cdot z\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, 1 + x\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x + 1 \cdot 1\right) \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1 \cdot 1\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot \left(z \cdot y\right) - \frac{1}{2}\right) \cdot y - z, y, x - -1\right) \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot y\right) - 0.5\right) \cdot y - z, y, x - -1\right)} \]

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot z\right) \cdot \left(-1 \cdot \frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z} + \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot z\right) \cdot \left(-1 \cdot \frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z} + \color{blue}{y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z} + \color{blue}{y} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(-z\right) \cdot \left(-1 \cdot \frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z} + \color{blue}{y} \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \left(-z\right) \cdot \left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right) + -1 \cdot \color{blue}{\frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z}}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(-z\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {y}^{2}\right) \cdot y + -1 \cdot \frac{\color{blue}{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}}{z}\right) \]

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

          \[\leadsto \left(-z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{6}, y \cdot y, 1\right), y, \mathsf{neg}\left(\frac{1 + \left(x + \frac{-1}{2} \cdot {y}^{2}\right)}{z}\right)\right) \]

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 50.5%

      \[\leadsto \left(-z\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right), y, -\frac{\mathsf{fma}\left(y \cdot y, -0.5, x\right) + 1}{z}\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto z \cdot \left(\frac{1}{z} + \color{blue}{\frac{x}{z}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{z} + \frac{x}{z}\right) \cdot z \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{z} + \frac{x}{z}\right) \cdot z \]

      3. div-add-rev N/A

         \[\leadsto \frac{1 + x}{z} \cdot z \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 + x}{z} \cdot z \]

      5. +-commutative N/A

         \[\leadsto \frac{x + 1}{z} \cdot z \]

      6. lower-+.f64 40.2

         \[\leadsto \frac{x + 1}{z} \cdot z \]

   10. Applied rewrites 40.2%

       \[\leadsto \frac{x + 1}{z} \cdot z \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 39.1%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.8% accurate, 72.9× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

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(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

   \[\left(x + \cos y\right) - z \cdot \sin y \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 42.8%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (x y z)
  :name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, B"
  :precision binary64
  (- (+ x (cos y)) (* z (sin y))))